Regularization of singular time-dependent Lagrangian systems

R egularization of singular time-dependent Lag r angian systems M. de León 1 , 3 , 4 , R. Izquierdo-López 1 , 5 , L. Schia v one 2 , 6 , P . Sot o-Martín 1 , 7 1 Institut o de Ciencias Matemáticas, Cam pus Cantoblanco, Consejo Superior de Inv estig aciones Cientíﬁcas, Calle Nicolás Cabrer a, 13–15, 28049, Madrid, Spain 2 Dipartimento di Matematica e Applicazioni Renato Caccioppoli, Univ ersità deg li Studi di Napoli F ederico II, V ia Cintia, Monte S. Angelo I, 80126, Napoli, Ital y 3 Real Academia de Ciencias Exactas, Físicas y Natur ales de España, C/V al ver de, 22, Madrid 28004, Spain 4 e-mail: mdeleon@icmat.es 5 e-mail: ruben.izquierdo@icmat.es 6 e-mail: luca.schiavone@unina.it 7 e-mail: pablo@soto.es March 26, 2026 Abstract One approach to studying t he dynamics of a singular Lag rangian system is to attemp t to regularize it, t hat is, to ﬁnd an equiv alent and regular system. In t he case of time- independent singular Lagrangians, an approach due to A. Ibort and J. Marín-Solano is to use t he coisotropic embedding t heorem prov ed by M.J. Gotay which states that an y pre-symplectic manifold can be coisotropicall y embedded in a symplectic manifold. In this paper , w e revisit t hese results and provide an alter nativ e approach—also based on t he coisotropic embedding t heorem—that emplo ys t he T ulczyje w isomorphism and almost product s tructures, and allow s for a slight g eneralization of the construction. In t his revision, w e also prov e uniqueness of the Lagrangian regularization to ﬁrst order . Further more, we extend our met hodology to the case of time-dependent singular Lagrangians. Contents 1 Introduction 2 2 Preliminaries 4 2.1 Distributions and foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 A T ulczyjew isomor phism for foliations . . . . . . . . . . . . . . . . . . . . . . 7 2.3 T ang ent structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Jet structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Regularization of autonomous systems 14 3.1 Symplectic Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1 3.2 Coisotropic regularization of pre-symplectic Hamiltonian systems . . . . . . . 15 3.3 Coisotropic regularization of degenerate Lagrangian systems . . . . . . . . . . 18 3.4 Existence and uniqueness of Lag rangian regularization . . . . . . . . . . . . . 24 4 Regularization of non-autonomous systems 33 4.1 Cosymplectic Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Coisotropic regularization of pre-cosymplectic Hamiltonian systems . . . . . 35 4.3 Coisotropic regularization of non-autonomous Lagrangian systems . . . . . . 36 4.4 Existence and uniqueness of Lag rangian regularization . . . . . . . . . . . . . 39 4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.5.1 T rivialized bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.5.2 Degener ate metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5 Conclusions and fur t her w ork 49 1 Introduction One of t he g reatest successes of symplectic g eometr y has been to ser v e as a setting for Hamiltonian mechanics as w ell as its Lagrangian description. Indeed, a Hamiltonian function is just a function H deﬁned on t he cotang ent bundle T ∗ Q of t he conﬁguration manifold Q , such t hat t he Hamiltonian dynamics is pro vided b y t he corresponding Hamiltonian v ector ﬁeld X H obtained using t he canonical symplectic form ω Q on T ∗ Q [ 1 , 18 ], sa y i X H ω Q = d H . In the Lagrangian picture, given a Lag rangian function L on the tangent bundle T Q , one obtains a diﬀerential 2-form ω L such t hat the equation i ξ L ω L = d E L pro vides the Euler-Lagrang e v ector ﬁeld ξ L [ 18 ]. W e need to make tw o clariﬁcations about t his last equation: (1) ω L is symplectic if and onl y if t he Lag rangian L is regular (t he Hessian matrix of L with respect to v elocities is regular), and (2) ξ L is a second-order diﬀerential equation (SODE, for short) such t hat its solutions (the projections onto Q of its integral cur v es) are t he solutions of t he Euler -Lagrange equations. Lagrangians giving rise to degener ate 2-forms ω L are usuall y referred to as singular . The Legendre transf ormation Leg : T Q − → T ∗ Q connects both descriptions in a natural w a y (see [ 18 , 22 ] for more details). One of the most interesting contributions in P .A.M. Dirac ’ s and P .G. Bergmann ’ s w or ks was the introduction of t he constr aint algorit hm for dealing wit h singular Lag rangians, now kno wn as the Dirac-Ber gmann algorithm (see [ 2 , 19 , 20 , 21 ]). That algorit hm has been dev eloped in geometric terms using t he notion of pre-symplectic manifolds b y M.J. Gotay and collaborators [ 24 , 25 , 26 ], incorporating in the Lagrangian picture the problem of t he second-order diﬀerential equation, a remar kable distinction betw een t he Lag rangian and t he Hamiltonian descriptions. The abov e algorit hm has also been constructed for t he case of singular Lag rangians depending explicitl y on time [ 5 ]. In t hat case, the geometric scenarios are usually taken to be T Q × R and T ∗ Q × R , and now , t he singular case uses t he notion of pre-cosymplectic structure. More gener all y , w e ma y regar d time-dependent (or non-autonomous) Lag rangians as functions deﬁned in J 1 π , where π : Q − → R is a (not necessarily trivialized) ﬁber bundle (see [ 11 , 12 , 32 ]). 2 In both time-independent and time-dependent cases, a P oisson brack et that provides t he dynamics (t he so-called Dirac bracket) and the dynamics themselv es, modulo the kernel of the forms, can be found on the so-called ﬁnal constr aint submanifold selected b y the algorit hm. On t he other hand, coisotropic submanifolds pla y an important role in classical mechanics and ﬁeld t heories because they allo w for t he dev elopment of a procedure for reducing dynamics. Indeed, a foundational result due to A. W einst ein [ 47 ] (see also [ 1 ]) establishes t hat t he quotient space of a coisotropic submanifold b y its characteris tic foliation naturall y inherits a symplectic structure, providing a rigorous geometric setting for Hamiltonian reduction. The reduction is then accomplished when w e consider dynamics interpreted as a Lagrangian submanifold. By taking the opposite road (unfolding vs reduction), coisotropic submanifolds are also relev ant in t he context of regularization problems. In fact, t he Dirac-Bergmann algorit hm is not t he only w a y w e could use to regularize a singular Lagrangian. Indeed, M.J. Got ay [ 23 ] prov ed a coisotropic embedding theor em , stating t hat any pre-symplectic manifold can be embedded into a symplectic manifold as a coisotropic submanifold, and that t his embedding is unique in a neighborhood of t he original manifold. This result has been gener alized to man y other relev ant g eometric scenarios b y the authors [ 30 , 42 ], by taking adv antage of t he point of view dev eloped in [ 43 ]. Using t he coisotropic embedding t heorem, as w ell as a natural classiﬁcation of Lagrangians [ 3 ], A. Ibort and J. Marín-Solano [ 27 ] w ere able to dev elop a regularization met hod for certain types of Lag rangian systems (called type II Lagrangians). This classiﬁcation for Lag rangian functions has been extended b y M. de León et al. (see [ 15 ]) and later reconsidered again by A. Ibort and J. Marín-Solano [ 28 ]. The main objectiv e of t his paper is to adv ance this prog ramme of regularization of singular Lagrangian systems, which will culminate in the case of classical ﬁeld t heories in a future w ork. Thus, w e begin by carefully re-examining t he results of A. Ibort and J. Marín-Solano in t he pre-symplectic case, using a new met hodology based on t he use of almost product structures and T ulczyjew triples [ 44 , 45 ]. Most notabl y , we emplo y a (to our know ledge) no v el gener alization of t he T ulczyjew triples adapted to a foliated manifold. The construction is presented in Section 2.2 . For completeness, let us recall t hat t he use of almost product structures to deal wit h singular Lag rangian systems was introduced in [ 17 ] (see also [ 18 ]). This approach has allow ed us to clarify some of t hese results and introduce an alternativ e for explicitl y constructing a regular Lagrangian equiv alent to t he original singular one by using an auxiliary connection. In particular , w e w ould like to highlight t he follo wing results in t he pre-symplectic case: • In Theorem 3.23 , w e giv e a global description of t he Lag rangian in t he regularized manifold, proving t hat t he regularization provides a g lobal Lagrangian, and not only locall y so. This description is more g eneral than the one pre viously found in the literature. Indeed, it depends on strictl y less geometric ing redients (a connection instead of a metric). • Finall y , t his regularization is prov ed to be unique and independent of all choices to ﬁrst order in Theorem 3.27 , a result which was not present in the literature. N ext, w e studied t he case of singular Lag rangians explicitly dependent on time, obtaining similar results using pre-cosymplectic geometry (namely Theorem 4.20 and Theorem 4.21 , respectiv ely). Mos t importantl y , t he Reeb v ector ﬁeld needs to be taken into consideration, which is a signiﬁcant diﬀerence from the autonomous case. In addition, our methodology does not explicitly relies on the mentioned classiﬁcation of Lagrangians (ev en if it requires some conditions, equiv alent to t hose used in [ 27 ], to be fulﬁlled b y the Lag rangian), and so opens a clear pat h to extend it to t he case of singular Lag rangian ﬁeld theories. 3 The paper is structured as follo ws. F ollowing the introduction, w e dev ote Section 2 to recall some w ell-known notions and results on distributions and foliations, as w ell as to prov e an extension of T ulczyjew’ s triple to the case of foliations deﬁned on a smooth manifold; tangent structures and stable tangent structures are also review ed. Section 3 is dev oted to reconsidering t he regularization of singular autonomous Lagrangians and dev eloping a new technique t hat diﬀers from that dev eloped in [ 27 ], as mentioned abov e. In Section 4 , w e extend t his regularization scheme to t he context of singular time-dependent Lag rangians. The construction presented in Section 4 is illustrated by studying the trivialized case and degener ate metrics. Finall y , w e include a section on conclusions and an outlook for future w ork to be carried out. 2 Preliminar ies 2.1 Distr ibutions and foliations In t his section, w e recall the basic deﬁnitions and geometric properties of distributions and foliations on smooth manifolds. Deﬁnition 2.1 ( Regular Distribution ) . Let Q be a smoot h manifold of dimension n . A distr ibution D on Q of rank r is a smoot h assignment of an r -dimensional subspace D q ⊂ T q Q to each point q ∈ Q . The distribution is said to be regular if the rank r is const ant over Q . Deﬁnition 2.2 ( Inv ol utivity and Integrability ) . A distribution D is said to be involutive if it is closed under the Lie br acket, i.e., [ X , Y ] ∈ D , ∀ X , Y ∈ Γ( D ) , (1) wher e Γ( D ) denot es the space of smoot h sections of D . A distribution is said to be integ rable if, for every point q ∈ Q , ther e exists an integr al submanifold of D passing t hrough q (i.e., a submanifold whose tang ent space at each point coincides with D ). Remark 2.3. Sometimes, one distinguishes between maximal integr al submanifolds and integr al submanifolds, when maximal dimension means the rank of the in volutive distribution. The fundamental link betw een t hese concepts is provided by the Frobenius Theorem [ 46 ]. Theorem 2.4 ( Frobenius Theorem ) . A regular distribution D on a smoot h manifold Q is integr able if and only if it is inv olutive. Deﬁnition 2.5 ( Regular Folia tion ) . A regular foliation F of dimension r (and codimension k = n − r ) on a manifold Q is a partition of Q into a famil y of disjoint, connected, immersed submanifolds {L α } α ∈ A called leaves , such that : 1. F or every q ∈ Q , t here exis ts a unique leaf L q containing q . 2. Around every point q ∈ Q , ther e exists a local coor dinat e chart ( U, φ ) wit h coor dinates ( x 1 , . . . , x k , f 1 , . . . , f r ) such that the connected components of t he inter section of any leaf with U ar e described by the equations x a = c a , a = 1 , . . . , k , (2) wher e t he const ants c a determine the local leaf. Such a chart is called a foliat ed char t or adapt ed char t . 4 Remark 2.6 ( Rela tion to Distributions ) . Every r egular foliation F deﬁnes a unique in volutive r egular distribution D , where D q = T q L q . Con versel y , by the Fr obenius Theorem, every regular in volutive distribution D g enerat es a regular foliation F D whose leav es are the maximal int egr al manifolds of D . In an adapted coordinate system ( x a , f A ) , t he distribution D is locally spanned by the v ector ﬁelds D = span ( ∂ ∂ f A ) A =1 ,...,r . (3) The coordinates x a serve as local coordinates on t he space of lea v es when it exists as a quotient manifold, while f A serve as coordinates along t he leaf. Deﬁnition 2.7 ( T an gent Bundle of a Folia tion ) . Let F be a regular foliation on Q . The tangent bundle of the foliation , denot ed by T F , is the disjoint union of the tang ent bundles of its leav es: T F := [ L ∈F T L . (4) This set carries the structur e of a smoot h vector bundle of r ank r over Q , and it is isomor phic to t he distribution D ⊂ T Q associat ed with F . Giv en an adapted chart ( x a , f A ) on Q , w e induce local coordinates on T F deno ted b y: n x a , f A , v f A o a =1 ,...,k ; A =1 ,...,r . (5) Here, a point in T F is locally represented as v ector v = v f A ∂ ∂ f A attached to t he point ( x a , f A ) . N ote t hat t he "transv erse v elocities" are identically zero, v x a = 0 . Deﬁnition 2.8 ( C ot angent Bundle of a Folia tion ) . The cotangent bundle of the foliation , denot ed by T ∗ F , is the disjoint union of the cot angent bundles of its leav es: T ∗ F := [ L ∈F T ∗ L . (6) It carries the structur e of a smoot h vect or bundle of rank r over Q . It is canonically isomorphic to t he dual of the distribution D , say D ∗ . In t he adapted chart deﬁned abov e, local coordinates on T ∗ F are denoted by : n x a , f A , p f A o a =1 ,...,k ; A =1 ,...,r . (7) A point in T ∗ F is locally represented by a cov ector α = p f A d f A restricted to t he tangent space of t he leaf. Remark 2.9. It is important t o not e that the sets T F and T ∗ F ar e smoot h vector bundles of rank r over the base manifold Q , being a subbundle and a quotient bundle of the tang ent and cot angent bundles of Q , respectiv el y . Indeed, by deﬁnition, t he ﬁber of T F at q is the tang ent space to the leaf passing thr ough q , i.e., ( T F ) q = T q L q . Since F is g enerat ed by the regular distribution D , we have T q L q = D q . Thus, T F coincides with the tot al space of t he dis tribution D . Since D is a regular distribution, it is by deﬁnition a vector subbundle of T Q . On the ot her hand, t he ﬁber of T ∗ F at q is the dual space of t he tang ent space t o the leaf, i.e., ( T ∗ F ) q = ( T q L q ) ∗ = D q ∗ . Consider t he annihilator of the distribution, denot ed by D ◦ ⊂ T ∗ Q , 5 which is the subbundle of covect ors t hat vanish on D . W e hav e a short exact sequence of vect or bundles over Q : 0 − → D ◦ − → T ∗ Q π − − → T ∗ F − → 0 , (8) wher e the map π is t he res triction of a covect or in T ∗ q Q to the subspace D q . Since this res triction is surjective with kernel D ◦ , by the ﬁrs t isomor phism t heorem for vect or spaces applied ﬁber -wise, we hav e the canonical isomorphism: T ∗ F ∼ = T ∗ Q / D ◦ . (9) Thus, T ∗ F carries the structur e of a (quo tient) vect or subbundle of T ∗ Q . Remark 2.10. As vect or bundles over Q , T F and T ∗ F ar e dual to each ot her . The duality pairing ⟨ · , · ⟩ : T ∗ F × Q T F → R (10) is deﬁned naturall y by the evaluation map. Let v ∈ ( T F ) q and α ∈ ( T ∗ F ) q . Since α is a linear functional on ( T F ) q , the pairing is simpl y ⟨ α, v ⟩ F = α ( v ) . In adapted local coordinat es ( x a , f A ) , a vect or v ∈ T F r eads v = v f A ∂ ∂ f A . (11) A covector in the ambient space T ∗ Q r eads e α = p a d x a + p f A d f A . Since the 1-forms d x a annihilate the distribution D = span { ∂ ∂ f A } , t he y form a local basis for t he annihilator D ◦ . Theref or e, the equiv alence class in t he quo tient T ∗ Q/D ◦ (which r epresents the element α ∈ T ∗ F ) is det ermined solely by the components p f A . The pairing is thus given explicitl y by : ⟨ α, v ⟩ =  p a d x a + p f A d f A  v f B ∂ ∂ f B ! = p f A v f A . (12) Deﬁnition 2.11 ( Almost -Pr oduct Str ucture ) . [ 18 ] An almost-product str ucture on a smoot h manifold Q is a smoot h point-wise splitting of its tang ent bundle into a direct sum of two complementary distributions. That is, for each q ∈ Q , the tang ent space decomposes as: T q Q = D q ⊕ H q , (13) wher e D and H are smooth subbundles of T Q . Equiv alentl y , such a splitting is uniquel y char acterized by a smoo th (1 , 1) -tensor ﬁeld P ∈ Γ( T ∗ Q ⊗ T Q ) t hat is idempo tent, namely P ◦ P = P . This tensor acts as a project or onto D along H , meaning that Im( P ) = D and ker( P ) = H . Given a pr e-exis ting r egular distribution D on Q , an almost-product structur e P is said to be adapt ed to D if its imag e coincides with t he distribution, Im( P ) = D . In this scenario, choosing P is equivalent to smoot hly assigning the complementary horizontal distribution H = k er( P ) . If the distribution D is integr able, it gener at es a regular foliation F . Let ( x a , f A ) be a sys tem of local adapt ed coor dinates, such that D = span n ∂ ∂ f A o . Since P must act as the identity on its image, we hav e P  ∂ ∂ f A  = ∂ ∂ f A . Thus, the most gener al local expr ession for an almost-pr oduct structur e P adap ted to D is given by : P =  d f A − P A a ( x, f )d x a  ⊗ ∂ ∂ f A , (14) wher e t he local functions P A a ( x, f ) uniquel y determine the choice of t he complement ar y distribution k er( P ) = span n ∂ ∂ x a + P A a ∂ ∂ f A o . 6 2.2 A T ulczyjew isomor phism for foliations Since it will be relev ant for the whole manuscript, we dev ote t his section to adapting t he notion of one of t he T ulczyjew isomor phisms to t he context of regular foliations. The isomorphism w e are interested in is t he one existing betw een the iterated bundles TT ∗ Q and T ∗ T Q o v er a smooth diﬀerential manifold Q t hat w e recall in the follo wing lines [ 18 , 44 , 45 ]. Consider an n -dimensional smooth diﬀerential manifold Q with t he system of local coordinates n q j o j =1 ,...,n . (15) Its tangent bundle π Q : T Q → Q , (16) inherits t he natural system of local coordinates n q j , ˙ q j o j =1 ,...,n , (17) where π Q ( q j , ˙ q j ) = q j . (18) N ow , consider t he double bundle TT Q with t he system of local coordinates n q j , ˙ q j , v q j , v ˙ q j o j =1 ,...,n . (19) It can be giv en two structures of v ector bundle o v er T Q , namely TT Q T Q T Q Q π T Q T π Q π Q π Q (20) where π T Q ( q j , ˙ q j , v q j , v ˙ q j ) = ( q j , ˙ q j ) , (21) and T π Q ( q j , ˙ q j , v q j , v ˙ q j ) = ( q j , v q j ) . (22) There exists a natural isomor phism of ﬁber bundles of π T Q and T π Q deﬁned categorically as t he unique double v ector bundle isomorphism δ : TT Q → TT Q that interchanges t he tw o v ector bundle projections—meaning it satisﬁes π T Q ◦ δ = T π Q and T π Q ◦ δ = π T Q , while acting as t he identity map on t he core of the double v ector bundle (which is canonicall y isomorphic to T Q ). In local coordinates, it reads: δ : TT Q → TT Q : ( q j , ˙ q j , v q j , v ˙ q j ) 7→ ( q j , v q j , ˙ q j , v ˙ q j ) . (23) Consider t he iterated bundle T ∗ T Q , wit h the system of local coordinates n q j , ˙ q j , p q j , p ˙ q j o j =1 ,...,n . (24) It is t he dual v ector bundle to π T Q with respect to t he pairing ⟨ ρ, ξ ⟩ = p q j v q j + p ˙ q j v ˙ q j , (25) 7 where ρ is an element of T ∗ T Q and ξ is an element of TT Q . On t he other hand, t he iterated bundle TT ∗ Q , wit h the system of local coordinates n q j , p j , ˙ q j , ˙ p j o j =1 ,...,n , (26) is canonically t he dual v ector bundle to T π Q . The duality pairing is deﬁned intrinsically as the tangent lift of t he canonical pairing betw een T ∗ Q and T Q . Speciﬁcall y , if w e consider a cur v e γ ( t ) = ( q j ( t ) , ˙ q j ( t )) in T Q and a cur v e λ ( t ) = ( q j ( t ) , p j ( t )) in T ∗ Q projecting to t he same base cur v e on Q , t he pairing is t he time deriv ativ e of t he contraction ⟨ λ ( t ) , γ ( t ) ⟩ . In local coordinates, this operation reads: d d t ( p j ( t ) ˙ q j ( t )) = ˙ p j ( t ) ˙ q j ( t ) + p j ( t ) ¨ q j ( t ) , (27) yielding t he pairing: ⟨ η , ψ ⟩ ′ = ˙ p j v q j + p j v ˙ q j , (28) where η is an element of TT ∗ Q and ψ is an element of TT Q . The transpose map of δ with respect t o t he pairing ⟨ · , · ⟩ ′ is the T ulczyjew isomor phism betw een TT ∗ Q and T ∗ T Q . It reads locally α = δ T : TT ∗ Q → T ∗ T Q : ( q j , p j , ˙ q j , ˙ p j ) 7→ ( q j , ˙ q j , p q j = ˙ p j , p ˙ q j = p j ) . (29) N o w , given a regular foliation F K e on Q gener ated b y a regular integ rable distribution K f on Q , let us consider the tangent distribution K = K f C . That is, the distribution gener ated b y the v ertical and t he complete lifts of t he v ector ﬁelds generating K f 1 . It is a regular distribution on T Q t hat pro vides a regular foliation F K on T Q . Denote b y n x a , f A o a =1 ,...,l ; A =1 ,...,r , (30) a system of local coordinates on Q adapted to t he foliation F K e , and by n x a , ˙ x a , f A , ˙ f A o a =1 ,...,l ; A =1 ,...,r , (31) a system of local coordinates on T Q adapted to the foliation F K = F K e C . The set of coordinates { x a } a =1 ,...,l , (32) and { x a , ˙ x a } a =1 ,...,l , (33) represent systems of coordinates on t he spaces of lea v es of F K e and F K , which ma y be treated as smooth manifolds locally. Let us denote by n x a , ˙ x a , f A , ˙ f A , v f A , v ˙ f A o a =1 ,...,l,A =1 ,...,r , (34) a system of local coordinates on T F K e C = T F K , and by n x a , f A , ˙ f A , ˙ x a , v f A , v ˙ f A o a =1 ,...,l,A =1 ,...,r , (35) 1 W e refer to [ 18 , 48 ] or to Deﬁnition 2.12 for t he deﬁnition of vertical and complete lifts of a v ector ﬁeld. 8 a system of local coordinates on T T F K e . As for t he bundles π T Q and T π Q , an isomor phism betw een t he bundles T F K e C and T T F K e exists. Intrinsicall y , t his isomor phism is exactl y the restriction of the canonical in v olution δ : TT Q → TT Q to t he subbundle T F T K e . Indeed, since T F K e is a smooth submanifold of T Q (being t he total space of t he distribution K f ), its tangent bundle T T F K e embeds naturall y into TT Q . A t t he same time, T F K e C is naturall y a subbundle of TT Q . It is straightf or w ard to show t hat the canonical inv olution δ maps t his subbundle exactly onto T T F K e . W e can t herefore deﬁne: δ F := δ    T F K e C : T F K e C → T T F K e , (36) which locally reads: δ F ( x a , ˙ x a , f A , ˙ f A , v f A , v ˙ f A ) 7→ ( x a , f A , ˙ f A = v f A , ˙ x a , v f A = ˙ f A , v ˙ f A ) . (37) Similar ly to what happens for the iter ated bundles considered b y T ulczyjew , t he bundle T ∗ F K e C , where w e chose t he system of local coordinates n x a , ˙ x a , f A , ˙ f A , µ f A , µ ˙ f A o a =1 ,...,l,A =1 ,...,r , (38) is t he dual bundle to T F K e C with respect to t he pairing ⟨ ρ, ξ ⟩ = µ f A v f A + µ ˙ f A v ˙ f A , (39) where ρ is an element of T ∗ F K e C and ξ is an element of T F K e C . Similar ly , as for the bundle TT ∗ Q , t he bundle T T ∗ F K e is t he dual v ector bundle to T T F K e . The duality pairing is giv en by the tangent lif t of the canonical pairing Eq. (12) betw een T ∗ F K e and T F K e . Explicitly , this means t hat for an y cur v e γ ( t ) in T F K e and an y curv e λ ( t ) in T ∗ F K e projecting to t he same base cur v e on Q , the pairing of t heir tangent v ectors is the time deriv ativ e of t heir contraction: d d t ⟨ λ ( t ) , γ ( t ) ⟩ = d d t  µ A ( t ) f A ( t )  = ˙ µ A f A + µ A ˙ f A , (40) yielding t he pairing ⟨ η , ψ ⟩ ′ = ˙ µ A f A + µ A ˙ f A , (41) where η is an element of T T ∗ F K e (with coordinates ( x a , f A , µ A , ˙ x a , ˙ f A , ˙ µ A ) ) and ψ is an element of T T F K e . The transpose map of δ F with respect to t he pairing ⟨ · , · ⟩ ′ is an isomorphism betw een T T ∗ F K e and T ∗ F K e C . It reads α = δ F T : T T ∗ F K e → T ∗ F K e C : ( x a , f A , µ A , ˙ x a , ˙ f A , ˙ µ A ) 7→ ( x a , ˙ x a , f A , ˙ f A , µ f A = ˙ µ A , µ ˙ f A = µ A ) . (42) 2.3 T angent str uctures In this section w e introduce tang ent structures, which correspond to t he g eometric structures gener alizing the local picture of T Q , f or some conﬁguration manifold Q (see [ 18 , 48 ] for further details). F irst, let us deﬁne some elementary operations on the tangent bundle T Q of a conﬁguration manifold Q and study its geometry . 9 Deﬁnition 2.12 ( Lifts of vector fields ) . Let X ∈ X ( Q ) be a vector ﬁeld on Q . • The ver tical lif t of X , denot ed by X V ∈ X ( T Q ) , is the unique vector ﬁeld on T Q such that for any 1-form α on Q , X V ( i α ) = α ( X ) ◦ τ , where i α is the ﬁber -wise linear function on T Q induced by α (locally v i α i ). In local coordinat es ( q i , v i ) , if X = X i ( q ) ∂ ∂ q i , then: X V = X i ( q ) ∂ ∂ v i . (43) • The comple te lif t (or tangent lif t ) of X , denoted by X C ∈ X ( T Q ) , is the vect or ﬁeld on T Q whose ﬂow is the tang ent lif t of t he ﬂow of X . That is, if φ t is the ﬂow of X , then Φ t = T φ t is t he ﬂow of X C . In local coordinat es, it reads: X C = X i ( q ) ∂ ∂ q i + v k ∂ X i ∂ q k ∂ ∂ v i . (44) The mapping X 7→ X C is a Lie algebr a homomorphism from X ( Q ) to X ( T Q ) , while the vertical lift is commutativ e. Speciﬁcally , for any X , Y ∈ X ( Q ) , the following br acket relations hold: [ X V , Y V ] = 0 , (45) [ X C , Y V ] = [ X , Y ] V , (46) [ X C , Y C ] = [ X , Y ] C . (47) Remark 2.13. No tice that complete lif ts of vect or ﬁelds X C g enerat e T Q point wise, for every v ∈ T Q . As a consequence, we may use the complet e lif t t o comput e t he lift of tensor s of diﬀer ent degree. A ﬁrst exam ple of the idea presented in Remar k 2.13 is t he lif t of almost-product structures to the tangent bundle as w ell. The study of these lifts will result useful in t he sequel. For t he sake of exposition, we restrict to the case of almost product structures complementing an integrable distribution. Deﬁnition 2.14 ( C omplete lift of an almost product str ucture ) . Let P e be an almost product structur e on t he conﬁguration manifold Q complementing an integr able distribution. Locall y , with adapt ed coordinat es, the project or r eads as: P e =  d f A − P e A a ( x, f )d x a  ⊗ ∂ ∂ f A , (48) wher e ( x a , f A ) denot e local coordinat es on Q adapt ed to the foliation induced by P e . The complete lift of P e to the tang ent bundle T Q is deﬁned by t he condition P e C ( X C ) = h P e ( X ) i C , ∀ X ∈ X ( Q ) . (49) In the induced local coor dinates ( x a , f A , v x a , v f A ) on T Q , the complet e lif t P takes the speciﬁc form: P = P A f ⊗ ∂ ∂ f A + P A v ⊗ ∂ ∂ v f A , (50) wher e t he projection 1-forms are given by : P A f = d f A − P e A a d x a , (51) P A v = d v f A − P e A a d v x a − v x b ∂ P e A a ∂ x b + v f B ∂ P e A a ∂ f B ! d x a . (52) 10 Remark 2.15. If P e complements t he integr able distribution K f , its complet e lift P e C complements t he complet e lif t of the distribution K f C . Deﬁnition 2.16 ( The geometr y of the t angent bundle ) . Given a smoot h d -dimensional diﬀer ential manifold Q , wit h local coor dinates { q j } j =1 ,...,d , t he geometry of its tang ent bundle T Q is char acterized by two canonical objects (see [ 18 , 22 ]): • The ver tical endomor phism (or solder ing for m ) S , which is a (1 , 1) -tensor ﬁeld S on T Q t hat locall y , using the sys tem of coordinat es { q j , v j } j =1 ,...,d for T Q , r eads S = d q j ⊗ ∂ ∂ v j . (53) It deﬁnes the vertical distribution V ( T Q ) = Im( S ) . Identiﬁed as a map S : T Q − → T Q , it can be intrinsically deﬁned as the unique map satisfying S ( X ( v )) := (( π Q ) ∗ X ( v )) V , (54) for every X ∈ X ( T Q ) . • The Liouville vector ﬁeld ∆ , which is the inﬁnitesimal gener ator of dilations along the ﬁbers. Locall y , ∆ = v j ∂ ∂ v j . (55) These tensors satisfy t he following properties: Im S = ker S , (56) N S = 0 , (57) ∆ ∈ Im S , (58) L ∆ S = − S , (59) wher e N S = [ S , S ] F − N (wher e [ · , · ] F − N denot es t he F rolic her -Nĳenhuis brac kets [ 31 ]) is t he Nĳenhuis tensor of S . Properties ( 56 ) - ( 59 ) are not incidental; t hey uniquely characterize t he tangent bundle structure (see Theorem 2.19 ). W it h t he abo v e discussion in mind, t he follo wing deﬁnition is natural. Deﬁnition 2.17 ( (Almost) t angent str ucture ) . An almost t angent str ucture on a manifold M is a (1 , 1) -tensor ﬁeld S such t hat, when identiﬁed as an endomor phism S : T M − → T M , it satisﬁes Im S = ker S . An almost tang ent structur e S on M is called a tangent str ucture or inv olutiv e if t he endomorphism S satiﬁes N S = [ S, S ] F − N = 0 . Remark 2.18. Let Q be an arbitrary conﬁguration manifold. In light of Deﬁnition 2.16 , we have that M = T Q admits a canonical tang ent structur e. In fact, an y almost tangent structure on a manifold M , say S ∈ Γ ( T ∗ M ⊗ T M ) has t he local expression of Eq. ( 53 ) if and only if [ S, S ] F − N = 0 (see [ 31 ]). Whether a tangent structure S ∈ Γ ( T ∗ M ⊗ T M ) is isomor phic to t he canonical tangent structure on T Q is characterized b y the existence of a Liouville vector ﬁeld satisfying properties Eq. (58) -( 59 ). Indeed, w e ha v e: 11 Theorem 2.19 ( Chara cteriza tion of t angent bundles ) . [ 8 , 9 , 37 ] Given a 2 d -dimensional manifold M , equipped wit h a (1 , 1) -tensor S and a vector ﬁeld ∆ suc h that : • The vector ﬁeld ∆ is complet e. • The set of zeroes of ∆ , Q := { m ∈ M | ∆ m = 0 } , is a smoot h d -dimensional embedded submanifold of M . • The limit of the ﬂow of ∆ , lim t →−∞ F ∆ t ( m ) , exis ts for all m ∈ M (and deﬁnes the projection onto Q ). • S and ∆ satisfy the relations in Eqs. (56) to (59) . Then the manifold M is diﬀeomorphic to the tang ent bundle of Q , M ∼ = T Q . 2.4 Jet str uctures In order to deal wit h t he regularization of time-dependent (or non-autonomous) singular Lagrangian systems, t he geometry of t he so-called jet bundle of a ﬁber bundle ov er t he real line Q − → R will take a primary role. The ﬁrst jet of the manifold is deﬁned similar ly to t he tangent bundle: Deﬁnition 2.20. Let π : Q − → R be a ﬁber bundle. As a set, its ﬁrst jet bundle J 1 π is deﬁned as the equiv alence class of sections γ : R − → Q to ﬁrs t or der . It can be naturall y endowed with a smoot h structur e (see [ 39 ]). Remark 2.21 ( N a tural coordin a tes ) . Ther e are natural coordinat es on J 1 π , for any given ﬁbered coor dinates ( t, q i ) on π : Q − → R , which read as ( q i , ˙ q i , t ) , repr esenting the class of sections thr ough ( q i , t ) with velocity ∂ ∂ t + ˙ q i ∂ ∂ q i . On J 1 π , t he notion of v ertical and complete lifts ma y be deﬁned similarl y to t he case of T Q , and take the same local expression as in Deﬁnition 2.12 when the canonical coordinates ( q j , ˙ q i , t ) are chosen. In particular , one ma y deﬁne it using the follo wing embedding: Remark 2.22 ( Embedding of jets into t angent bundle ) . F or a jet bundle J 1 π , of some (arbitrary) conﬁgur ation bundle π : Q − → R , we hav e a canonical embedding i π : J 1 π  → T Q . (60) This embedding is deﬁned using the g lobal vect or ﬁeld ∂ ∂ t on R as follow s: i π ( j t γ ) := γ ∗ ∂ ∂ t ! . (61) Deﬁning natural coordinat es ( t, q i , ˙ t, ˙ q i ) on T Q , it r eads as i ∗ π ( t, q i , ˙ t, ˙ q i ) = ( t, q i , 1 , ˙ q i ) . Then, giv en a v ertical v ector ﬁeld X ∈ X ( Q ) with respect to t he projection π : Q − → R , w e can deﬁne t he v ertical and complete lift simpl y b y restricting t he v ertical and complete lif t on T Q to J 1 π (which are tangent v ectors). On jet bundles one usually deﬁnes t he so-called 1st order jet prolongation of a vect or ﬁeld on Q . It can be deﬁned for v ertical v ector ﬁelds like those w e are considering here (and, indeed, in t hat case it is t he same as what w e are here calling complete lif t), and, more gener ally , for projectable v ector ﬁelds. Remark 2.23 ( Local expressions of ver tical and complete lifts ) . Let X = X i ( q , t ) ∂ ∂ q i be a 12 vertical vect or ﬁeld on π : Q − → R . Then t he vertical and complet e lift take t he following expr essions: X v = X i ∂ ∂ ˙ q i and X C = X i ∂ ∂ q i + ∂ X i ∂ t + ∂ X i ∂ q j ˙ q j ! ∂ ∂ ˙ q i . (62) Deﬁnition 2.24 ( The geometr y of the jet bundle ) . Let π : Q − → R denot e a ﬁber bundle over R , wher e t he st andar d ﬁber has dimension d . The geometry of the ﬁrs t jet bundle is char acterized by the following ingr edients: • A closed 1 -form, τ = d t . • The vertical endomor phism S , which is a (1 , 1) -tensor ﬁeld on T Q × R t hat, employing the canonical set of coor dinates { q j , ˙ q j , t } j =1 ,...,d , reads as S = (d q i − ˙ q i d t ) ⊗ ∂ ∂ ˙ q i (63) and satisﬁes that Im S is an integr able distribution with S 2 = 0 , rank S = d , d t (Im S ) = 0 , and [ S, S ] F − N = 2d t ∧ S . (64) • It is an aﬃne bundle over Q modeled on the vector bundle k er d π − → Q . Remark 2.25 ( Trivialized bundles ) . When t he bundle is trivialized Q = Q × R − → R , we ha ve a canonical diﬀeomorphism J 1 π ∼ = T Q × R . Since every ﬁber bundle over R is trivializable, it follow s t hat J 1 π ∼ = T Q × R , for some Q (which is the st andard ﬁber). Howev er , this diﬀeomorphism depends on the trivilization, and breaks the jet geometry . The discussion that we present applies to the trivialized ver sion as well, and has t he advant age of being readibil y gener alizible to mor e gener al variational problems, where the bundle may not be trivial. When dealing wit h trivialized bundles, the geometry of J 1 π is that of st able tang ent structur es (see [ 16 ]). In t his case, t he v ertical endomor phism canonically splits as S = S − d t ⊗ ∆ , (65) where ∆ is t he Liouville v ector ﬁeld inherited b y t he v ector bundle structure on T Q . In t his case, one ma y introduce t he follo wing (1 , 1) -tensor e S = S + d t ⊗ ∂ ∂ t . (66) These objects (and t heir relations) completel y characterize stable tangent structures. Similar ly to almost tangent structures, one can deﬁne a almost stable tangent structure on a manifold M to be a collection of objects t hat satisfy point wise t he same properties t hat t hose canonical objects on T Q × R satisfy . F or completeness, w e collect t he deﬁnitions and results here. Deﬁnition 2.26 ( (Almost) st able t angent str ucture ) . An almost stable t angent str ucture on a (2 d + 1) -dimensional manifold M is a tuple ( e S , τ , ξ ) consisting of a (1 , 1) -tensor e S ∈ Γ ( T ∗ M ⊗ T M ) , a 1 -form τ ∈ Ω 1 ( M ) and a vect or ﬁeld ξ ∈ X ( M ) satisfying e S 2 = τ ⊗ ξ , rank e S = d + 1 , and τ ( ξ ) = 1 . (67) If, in addition, d τ = 0 and [ e S , e S ] F − N = 0 , we say that ( e S , τ , ξ ) is in volutiv e or t hat it deﬁnes a stable tangent str ucture . 13 Locall y , ev er y stable tangent structure looks like T Q × R . T o obtain a global isomorphism w e need t he linear structure, which is characterized b y t he existence of a Liouville v ector ﬁeld (as in t he tangent case). Theorem 2.27 ( C hara cteriza tion of st able t angent bundles ) . [ 16 ] Let M be a (2 d + 1) - dimensional manifold equipped wit h a st able tangent structur e ( S , τ , ξ ) and a vect or ﬁeld ∆ ∈ X ( M ) satisfying: • The vector ﬁeld ∆ is complet e. • The closed form τ is exact τ = d t , for certain surjective function t : M − → R . • The following set Q := { m ∈ M : ∆ | m = 0 and t ( m ) = 0 } is a smoot h d -dimensional embedded submanifold. • Denoting by F ∆ t t he ﬂow of M , the limit lim t →−∞ F ∆ t ( m ) exis ts for all m ∈ M and is a surjective submersion onto Q . • ∆ satisﬁes the following: £ ∆ e S = − S , and e S (∆) = 0 (68) Then, M is diﬀeomorphic to a st able tang ent bundle T Q × R , for some conﬁguration manifold Q . It is unkno wn to the aut hors if a similar characterization for jet structur es on a manifold M exists. How ev er , to deal wit h uniqueness of t he Lag rangian regularization of non-autonomous systems, w e need to introduce t his notion abstr actly . In principle, t here ma y be a regularization which could be considered Lag rangian locally (due to t he presence of a v ertical endomorphism and 1 -form d t ), but not considered Lag rangian globally , in t he sense t hat it is not diﬀeomorphic to a jet bundle itself. The notion t hat w e w or k with is t he follo wing: Deﬁnition 2.28 ( Almost jet str ucture ) . Let M be a (2 d + 1) -dimensional manifold. An almost jet str ucture on M is a pair ( S, ξ ) , where S is a (1 , 1) -tensor and ξ is a nowher e zero 1 -form, t hat satisfy the following properties. S 2 = 0 , rank S = d , ξ (Im S ) = 0 . (69) When the equalities d ξ = 0 , [ S, S ] F − N = 2 ξ ∧ S hold and Im S is an integr able distribution, we call it a jet str ucture . 3 Regularization of autonomous systems 3.1 Sym plectic Hamiltonian systems Deﬁnition 3.1 ( Symplectic manifold ) . A symplectic manifold is a pair ( M , ω ) , where M is a smoot h manifold of even dimension 2 n and ω ∈ Ω 2 ( M ) is a closed and non-degener ate diﬀer ential 2-form, called t he symplectic for m . Theorem 3.2 ( D arboux’s Theorem ) . Let ( M , ω ) be a symplectic manifold of dimension 2 n . Around every point m ∈ M , ther e exists local coor dinates ( q 1 , . . . , q n , p 1 , . . . , p n ) , called Darboux coordinates , such that the symplectic form locally reads: ω = d q i ∧ d p i . Deﬁnition 3.3 ( Hamil tonian sy stem ) . A Hamiltonian system is a triple ( M , ω , H ) , where ( M , ω ) is a symplectic manifold (the phase space) and H ∈ C ∞ ( M ) is a smoot h function (the Hamiltonian). 14 Remark 3.4 ( Poisson manifolds ) . [ 18 , 33 , 36 ] Some author s deﬁne a Hamiltonian syst em mor e g enerall y as a pair ( M , Λ) , where M is a manifold and Λ is a P oisson tensor (a bivect or ﬁeld whose Schout en-Nĳenhuis br acket [Λ , Λ] vanishes), tog ether with a Hamiltonian function H ∈ C ∞ ( M ) . Every symplectic manifold ( M , ω ) is a P oisson manifold, with the P oisson t ensor Λ being t he bivect or ﬁeld ω ♯ associated with ω . The resulting P oisson brac ket is given by { f , g } = Λ(d f , d g ) = ω ( X f , X g ) . The con verse, howev er , is not true, as a P oisson tensor may be deg enerat e (i.e., the map Λ ♯ : T ∗ M → T M is not an isomor phism). Throughout this paper , we will adhere to t he deﬁnition given in Deﬁnition 3.3 , and the term Hamiltonian sys tem will alway s r efer to a sys tem deﬁned on a symplectic manifold. Since t he 2 -form ω is non-degenerate, t he musical mor phism ω ♭ : T M → T ∗ M , giv en b y ω ♭ ( v ) = i v ω , actually deﬁnes a v ector bundle isomor phism. This guarantees t he existence of a unique vect or ﬁeld X H ∈ X ( M ) , t he Hamiltonian vect or ﬁeld , satisfying t he intrinsic Hamilton ’ s equations : i X H ω = d H . (70) In Darboux coordinates ( q i , p i ) , t he v ector ﬁeld takes t he local form: X H = ∂ H ∂ p i ∂ ∂ q i − ∂ H ∂ q i ∂ ∂ p i , (71) and its integral curv es γ ( t ) = ( q i ( t ) , p i ( t )) are the solutions of the Hamiltonian system, satisfying t he standar d Hamilton ’s equations: d q i d t = ∂ H ∂ p i , d p i d t = − ∂ H ∂ q i . (72) The non-degeneracy of ω ensures local existence and uniqueness of solutions for an y giv en initial condition m ∈ M . 3.2 Coisotropic regular ization of pre-sym plectic Hamiltonian systems In gener al, when w orking wit h singular theories (such as gaug e Hamiltonian theories and singular time-independent Lag rangian t heories), w e w or k on a pre-symplectic manifold, rather t han on a symplectic one, sa y ( M , ω ) . Deﬁnition 3.5 ( Pre-symplectic Hamil tonian sys tem ) . A pr e-symplectic Hamiltonian syst em is a triple ( M , ω , H ) , where ( M , ω ) is a pre-sym plectic manifold and H ∈ C ∞ ( M ) is a Hamiltonian. In t his case, the char acteris tic distribution V := k er ω is non-trivial. The dynamics is still formall y governed by t he equation i X ω = d H . (73) Howev er , this equation poses two distinct problems: 1. Existence: A vect or ﬁeld X satisfying the equation may not exis t. 2. Uniqueness: If a solution X exis ts, it is not unique (it is deﬁned only up to t he addition of any vect or ﬁeld Y ∈ V ). These tw o problems identify two classes of pre-symplectic systems: Inconsistent Hamiltonian Systems. A system is inconsis tent if t he existence condition fails, i.e., d H is not in t he imag e of ω ♭ . This happens at points m where d H m does not annihilate the kernel: k er ω ⊆ ker d H . F or these systems, one must ﬁrst ﬁnd t he submanifold of M where a consistent dynamical ev olution exists. This is achiev ed by t he pre-sym plectic constr aint 15 algorit hm (PCA) , dev eloped b y M.J. Gotay , J.M. Nes ter , and G. Hinds [ 26 ] (see also the papers [ 24 , 25 ]). The algorit hm proceeds iterativ ely . W e deﬁne M 0 := M and deﬁne the ﬁrs t cons traint manifold M 1 as t he locus where d H is compatible wit h ω : M 1 := { m ∈ M 0 | (d H ) m ( Y ) = 0 , ∀ Y ∈ ( T m M 0 ) ⊥ ω } , (74) where ( T m M 0 ) ⊥ ω := V m = ker ω m . Assuming M 1 is a smooth submanifold, the algorithm imposes solutions of ( 73 ) on M 1 to be tangent to M 1 , which is a new consistency requirement. Ev entually , at each step k ≥ 2 , one ﬁnds the submanifold M k ⊂ M k − 1 : M k := { m ∈ M k − 1 | (d H ) m ( Y ) = 0 , ∀ Y ∈ ( T m M k − 1 ) ⊥ ω } . (75) where ( T m M k − 1 ) ⊥ ω := { Y ∈ T m M | ω m ( Y , Z ) = 0 , ∀ Z ∈ T m M k − 1 } . Assuming t hat all of these subsets are actuall y smooth submanifolds, w e hav e tw o main possibilities: • There is a certain k for which M k = ∅ , in which case t he dynamics are globally not w ell-deﬁned. • The algorit hm stabilizes on a ﬁnal constr aint manifold M f  = ∅ , meaning t here exists k ∈ N such that M k = M k − 1 =: M f . If t he algorithm stabilizes, t hen t he pre-symplectic Hamiltonian system ( M f , ω f , H f ) (where ω f = i ∗ f ω , H f = i ∗ f H ) is, by construction, no longer inconsistent. Equations i Γ f ω f = d H f , (76) are now w ell-posed and the integ ral cur v es of Γ f , embedded into t he starting manifold M , are t he solutions of t he original pre-symplectic Hamiltonian system. Remark 3.6. The abov e (PCA) algorithm is a geometrization of t he so-called Dirac-Ber gmann constr aint algorithm developed by bot h authors in an independent manner (see [ 2 , 19 , 20 , 38 ]). The r eader can ﬁnd a more complet e information in P .A.M. Dirac’ s monogr aph [ 21 ] as well as in these two papers by M.J. Got ay and J.M. Nes t er [ 24 , 25 ]. Consistent Hamiltonian Systems. A system is consistent if it admits a global dynamics, i.e., k er ω ⊆ k er d H . This corresponds to a system t hat either started consistent (like a pure gaug e theor y) or is t he result ( M f , ω f , H f ) of applying t he PCA . In t his case, t he existence problem is solv ed, but t he uniqueness problem (gauge ambiguity) remains, as the dynamics X is only deﬁned up to X 7→ X + Y for Y ∈ ker ω f . T o sol v e this remaining ambiguity , one can regularize t he sys tem using the coiso tropic embedding t heorem. Theorem 3.7 ( The c oisotr opic embedding theorem ) . Let ( M , ω ) be a pr e-symplectic manifold wit h char acteris tic distribution V = ker ω . Ther e exists a symplectic manifold ( f M , e ω ) and an embedding i : M  → f M , (77) such t hat i ∗ e ω = ω and i ( M ) is a closed coisotropic submanifold of ( f M , e ω ) . The pair ( f M , e ω ) is called a symplectic thick ening of ( M , ω ) . F urthermor e, this thic kening is unique up to a neighbor hood equiv alence [ 23 ]. 16 Remark 3.8 ( C onstr uction of the symplectic thickenin g ) . The construction of f M (see [ 30 , 43 ]) r equir es choosing an almost product structur e P of t he type ( 2.11 ) . Locall y , using Darboux coor dinates ( q a , p a , f A ) such t hat ω = d q a ∧ d p a and V = span { ∂ ∂ f A } , t he project or P (which deﬁnes H = ker P ) is given by P = P A ⊗ ∂ ∂ f A =  d f A − P q A a d q a − P p Aa d p a  ⊗ ∂ ∂ f A . (78) The t hickening f M is a neighbor hood of the zero section in t he dual bundle V ∗ , wit h coor dinates ( q a , p a , f A , µ A ) . The symplectic form is e ω := τ ∗ ω + d ϑ P , where ϑ P = µ A P A is the tautological 1-form. In local coordinat es: e ω = d q a ∧ d p a + d µ A ∧ P A − µ A d P A . (79) It is symplectic only in a tubular neighbor hood of t he zer o-section of τ ( µ A ≈ 0 , namely µ A approac hing zero), unless P has a vanishing Nĳenhuis tensor [ 18 ]. The coiso tropic embedding theorem pro vides the tool t o regularize an y consistent pre- symplectic Hamiltonian system. If w e start wit h an inconsistent system, we ﬁrst apply the PCA to get t he consistent system ( M f , ω f , H f ) . W e t hen apply Theorem 3.7 to t his M f . In both scenarios, the procedure is the same: w e embed the consistent pre-sym plectic manifold ( M , ω ) (which could be M 0 or M f ) into its symplectic t hickening ( f M , e ω ) . W e extend the Hamiltonian H to f H ∈ C ∞ ( f M ) as f H = τ ∗ H . The new system ( f M , e ω , f H ) is regular (symplectic), and its unique Hamiltonian v ector ﬁeld X e H is easily shown to be tangent to M , pro viding a (gaug e-ﬁxed) unique dynamical ev olution for t he original system. Indeed, let us introduce t he adapted basis of v ector ﬁelds on f M : ( H a = ∂ ∂ x a + P A a ∂ ∂ f A , V A = ∂ ∂ f A , W A = ∂ ∂ µ A ) , (80) and its dual coframe of 1-forms: n d x a , P A = d f A − P A a d x a , d µ A o . (81) N otice that the characteris tic distribution of ω is locally spanned b y V A , meaning k er ω = span { V A } , and t he original pre-symplectic form locally reads ω = 1 2 ω ab d x a ∧ d x b . Since t he extended Hamiltonian f H = τ ∗ H depends only on t he base coordinates ( x a , f A ) , its exterior deriv ativ e can be naturall y expanded in t he dual coframe as: d f H = ∂ H ∂ x a d x a + ∂ H ∂ f A d f A = H a ( H )d x a + V A ( H ) P A . (82) On the other hand, t he thickened symplectic form is deﬁned as e ω = π ∗ ω + d  µ A P A  . As discussed in Theorem 3.7 it is symplectic only in a tubular neighborhood of the zero-section of τ (where µ A ≈ 0 ), and it reads e ω = 1 2 ω ab d x a ∧ d x b + d µ A ∧ P A . (83) The Hamiltonian v ector ﬁeld in t he adapted basis considered reads X e H = X a H a + X A V A + X µ A W A . Computing its interior product wit h e ω giv es: i X e H e ω = X a ω ab d x b + X µ A H A − X A d µ A . (84) 17 Imposing the Hamiltonian condition i X e H e ω = d f H and matching t he coeﬃcients of the linearl y independent 1-forms, w e obtain t he system: X a ω ab = H b ( H ) , (85) − X A = 0 = ⇒ X A = 0 , (86) X µ A = V A ( H ) = ∂ H ∂ f A . (87) Since t he original pre-symplectic Hamiltonian system is consistent b y hypo thesis, the original Hamiltonian H must annihilate t he ker nel of ω , meaning d H ( V A ) = 0 , which translates to ∂ H ∂ f A = 0 . Substituting t his into Eq. (87) , w e ﬁnd t hat X µ A = 0 ev er ywhere on M . Since the transv ersal components X µ A along t he directions W A v anish on the zero section, the dynamical v ector ﬁeld X e H does not point outside t he submanifold, meaning it is strictl y tangent to M . 3.3 Coisotropic regular ization of degenerate Lag rangian systems W e now shif t our focus to t he Lagrangian formalism. Using t he intrinsic geometry of t he tangent bundle, w e recall how the dynamics of any Lag rangian system (regular or degenerate) can be formulated in a "Hamiltonian-like" manner on t he v elocity phase space T Q . The presence of the tangent bundle structure ( S, ∆) requires slightly modifying t he coisotropic regularization scheme presented for Hamiltonian systems. Firs t, using the geometry of the tangent bundle described in Section 2.3 , let us recall t he geometric deﬁnition of a second-order diﬀerential equation and let us deﬁne t he geometric structures associated wit h any Lag rangian L ∈ C ∞ ( T Q ) . Deﬁnition 3.9 ( Second Order Differential Equ a tion (SODE) ) . A vect or ﬁeld X ∈ X ( T Q ) is a Second Order Diﬀerential Eq uation (SODE) ﬁeld if its integr al curves γ ( t ) = ( q j ( t ) , v j ( t )) corr ectly relat e the position and velocity coordinat es, i.e., t he y satisfy the kinematic condition d q j d t = v j . In local coordinat es ( q j , v j ) , a gener al vector ﬁeld X r eads X = A j ( q , v ) ∂ ∂ q j + B j ( q , v ) ∂ ∂ v j . (88) Its integr al curves satisfy ˙ q j = A j . Ther efor e, for a SODE we must have A j = v j . Intrinsicall y , this condition is expressed by S ( X ) = ∆ . (89) Deﬁnition 3.10 ( Regular La grangian sys tem ) . A Lag rangian system is a pair ( Q, L ) . W e deﬁne: • The Poincaré-Car tan 1-for m θ L := d S L = S ∗ (d L ) , locally reading θ L = ∂ L ∂ v j d q j . (90) • The Lag rangian 2-for m ω L := − d θ L , locally reading ω L = ∂ 2 L ∂ v j ∂ v k d q j ∧ d v k − ∂ 2 L ∂ q k ∂ v j d q k ∧ d q j . (91) • The Lag rangian ener gy E L := ∆( L ) − L . 18 A syst em is regular if ω L is symplectic (i.e., its Hessian matrix  W ij = ∂ 2 L ∂ v i ∂ v j  is non-singular). The follo wing theorem giv es suﬃcient (which are, triviall y , also necessar y) conditions for a 2 -form ω on T Q to be Lag rangian. Theorem 3.11 ( Chara cteriza tion of La grangian 2-f orms ) . A 2-form ω on T Q is a (local) Lagr angian 2-form, i.e., ω = ω L = − dd S L for some L ∈ C ∞ ( T Q ) , if and only if it satisﬁes the following conditions: d ω = 0 , (92) ω ( S X , Y ) = ω ( S Y , X ) , ∀ X , Y ∈ X ( T Q ) . (93) Proof. ( = ⇒ ) Condition ( 92 ) implies t hat ω can be locally written as ω = d  A j d q j + B j d v j  . (94) Condition ( 93 ) for any pair of t he type X = ∂ ∂ q j , Y = ∂ ∂ v k giv es ω ∂ ∂ v j , ∂ ∂ v k ! = 0 , ∀ j, k = 1 , ..., n , (95) namel y that ∂ B [ j ∂ v k ] = 0 , ∀ j, k = 1 , ..., n (96) (square brackets denoting skew -symmetrization) i.e., B j = ∂ B ∂ v j (97) for some B ∈ C ∞ ( T Q ) . Thus, ω locall y reads ω = d A j d q j + ∂ B ∂ v j d v j ! = d A j d q j + d B − ∂ B ∂ q j d q j ! = d  C j d q j  , (98) for C j = A j − ∂ B ∂ q j . (99) On t he other hand, condition ( 93 ) for an y pair of t he type X = ∂ ∂ q j , Y = ∂ ∂ q k , giv es ω ∂ ∂ v j , ∂ ∂ q k ! = ω ∂ ∂ v k , ∂ ∂ q j ! . (100) It is easy to see t hat t his latter condition implies t hat ∂ C [ j ∂ v k ] = 0 , ∀ j, k = 1 , ..., n , (101) i.e., C j = ∂ L ∂ v j , (102) for some L ∈ C ∞ ( T Q ) . This prov es that there exis ts a local function L on T Q such t hat ω = dd S L . ( ⇐ = ) The necessity of condition ( 92 ) - ( 93 ) is trivial and follo ws from a str aightforwar d computation. 19 The solutions of a regular system are the integ ral cur v es of the unique SODE (Second Order Diﬀerential Equation) ﬁeld X L (namel y , satisfying S ( X L ) = ∆ ), which is t he unique solution to t he intrinsic Euler-Lagrang e equation i X L ω L = d E L . (103) Deﬁnition 3.12 ( Degenera te La grangian sys tems ) . A Lagrangian syst em ( Q, L ) is degenerate if ω L is degener at e (pr e-symplectic). A degener ate Lagrangian sys tem is precisel y a pr e-symplectic Hamiltonian syst em ( T Q, ω L , E L ) , but one which carries t he "extr a" kinematic constr aint t hat its physical dynamics mus t be a SODE ﬁeld. As in t he Hamiltonian case, t his pre-symplectic system ( T Q, ω L , E L ) can be eit her inconsistent or consistent. Inconsistent Lag rangian Systems. In t his case k er ω L ⊆ ker d E L , and, t hus, the equation i X ω L = d E L is not sol vable on t he entire T Q , meaning no global dynamical ﬁeld X exists. T o ﬁnd the (sub)manifold where consis tent solutions exist, one must apply a cons traint algorithm. His toricall y , this problem was tackled b y the celebrated Dir ac-Bergmann algorithm [ 21 ]. This ter m often encompasses tw o related but distinct procedures. Dir ac’ s Hamiltonian Algorithm , dev eloped by P .A.M. Dirac and P .G. Bergmann [ 21 ], follow s t he follo wing steps: • The Legendre map F L : T Q → T ∗ Q , deﬁned by the ﬁber deriv ativ e ⟨F L ( v ) , w ⟩ := d d s L ( v + sw )     s =0 , (104) and locally reading ( q j , v j ) 7→ q j , p j = ∂ L ∂ v j ! , (105) is assumed to be almost-r egular (meaning t he Hessian W ij has constant rank). Conse- quentl y , its image M 1 := F L ( T Q ) is a submanifold of T ∗ Q deﬁned by a set of primary constr aints Φ (1) a ( q , p ) ≈ 0 (where ≈ means t hat the equality should be fulﬁlled along solutions of t he equations of motion). • A canonical Hamiltonian H C ∈ C ∞ ( M 1 ) is deﬁned (as it can be shown t hat E L is constant on t he ﬁbers of F L [ 24 ]). This H C is t hen extended to a Hamiltonian H E ∈ C ∞ ( T ∗ Q ) on the ambient space. • The tot al Hamiltonian is deﬁned as H T = H E + µ a Φ (1) a , where µ a are arbitrary functions (Lagrange multipliers). • The algorithm imposes the consistency condition ˙ Φ (1) a ≈ { Φ (1) a , H T } ω Q ≈ 0 . This procedure iterativ ely gener ates a set of secondar y (and tertiary , etc.) constr aints, deﬁning a ﬁnal constraint manifold M f ⊂ T ∗ Q . Remark 3.13 ( Rela tion between Dira c’s algorithm and the PCA ) . W e can now clarify the r elationship between Dirac’ s algorithm and the geometric PCA (as deﬁned in Section 3.2 ). Dir ac’ s algorit hm is, in essence, t he local coordinat e version of t he g eometric PCA. Indeed, as it is prov en in [ 24 ], Dirac’ s k -ary constr aints Φ ( k ) ≈ 0 locall y select the submanifold M k ⊂ T ∗ Q . F urthermor e, the Hamiltonian vector ﬁeld X Φ ( k ) (associated with a k -ary constr aint via the symplectic structur e ω T ∗ Q of T ∗ Q ) is tang ent to the constr aint submanifold M k and belongs to t he space ( T M k ) ⊥ ω T ∗ Q used in the 20 PCA iter ation. On the ot her hand, it is also prov en that any vector ﬁeld Y ∈ ( T m M k − 1 ) ⊥ ω T ∗ Q (t he space used by the PCA) gives rise to a local constr aint condition (d H E )( Y ) ≈ 0 , which is precisel y how Dir ac gener ates the next set of cons traints. Theref or e, one can conclude that the constr aint conditions imposed by Dirac are t he local coordinat e expressions of the geometric conditions t hat select t he submanifolds M k of the PCA. Howev er , a conceptual diﬀerence between the two approac hes exists. Dirac’ s algorithm wor ks on the whole Phase Space T ∗ Q along the constr aint submanifolds M k , and deﬁnes solutions (vect or ﬁelds X T ) on the whole T ∗ Q . In contr ast, one could appl y the PCA (as deﬁned in Section 3.2 ) intrinsicall y , st arting from the pre-sym plectic manifold ( M 1 , ω 1 = ω T ∗ Q | M 1 ) as the "ambient" space. As not ed in [ 24 ], these two procedur es ar e equiv alent and st abilize on the same ﬁnal constr aint manifold M f . In this r espect, it is prov en in [ 24 ] that t he intrinsic solution X f (found on M f and satisfying i X f ω f = d H f ) can be lifted to a solution X T on the ambient space T ∗ Q satisfying the equations for the to tal Hamiltonian, i X T ω Q = d H T . On t he other hand, Ber gmann’ s Lagrangian Algorithm , dev eloped b y P .G. Bergmann [ 2 , 38 ], operates entirel y on the tangent bundle (the v elocity space) T Q . It follo ws the steps: • The Euler-Lagrang e equations are W ij ¨ q j + ( ∂ 2 L ∂ v i ∂ q j v j − ∂ L ∂ q i ) = 0 , where ( W ij = ∂ 2 L ∂ v i ∂ v j ) is the singular Hessian. • Contracting with a v ector Y i in t he kernel of W ( Y i W ij = 0 ) annihilates t he ¨ q term, yielding t he primary constr aints Φ (1) a ( q , v ) ≈ Y i a ( ∂ 2 L ∂ v i ∂ q j v j − ∂ L ∂ q i ) ≈ 0 . • The algorit hm imposes consistency b y diﬀerentiating t hese constraints, d d t Φ (1) a ( q , v ) ≈ 0 . This introduces ¨ q terms, which are t hen replaced using t he "ev olutiv e" part of the E-L equations, generating secondary constr aints Φ (2) b ( q , v ) ≈ 0 . Remark 3.14 ( Rela tion between Ber gmann’s algorithm and the PCA ) . In our intrinsic formulation ( Deﬁnition 3.12 ), the degener ate sys tem is the pre-sym plectic sys tem ( T Q, ω L , E L ) . It can be prov en that Ber gmann’ s algorithm is exactl y the application of t he PCA to this sys tem. Indeed, t he PCA begins by deﬁning M 1 = { m ∈ T Q | (d E L ) m ( Y ) = 0 , ∀ Y ∈ k er( ω L ) m } . Let us identify t he kernel K = k er ω L . It can be locall y decomposed as K = K V ⊕ K H , where K V = K ∩ Im S (t he vertical kernel, relat ed to k er W ) and K H is a horizontal complement (r elated to k er W in the ∂ /∂ q dir ections). W e tes t the PCA condition (d E L )( Y ) = 0 on bot h parts: • F or the vertical kernel Y Z = Z i ∂ ∂ v i ∈ K V (wher e W ij Z j = 0 ): (d E L )( Y Z ) = ∂ E L ∂ v i ! Z i = ( v k W ki ) Z i = v k ( W ik Z i ) = 0 . (106) This condition is satisﬁed identicall y because Z ∈ k er W . The vertical kernel gener ates no constr aints. • F or the horizontal kernel Y Y = Y i ∂ ∂ q i ∈ K H (wher e W ij Y j = 0 ): (d E L )( Y Y ) = ∂ E L ∂ q i ! Y i = v j ∂ 2 L ∂ q i ∂ v j − ∂ L ∂ q i ! Y i ≈ 0 . (107) This last equation is precisel y the set of primar y constr aints Φ (1) a ( q , v ) ≈ 0 derived from Bergmann ’ s Lagr angian algorit hm. Since the subsequent st eps of bot h algorit hms are deﬁned by the same iter ative tang ency req uirement, t he two algorithms are equiv alent. 21 The PCA onl y checks for t he existence of some v ector ﬁeld X satisfying i X ω L = d E L . It does not check if t his X is a SODE ﬁeld (i.e., S ( X ) = ∆ ). A system can be Hamilton-consis tent ( M f  = ∅ ) but Lag rangian-inconsis tent if none of t he solutions X on M f are SODEs. T o solv e t his, one should use a "SODE-compatible" PCA . A t each step k , one deﬁnes t he next manifold F k ⊂ F k − 1 as t he locus of points m where t here exists a v ector X m ∈ T m ( T Q ) that satisﬁes all thr ee conditions: 1. Hamiltonian condition: ( i X m ω L + d E L ) m ( Y ) = 0 for all Y ∈ ( T m F k − 1 ) ⊥ ω L . 2. SODE condition: S ( X m ) = ∆ m . 3. T angency condition: X m ∈ T m F k − 1 . If this algorit hm con v erg es, it ﬁnds a ﬁnal constr aint manifold F f where the Lag rangian system becomes consistent, and which is, in gener al, a subset of the Hamiltonian one, F f ⊆ M f . The case of inconsistent Lag rangian systems forces us to ﬁrst apply the L CA to ﬁnd t he ph ysical manifold F f . In gener al, t here is no reason to expect F f to be a tangent bundle itself. A t t his stage, t he system is ( F f , ( ω L ) f , ( E L ) f ) , which is a consistent pre-symplectic system (genericall y with gaug e freedom k er( ω L ) f  = { 0 } ), but it is no longer a Lagrangian system. T o regularize t he remaining gaug e freedom, one can apply the coisotropic embedding ( Theorem 3.7 ) to ( F f , ( ω L ) f ) . The result is a symplectic manifold ( f F f , f ω f ) . This manifold is generic, non-Lagrangian, and has lost t he original physical tangent structure. This path solv es the constraint problem but "destro ys" t he Lagrangian structure. Exam ple 3.15 ( Affine La grangians ) . A particular l y enlightening example of the problem present ed above is t hat of aﬃne Lagrangians. Indeed, let α ∈ Ω 1 ( Q ) be a 1 -form and f ∈ C ∞ ( Q ) be a function. Let L : T Q − → R , L ( v ) := α ( v ) + f ( π Q ( v )) . (108) Locall y , if α = α i d q i , we ha ve L = α i ˙ q i + f ( q ) . The P oincaré–Cart an form in this scenario is ω L = − d  α i d q i  = − d α . (109) And the Lagrangian energy is E L = − f . Then, the equations of motion are ι X d α = d f , so that it is actually a ﬁrs t order equation on Q , and may be r egar ded as a pr e-symplectic sys tem. Then, t he constr aint algoritm on ( T Q, L ) is π Q -r elated to the algorithm on the pr e-symplectic syst em ( Q, d α, f ) as follows: F f · · · F 1 T Q M f · · · M 1 Q , (110) and we ha ve the equality F k = { v ∈ T p M k − 1 : p ∈ M k and ι v d α = d f } . (111) In particular , ( M f , d α, d f ) is a consist ent pre-symplectic Hamiltonian syst em and F f − → M f is an aﬃne bundle modeled over K f = k er d α . Then, the char at eristic distribution on F f is K f C | F f , and the t hinkenning is (an open subset of) e F f =  K f C  ∗ | F f − → F f . (112) In particular , by taking adapt ed coor dinat es ( x a , f A ) to K f , the coor dinates on e F f ar e ( x a , f A , ˙ f A , µ A , ˙ µ A ) , and ther e is no reason to expect e F f to be a tang ent bundle. 22 Remark 3.16 ( Got a y and Nester’s La grangian constraint algorithm ) . There ar e diﬀer ent ways of performing the constr aint algorithm to ensur e t hat the ﬁnal constr aint not only has well-deﬁned (tang ent) dynamics, but the dynamics can be chosen to satisfy the SODE condition. T o our knowledg e, t he most st andar d algorit hm is the Lagrangian constr aint algorithm by Got ay and Nes ter [ 25 ]. The algorit hm proceeds as follows. Let L ∈ C ∞ ( T Q ) be a singular Lagr angian. Then, inst ead of req uiring t he SODE condition at each step, one follow s the Hamiltonian version of the algorit hm, namely , by setting M 0 = T Q and then M k := { m ∈ M k − 1 | (d H ) m ( Y ) = 0 , ∀ Y ∈ ( T m M k − 1 ) ⊥ ω } . (113) Suppose t hat the sequence M 0 , M 1 , . . . st abilizes at M f . Then, ther e is a vect or ﬁeld X ∈ X ( M f ) such t hat i X ω f = d E f , (114) wher e ω f = ( ω L ) | M f and E f := ( E L ) | M f . Howev er , t he vect or ﬁeld X may not satisfy the SODE condition. The idea by Got ay and Nes ter is to ﬁnd a submanifold of M f , in which X can be chosen to satisfy it. The construction employ s the Legendr e transf ormation, and that t he algorit hms (in the Lagr angian and Hamiltonian side) are conv eniently r elated by it. Indeed, when L is almost r egular , by deﬁning P 0 := Leg L ( T Q ) ⊂ T ∗ Q , (115) one has a ﬁbration Leg L : T Q − → P 0 . One can show t hat the Ener gy E L is cons tant along this ﬁbers, so that ther e is a well deﬁned Hamiltonian T Q P 0 R Leg L E L H 0 . (116) Then, by denoting P 0 , P 1 , . . . t he submanifolds obtained by the pre-sym plectic constr aint algorihtm applied to ( P 0 , ( ω Q ) | P 0 , H 0 ) , one has that t he y are relat ed by the Legendr e transf ormation: M f · · · M 1 T Q P f · · · P 1 P 0 T ∗ Q Leg L Leg L Leg L . (117) U nder mild r egularity conditions, Leg L : M f − → P f deﬁnes a ﬁber bundle. Gotay and N est er t hen sol ve t he SODE problem as follows: Choose a vector ﬁeld X ∈ X ( P f ) sol ving i X ( ω L ) P f = d( E L ) P f , (wher e ( ω L ) P f r eads the pull-back of ( ω Q ) | P 0 to P f and ( E L ) P f is t he pull-back of E L to P f )) and Y ∈ X ( M f ) which is Leg L -r elated t o X , namel y (Leg L ) ∗ Y = X . Then, Y sol ves t he equation i Y ( ω L ) M f = d( E L ) M f wher e ( ω L ) M f is the pull-back of ω L to M f and ( E L ) M f is the pull-back of E L to M f ), but, as discussed, does not necessarily satisfy the SODE condition S ( Y ) = ∆ . Hence, one studies the defect Y ∗ = S ( Y ) − ∆ . (118) Let us show t hat lim t →∞ ψ Y ∗ t (wher e ψ Y ∗ t denot es the local ﬂow of Y ∗ ) exis ts and deﬁnes a submanifold S f  → M f which is diﬀeomorphic to P f t hrough the Legendr e tr ansformation. Indeed, if Y = a i ∂ ∂ q i + b i ∂ ∂ ˙ q i , we ha ve Y ∗ = ( a i − ˙ q i ) ∂ ∂ ˙ q i . (119) Since Y is Leg L -project able, a i is cons tant on the ﬁbers of M f − → P f and, furt hermore, since we hav e i S ( X ) ω L = i ∆ ω L for every vect or ﬁeld Y , in particular we have Y ∗ ∈ k er ω L . This, tog et her with the 23 fact t hat k er d Leg L = k er d π Q ∩ ker ω L , implies that Y ∗ is tang ent to the ﬁbers. N ow it is clear that t he integr al curve of Y ∗ t hrough the point ( q i 0 , ˙ q i 0 ) is γ ( t ) = ( q i 0 , a i + e − t ( ˙ q i 0 − a i )) , (120) so that its limit exis ts, and is t he point wit h coor dinates ( q i 0 , a i ) . In particular , S f is the image of a section σ : P f − → M f S f M f · · · M 1 T Q P f · · · P 1 P 0 T ∗ Q σ . (121) By deﬁnition, Y ∗ vanishes on S f , so that Y sol ves the equation and satiﬁes t he SODE condition S ( Y ) = ∆ . However Y need not be tang ent to S f , but it is enough to consider e Y := σ ∗ ( X ) , which • Is tang ent to S f . • Sol ves t he equations by deﬁnition. • Satisﬁes the SODE condition. Indeed, Y satisﬁes it, and Y − e Y is a vertical vect or ﬁeld (with r espect to t he projection π Q : T Q − → Q ), since ker(d Leg L ) = ker d π ∩ ker ω L . Remark 3.17 ( Rela tion between both algorithms ) . The algorithm present ed is relat ed t o the one by Gotay and Nes t er as follows. Deno te by F 0 , F 1 , . . . t he constr aint submanifold obatined by req uiring t he SODE condition at each st ep. Then, it is clear t hat at each st ep F k ⊆ M k , so that F f ⊆ M f . F urthermor e, S f ⊂ F f , as we clear l y hav e S f ⊆ F k , for every k . Indeed, the inclusion S f ⊆ F 0 is clear , and the subsequent ones are obtained iter atively by deﬁnition. Consistent Lag rangian Systems. If t he system is consistent but has gauge ambiguities, we can bypass the constr aint algorithm and appl y the coisotropic embedding theorem directly to the initial, degenerate Lagrangian system ( T Q, ω L ) . This approach raises the true "tangent structure problem": does t his procedure preser v e t he tangent structure? Speciﬁcall y: 1. Is t he regularized symplectic manifold f M diﬀ eomor phic to a tangent bundle T e Q ? 2. If so, is the new symplectic form e ω a r egular Lagrangian 2-form ω e L ? In t he next section, w e show t hat for a speciﬁc, physicall y relev ant class of degener acies, the answ er to the ﬁrst question is y es, while the answ er to the second one is no, unless t he coisotropic regularization scheme used in the Hamiltonian setting is slightly modiﬁed. 3.4 Existence and uniqueness of Lag rangian regular ization The objectiv e of t his section is to prov e the existence of an autonomous Lagrangian regu- larization, under speciﬁc conditions on t he gauge ambiguities of L . W e also discuss the matter of uniqueness. Although global uniqueness is not guar anteed, as a plet hora of extended Lag rangians ma y be considered, w e pro v e t hat an y tangent structure on a particular symplectic regularization f M must be “isomor phic on T Q ” to t he one w e build. Namel y , the ﬁrst-or der germ of t he extension is unique. The main assum ption t hat w e will make to endo w t he regularization wit h a Lagrangian structure (as in [ 27 , 29 ]) is t hat the characteristic distribution K = ker ω L pro viding the characteristic bundle K ⊂ T ( T Q ) is t he complete lift (tangent distribution) of an integ rable distribution K f on Q , which deﬁnes a regular foliation F K e . Let Q ha v e local coordinates ( x a , f A ) , where x a are coordinates on t he lea v es of F K e and 24 f A parameterize t he ﬁbers (the distribution K f ). The tangent bundle T Q has coordinates ( x a , f A , ˙ x a , ˙ f A ) . U nder t his h ypothesis, t he kernel of ω L is K = K f C , locally spanned b y span    ∂ ∂ f A ! C = ∂ ∂ f A , ∂ ∂ f A ! V = ∂ ∂ ˙ f A    . (122) The symplectic t hickening ( f M , e ω ) is constructed as a neighborhood of the zero section in the dual bundle K ∗ → T Q , as described in Theorem 3.7 . As discussed in [ 27 ], such thickening coincides with t he whole K ∗ if the almost product structure P can be chosen to hav e v anishing Nĳenhuis tensor . W e assume this is t he case from no w on. On t he other hand, the t hickened space f M = K ∗ can be identiﬁed as t he cotangent bundle of the foliation F K in t he sense of t he follo wing proposition: Proposition 3.18. The following canonical isomor phism exis ts: f M = K ∗ ≃ T ∗ F K := G F ∈F K T ∗ F , (123) wher e F deno tes a leaf of F K . Proof. A point p ∈ K ∗ is, b y deﬁnition, an element of t he dual bundle to K . It consists of a pair ( m, α m ) , where m ∈ T Q is t he base point, and α m ∈ K ∗ m is a linear functional on t he ﬁber K m = k er( ω L ) m . Thus, α m : K m → R . On t he other hand, a point q ∈ T ∗ F K is, by its deﬁnition as a disjoint union, an element of the cotang ent bundle of a leaf F ∈ F K . This point consists of a pair ( m, β m ) , where m ∈ F is the base point, and β m ∈ T ∗ m F is a linear functional on t he tangent space to t hat leaf, T m F . Thus, β m : T m F → R . The foliation F K is, by construction, t he integ ral foliation of the distribution K . This means that, at any point m ∈ T Q , t he tangent space to the unique leaf F passing through m is precisel y the subspace K m : T m F = K m . (124) Since the domain spaces K m and T m F are the same v ector space, their dual spaces K ∗ m and T ∗ m F are also canonicall y identical. Therefore, there is a natural, ﬁber-preserving isomorphism Φ : K ∗ → T ∗ F K giv en by Φ( m, α m ) = ( m, α m ) , which simpl y re-inter prets t he co v ector α m ∈ K ∗ m as an element of T ∗ m F . This establishes the identity K ∗ ∼ = T ∗ F K . Remark 3.19. In the case where the almost product structur e P does not hav e vanishing Nĳenhuis tensor , t he identiﬁcation of K ∗ as a g lobal tang ent manifold does not makes sense anymor e. Indeed, t he coisotr opic embedding theor em for ces us to res trict to a tubular neighbor hood of T Q in K ∗ for the form to be symplectic. This neighbor hood may no longer be a vector bundle, thus destr oying the global tang ent structur e. Howev er , it inherits a natur al tang ent structur e, and may be considered a tang ent bundle locally . F or con venience, we res trict to the case in which e ω is globall y symplectic, although wit hout much diﬃculty all the constructions ext end to the situation where one must wor k on an open subset (indeed, t he local expr ession and constructions that we show wor k for any almost product structur e). Moreov er , if the Hamiltonian vector ﬁelds are complet e (in particular , if the Hamiltonian vect or ﬁeld in the thic kening is complete), t he dynamics alway s r emain in t his open subset, as t he ﬂow maintains r egularity . 25 The coordinates of f M are ( x a , f A , ˙ x a , ˙ f A , µ f A , µ ˙ f A ) , where ( µ f A , µ ˙ f A ) are the ﬁber coordinates dual to the kernel gener ators    ∂ ∂ f A , ∂ ∂ ˙ f A    . (125) W e no w deﬁne a new conﬁguration manifold e Q . W e identify e Q as t he cotang ent bundle of t he foliation F K e , denoted e Q := T ∗ F K e ≡ K f ∗ , and deﬁned as e Q = T ∗ F K e := G F e ∈F K e T ∗ F e , (126) where by F e w e denote a leaf of F K e . The manifold e Q has local coordinates ( e q ) = ( x a , f A , µ A ) . The tangent bundle of t his new space is T e Q = T ( T ∗ F K e ) , wit h local coordinates ( e q , ˙ e q ) = ( x a , f A , µ A , ˙ x a , ˙ f A , ˙ µ A ) . Proposition 3.20. Ther e exis ts a canonical isomorphism α t hat relat es T e Q to the thic kened space f M ≃ T ∗ T F K e α : T e Q → f M (127) Proof. The isomorphism is the T ulczyjew isomor phism for foliations deﬁned in Section 2.2 α : T e Q = T T ∗ F K e → T F T K e = T F K : ( x a , f A , µ A , ˙ x a , ˙ f A , ˙ µ A ) 7→ ( x a , f A , ˙ x a , ˙ f A , µ f A = ˙ µ A , p ˙ f A = µ A ) . (128) The isomor phism α is evidently diﬀerentiable, thus deﬁning a diﬀeomorphism, and allow s us to endow t he regularized manifold f M with t he structure of a tangent bundle, by "pushing forw ard" t he canonical tangent structure ( S T e Q , ∆ T e Q ) from T e Q to f M . W e deﬁne the tangent structure ( e S , e ∆) on f M as: e ∆ := α ∗ (∆ T e Q ) (129) e S := α ∗ ( S T e Q ) . (130) Intrinsicall y , e ∆ is the v ector ﬁeld on f M that is α -related to the canonical Liouville ﬁeld ∆ T e Q . The tensor e S is the unique (1 , 1) -tensor on f M satisfying e S ◦ α ∗ = α ∗ ◦ S T e Q . Since α is a diﬀeomorphism, this new structure ( f M , e S , e ∆) automaticall y satisﬁes the tangent bundle axioms ( Eq. (56) ). T o see t his structure explicitly , w e com pute its local form. The canonical structure on T e Q (with coordinates ( x a , f A , µ A , ˙ x a , ˙ f A , ˙ µ A ) ) is: ∆ T e Q = ˙ x a ∂ ∂ ˙ x a + ˙ f A ∂ ∂ ˙ f A + ˙ µ A ∂ ∂ ˙ µ A , (131) S T e Q = d x a ⊗ ∂ ∂ ˙ x a + d f A ⊗ ∂ ∂ ˙ f A + d µ A ⊗ ∂ ∂ ˙ µ A . (132) A t this point, a fundamental issue arises. Let us pull-back t he standard regularized symplectic form e ω from f M to T e Q via t he T ulczyjew isomorphism α , and check if the resulting 2-form b ω := α ∗ e ω is Lagrangian with respect to t he canonical tangent structure S T e Q . 26 Recall t hat the thickened symplectic form constructed via the standar d Hamiltonian coisotropic embedding is: e ω = τ ∗ ω L + d  µ f A P A + µ ˙ f A R A  , (133) where t he 1-forms P A and R A are t he 1-forms deﬁning an almost-product structure P adapted to K P = P A ⊗ ∂ ∂ f A + R A ⊗ ∂ ∂ ˙ f A , (134) where P A = d f A − P A a d x a − P ′ A a d ˙ x a , (135) R A = d ˙ f A − R A a d x a − R ′ A a d ˙ x a . (136) Appl ying the pull-back α ∗ , w e obtain: b ω = τ ∗ ω L + d ˙ µ A ∧ P A + ˙ µ A d P A + d µ A ∧ R A + µ A d R A . (137) F or b ω to be a regular Lag rangian 2-form on T e Q , it must satisfy t he symmetry condition ( 93 ) , namel y b ω ( S T e Q X , Y ) = b ω ( S T e Q Y , X ) for an y pair of vect or ﬁelds X , Y . Let us test this condition using t he canonical tangent structure S T e Q = d x a ⊗ ∂ ∂ ˙ x a + d f A ⊗ ∂ ∂ ˙ f A + d µ A ⊗ ∂ ∂ ˙ µ A , (138) and t he speciﬁc pair of coordinate v ector ﬁelds X = ∂ ∂ µ A and Y = ∂ ∂ f B . Appl ying t he v ertical endomorphism, w e ha v e S T e Q X = ∂ ∂ ˙ µ A and S T e Q Y = ∂ ∂ ˙ f B . Ev aluating t he lef t-hand side of t he symmetry condition yields: b ω ( S T e Q X , Y ) = b ω ∂ ∂ ˙ µ A , ∂ ∂ f B ! . (139) The only ter m in b ω containing d ˙ µ A is d ˙ µ A ∧ P A . Since P A  ∂ ∂ f B  = δ A B , w e obtain b ω ( S T e Q X , Y ) = δ A B . (140) Con v ersely , ev aluating the right-hand side yields b ω ( S T e Q Y , X ) = b ω ∂ ∂ ˙ f B , ∂ ∂ µ A ! = − b ω ∂ ∂ µ A , ∂ ∂ ˙ f B ! . (141) The only ter m in b ω containing d µ A is d µ A ∧ R A . Since R A  ∂ ∂ ˙ f B  = δ A B , w e obtain b ω ( S T e Q Y , X ) = − δ A B . (142) This implies t hat for t he 2-form to be Lagrangian, we w ould fundamentally need δ A B = − δ A B . This show s t hat b ω is nev er a Lag rangian 2-form, regar dless of t he choice of t he almost-product structure P . Therefore, t he standard coisotropic embedding inherited from the Hamiltonian setting fundamentally break s t he tangent bundle geometry , making it mandatory to slightly modify t he regularization scheme to preserv e t he Lag rangian nature of t he system. A t t his stage, ha ving prov ed that t he standard coisotropic embedding inevitabl y break s t he Lagrangian nature of t he system wit h respect to t he canonical tangent structure on T e Q , there are essentially tw o paths to proceed: 27 • Modifying t he isomorphism: One can abandon the canonical T ulczyjew isomorphism α and construct a diﬀerent bundle isomor phism betw een t he t hickened space f M and the tangent bundle T e Q . This is the approach adopted by A. Ibort and J. Marín-Solano in [ 27 ], where they introduce an arbitrary Riemannian metric on the ﬁbers of the v ector bundle K → T Q to build a non-canonical, metric-dependent isomor phism that correctly "twists" the variables to get a Lagrangian 2-form. • Modifying t he regularized 2-form: One can preser v e t he canonical, purel y geometric T ulczyjew isomorphism α and modify the deﬁnition of the regularized 2-form itself. In t he present w or k, w e adopt t he second approach. Speciﬁcally , rather t han rel ying on t he standar d Hamiltonian coisotropic form e ω , w e construct t he regularized Lagrangian 2-form b ω on T e Q b y taking t he pull-back of t he original degener ate Lag rangian form, α ∗ τ ∗ ω L , and adding a correction ter m t hat is Lag rangian b y construction. This ter m takes the form − dd S T e Q F , where F ∈ C ∞ ( T e Q ) is a globally deﬁned smooth function. The construction of such a function F requires ﬁxing an auxiliary connection on the bundle e Q → Q satisfying suitable properties (which are fulﬁlled, for example, by any linear connection). This new methodology presents tw o signiﬁcant advantag es ov er the existing literature: • It requires ﬁxing a less restrictiv e geometric structure (a connection) compared to t he requirement of a full Riemannian metric. • It yields a g loball y deﬁned regularized Lag rangian function e L that generates t he dynamics, unlike t he approach in [ 27 ] which only guarantees t he existence of local Lagrangian functions. The deﬁnition of t he function F is not canonical, and depends on the choice of tw o ingredients: • An Ehresmann connection ∇ on t he bundle e Q = K ∗ − → Q , giv en b y a splitting of the tangent bundle in v ertical and horizontal v ectors T e Q ∗ = V ⊕ H ∇ . (143) This connection is chosen so that t he splitting at Q (identiﬁed as a submanifold via t he zero section), is t he canonical splitting T e Q | Q = V ⊕ T Q , (144) in order for F to be zero at Q (and hence, to deﬁne an extension of L ). This can be achiev ed simpl y by choosing a linear connection, though it is not necessary . • An almost product structure P on Q , which complements t he distribution K f . Remark 3.21 ( C oordin a te expressions ) . Locally , we expr ess t he components of t he connection as H ∇ = span ( ∂ ∂ x a + Γ aA ∂ ∂ µ A , ∂ ∂ f B + Γ B A ∂ ∂ µ A ) . (145) The condition on ∇ inducing t he canonical splitting at t he zero section is reﬂect ed in t he Γ ’ s vanishing at Q (again identiﬁed via the zero section). On the ot her hand, we expr ess the project or deﬁning the almost product structur e as P = P A ⊗ ∂ ∂ f A =  d f A − P A a d x a  ⊗ ∂ ∂ f A . (146) 28 N ow , consider t he following maps f M T e Q ≃ V ⊕ H ∇ T Q V K f K f ∗ τ α p V P , (147) where p V denotes the projection from T e Q to V deﬁned by the connection ∇ chosen and t he arro w V − → K f ∗ is t he identiﬁcation of the v ertical bundle wit h t he ﬁber of a v ector bundle. Then, w e deﬁne the map F P, ∇ : T e Q − → R (148) for ξ ∈ T e Q via t he natural pairing betw een K f and K f ∗ F P, ∇ ( ξ ) = ⟨ ( P ◦ τ ◦ α )( ξ ) , p V ( ξ ) ⟩ . (149) Remark 3.22 ( Local expression of F P, ∇ ) . Using the coor dinate components of ∇ and P from Remar k 3.21 , we have t hat F P, ∇ =  ˙ µ A − ˙ x a Γ aA − ˙ f B Γ B A   ˙ f A − ˙ x a P A a  . (150) W e t hen hav e t he follo wing: Theorem 3.23 ( La grangian c oisotr opic embedding ) . Let L : T Q − → R be a singular Lagrangian. Suppose t hat L is consist ent and t hat the char acteris tic distribution K = k er ω L is t he complete lift of a distribution K f on Q . Then, given an Ehresmann connection ∇ on K ∗ and an almost product structur e P on Q as above, the embedding T Q  → K ∗ (151) is a coiso tropic embedding on a neighbor hood of T e Q for the symplectic s tructure ω e L , where e L = L + F P, ∇ . Proof. W e will ﬁrst show that T ( K ∗ )    T Q is a symplectic v ector bundle, so that e ω deﬁnes a symplectic structure on some neighborhood of T Q . Indeed, a quick computation show s t hat d S T e Q e L =d S L + ˙ f A d µ A + ˙ µ A d f A (152) −  Γ aA ( ˙ f A − ˙ x b P A b ) + P A a ( ˙ µ A − ˙ x b Γ bA − ˙ f B Γ B A )  d x a (153) −  Γ B A ( ˙ f A − ˙ x a P A a ) + ˙ x a Γ aA + ˙ f B Γ B A  d f A (154) − ˙ x a P A a d µ A . (155) Hence, taking the exterior diﬀerential and restricting to t he zero section (so that all Γ ’ s vanish), w e obtain t he follo wing 2 -form dd S T e Q e L = τ ∗ ω L + d ˙ f A ∧ d µ A + d ˙ µ A ∧ d f A + (semi-basic terms) . (156) N otice t hat t he ﬁrst t hree terms in the right-hand side deﬁne a symplectic structure. Since adding semi-basic terms (wit h respect to t he projection onto Q ) does not change regularity , w e hav e t hat T ( K ∗ )    T Q is a symplectic v ector bundle. Finall y , notice that it is a coisotropic embedding, as d S T e Q L    T Q = d S L . (157) 29 Ha ving established a constructiv e met hod for a regularized Lag rangian system, it is natural to ask to what extent t his regularization depends on the speciﬁc choices made (i.e., t he connection ∇ and t he almost-product structure P ). While t he global geometry of the t hickened space cannot be unique—since it heavil y depends on these arbitrary choices ev aluated a w a y from the zero section—its beha vior inﬁnitesimall y close to the original ph ysical system is completel y rigid. In mat hematical ter ms, w e can pro v e t hat its ﬁrs t-or der g erm along the original manifold T Q (that is, the regularized symplectic form and t he extended tangent structure ev aluated exactl y on the points of T Q ) is geometricall y unique, provided t he restricted tangent structures coincide. T o pro v e this, w e ﬁrst need a purel y algebr aic lemma regar ding symplectic v ector spaces equipped wit h nilpotent endomorphisms (which act as local models for tangent structures). Lemma 3.24. Let ( E , ω ) be a symplectic vector space, equipped wit h a (1 , 1) -tensor J satisfying J 2 = 0 and ω ( J x, y ) = ω ( J y , x ) (which is equiv alent to i J ω = 0 ). Let L ⊂ E be a Lagrangian subspace such t hat J ( L ) ⊆ L . If ther e exists a complementary subspace W such t hat E = L ⊕ W and J ( W ) ⊆ W , then we can construct a new complement A ( W ) which is Lagrangian and r emains in variant under J . Proof. Since L is a Lag rangian subspace, it is maximally isotropic, which means dim L = 1 2 dim E and ω ( l 1 , l 2 ) = 0 for an y l 1 , l 2 ∈ L . Because W is a complement ( E = L ⊕ W ), it immediatel y follo ws that dim W = dim E − dim L = 1 2 dim E . Further more, the non- degener acy of ω ensures that the map φ : L → W ∗ deﬁned b y φ ( l ) := ( i l ω ) | W is a linear isomorphism. W e deﬁne a deformation map A : W → L ⊕ W b y adding a speciﬁc correction ter m in L to ev er y v ector in W . Let l w := − 1 2 φ − 1 ( i w ω | W ) ∈ L , and deﬁne: A ( w ) := w + l w . (158) N otice t hat A is injectiv e: if A ( w ) = 0 , then w = − l w . Since w ∈ W and l w ∈ L , and t heir intersection is trivial ( L ∩ W = { 0 } ), it must be that w = 0 . Since A is an injective linear map, the image subspace A ( W ) has exactly the same dimension as W , namel y dim A ( W ) = 1 2 dim E . A dditionall y , w e must ensure t hat A ( W ) is a valid complement to L , meaning t heir intersection is trivial. Suppose a v ector v = w + l w ∈ A ( W ) also belongs to L . Since l w ∈ L , t his implies w = v − l w ∈ L . Ho w ev er , since the original sum E = L ⊕ W is direct, w e ha v e L ∩ W = { 0 } , which forces w = 0 . Consequentl y , l w = 0 , meaning t he only v ector in t he intersection is the zero v ector . Thus, A ( W ) intersects L triviall y and ser v es as a valid complementary subspace. N ext, w e explicitly show that A ( W ) is an isotropic subspace. Let w 1 , w 2 ∈ W and consider their images under A . Expanding the symplectic form using bilinearity , w e get: ω ( A ( w 1 ) , A ( w 2 )) = ω ( w 1 + l w 1 , w 2 + l w 2 ) = ω ( w 1 , w 2 ) + ω ( w 1 , l w 2 ) + ω ( l w 1 , w 2 ) + ω ( l w 1 , l w 2 ) . (159) The last term ω ( l w 1 , l w 2 ) = 0 , because L is isotropic. By deﬁnition of t he isomor phism φ , we ha v e ω ( l w 1 , w 2 ) = φ ( l w 1 )( w 2 ) . Substituting l w 1 , we get φ  − 1 2 φ − 1 ( i w 1 ω )  ( w 2 ) = − 1 2 ( i w 1 ω )( w 2 ) = − 1 2 ω ( w 1 , w 2 ) . W it h t his in mind w e get ω ( A ( w 1 ) , A ( w 2 )) = ω ( w 1 , w 2 ) − 1 2 ω ( w 1 , w 2 ) − 1 2 ω ( w 1 , w 2 ) + 0 = 0 . (160) 30 Therefore, A ( W ) is isotropic. Being an isotropic subspace with dimension exactl y half of dim E , A ( W ) is Lag rangian. Finall y , w e mus t ensure that A ( W ) is inv ariant under J . Let w ∈ W . It will be enough to sho w that J commutes wit h t he correction term, i.e., J ( l w ) = l J ( w ) , which is equiv alent to J ( φ − 1 ( w )) = φ − 1 ( J ( w )) . Let l = φ − 1 ( w ) , which by deﬁnition means ω ( l , x ) = ω ( w , x ) for all x ∈ W . Since W is in variant under J , t he v ector x = J ( w ′ ) also belongs to W for an y w ′ ∈ W . Substituting this into our deﬁning equation yields ω ( l , J ( w ′ )) = ω ( w, J ( w ′ )) . Using the symmetry of J with respect to ω ( i J ω = 0 ), w e can mov e J to t he ﬁrst slot, obtaining − ω ( J ( l ) , w ′ ) = − ω ( J ( w ) , w ′ ) . Since this holds for all w ′ ∈ W and t he map φ is an isomor phism, it follo ws t hat J ( l ) = φ − 1 ( J ( w )) , concluding the proof. Remark 3.25 ( The case of symplectic vector bundles ) . N otice that Lemma 3.24 applies as well to the case of symplectic vect or bundles ( E , ω ) − → M tog ether with a nilpot ent endomor phism J , an in variant Lagrangian subbundle L , and an inv ariant complement W . Indeed, the construction pr esented is g lobal and mantains smoot hness. Remark 3.26. F inally , notice that Lemma 3.24 req uires the exist ence of a J -in variant complement W . In the case of nilpotent endomor phisms, this may be built as follow s. Let f W be a complement of J ( L ) in J ( E ) , so that J ( E ) = J ( L ) ⊕ f W . Then, we can deﬁne W := J − 1 ( f W ) , which clear ly satisﬁes f W = J ( W ) ⊆ W ( J being nilpotent) and E = L ⊕ W . This holds as well in t he case of symplectic vect or bundles when J and J | L hav e constant r ank, which will certainly hold in our case. Using this algebraic tool, w e can now pro v e t he uniqueness t heorem for t he Lagrangian regularization. Theorem 3.27 ( Uniqueness of La grangian coisotr opic embedding to first order ) . Let ( f M 1 , e ω 1 ) and ( f M 2 , e ω 2 ) be two sym plectic regularizations of the deg ener ate Lagr angian sys tem ( T Q, ω L ) . Suppose that bot h regularizations admit a tang ent structur e e S i making the respectiv e forms Lagrangian (i.e., i e S i e ω i = 0 ), and that the embeddings i i : T Q  → f M i pr eserve the tang ent structur e: ( i i ) ∗ ◦ S T Q = e S i | T Q ◦ ( i i ) ∗ . Then, t her e exist tubular neighbor hoods U 1 , U 2 of T Q in f M 1 and f M 2 , r epsectivel y , tog ether wit h a local diﬀeomorphism ψ : U 1 → U 2 r estricting to the identity on T Q , such that its pushforw ar d provides an exact isomorphism of t he tang ent-symplectic structur es over T Q : ψ ∗ ( e ω 1 , e S 1 )    T Q = ( e ω 2 , e S 2 )    T Q . (161) Proof. Let n + r = dim Q , where K = k er ω L is t he characteristic distribution on T Q , wit h rank 2 r (because of t he h ypothesis t hat K is t he tangent distribution to a rank r distribution K f on Q ). W e can decompose the tangent space of T Q as T ( T Q ) = K ⊕ W , where t he com plementar y subbundle W must be chosen to be in v ariant under t he v ertical endomorphism S T Q . Such a complement naturall y arises from t he geometry of the tangent bundle: w e can choose a distribution H f complementary to K f on t he base manifold Q (so t hat T Q = K f ⊕ H f ), and deﬁne W as its tangent distribution. Since W is pointwise spanned by the complete lifts X C and vertical lifts X V of vect or ﬁelds X ∈ H f , t he fundamental properties S T Q ( X C ) = X V and S T Q ( X V ) = 0 intrinsicall y guarantee that S T Q ( W ) ⊆ W . F urt hermore, since K is t he ker nel of ω L , the restriction of ω L to W is non-degenerate, making ( W , ω L | W ) a symplectic v ector bundle of rank 2 n . Inside t he tang ent space of t he thickened manifold T f M i | T Q , w e consider the sym plectic orthogonal to W , deﬁned as W ⊥ , e ω i = { v | e ω i ( v , W ) = 0 } , which is again a symplectic v ector bundle. Crucially , W ⊥ , e ω i is inv ariant under e S i . Indeed, taking v ∈ W ⊥ , e ω i and testing it against 31 W : e ω i ( e S i ( v ) , W ) = e ω i ( e S i ( W ) , v ) = e ω i ( S T Q ( W ) , v ) = 0 , (162) where w e used t he symmetr y of e S i , the h ypothesis that t he embedding preser v es t he structure ( e S i | W = S T Q | W ), and t he fact that S T Q preserves W . By dimensional counting on the coisotropic embedding, dim f M i = dim ( T Q ) + dim K = (2 n + 2 r ) + 2 r = 2 n + 4 r , where 2 n is t he dimension of W . Since W is symplectic, then dim f M i = dim W + dim W ⊥ , e ω i , impl ying dim W ⊥ , e ω i = 4 r . By deﬁnition, K ⊂ W ⊥ , e ω i is isotropic. Since dim K = 2 r = 1 2 dim W ⊥ , e ω i , K is a Lag rangian subbundle of W ⊥ , e ω i . Appl ying Lemma 3.24 (together with Remar k 3.25 and Remar k 3.26 ) ﬁber-wise, w e can construct a J -in v ariant Lag rangian complement for K , allowing us to identify W ⊥ , e ω i ∼ = K ⊕ K ∗ . This provides a global, structure-preserving local isomorphism, sa y ψ : T f M i    T Q ∼ = W ⊕ K ⊕ K ∗ . (163) W e can therefore choose local coordinates ( x a , ˙ x a ) for W , ( f A , ˙ f A ) for K , and ﬁber coordinates ( µ A , ˙ µ A ) for K ∗ . In this univ ersal adapted frame evaluated precisely on T Q (where µ A = 0 , ˙ µ A = 0 ), an y regularized symplectic form must locall y read: ω = ω L + d µ A ∧ d f A + d ˙ µ A ∧ d ˙ f A . (164) It remains to show that t he extension of the tangent structure e S i is also uniquely deter mined on T Q . Let b S be an arbitrary (1 , 1) -tensor extending S T Q . In our univ ersal adapted frame ev aluated on T Q (where µ A = 0 , ˙ µ A = 0 ), the most g eneral matrix form for b S that acts as S T Q on t he base manifold is: b S =  d x a + F aA d µ A + ˙ F aA d ˙ µ A  ⊗ ∂ ∂ ˙ x a (165) +  d f A + G AB d µ B + ˙ G AB d ˙ µ B  ⊗ ∂ ∂ ˙ f A (166) +  H B A d µ B + ˙ H B A d ˙ µ B  ⊗ ∂ ∂ µ A (167) +  I B A d µ B + ˙ I B A d ˙ µ B  ⊗ ∂ ∂ ˙ µ A . (168) W e must impose the Lag rangian condition i b S ω = 0 . Recall t hat t he interior product of a (1 , 1) -tensor wit h a 2-form acts as ( i b S ω )( X , Y ) = ω ( b S X , Y ) + ω ( X, b S Y ) , which extends to w edge products as i b S ( α ∧ β ) = ( b S ∗ α ) ∧ β + α ∧ ( b S ∗ β ) . Applying the pull-back b S ∗ to t he basic 1-forms yields: b S ∗ (d x a ) = 0 , (169) b S ∗ (d f A ) = 0 , (170) b S ∗ (d ˙ x a ) = d x a + F aA d µ A + ˙ F aA d ˙ µ A , (171) b S ∗ (d ˙ f A ) = d f A + G AB d µ B + ˙ G AB d ˙ µ B , (172) b S ∗ (d µ A ) = H B A d µ B + ˙ H B A d ˙ µ B , (173) b S ∗ (d ˙ µ A ) = I B A d µ B + ˙ I B A d ˙ µ B . (174) 32 W e no w ev aluate i b S term by ter m on t he symplectic form ω = ω L + d µ A ∧ d f A + d ˙ µ A ∧ d ˙ f A . F or the base form ω L , since its kernel is exactly K = span { ∂ ∂ f A , ∂ ∂ ˙ f A } , it only contracts non-triviall y with coordinates ( x, ˙ x ) . Because ω L is already Lag rangian with respect to t he base tangent structure ( i S T Q ω L = 0 ), appl ying i b S onl y extracts t he new ly added transv erse coeﬃcients i b S ω L =  F aA d µ A + ˙ F aA d ˙ µ A  ∧ i ∂ ∂ ˙ x a ω L . (175) F or the second term, applying the product rule and using b S ∗ (d f A ) = 0 one gets i b S (d µ A ∧ d f A ) =  H B A d µ B + ˙ H B A d ˙ µ B  ∧ d f A , (176) whereas, for the t hird term one obtains i b S (d ˙ µ A ∧ d ˙ f A ) =  I B A d µ B + ˙ I B A d ˙ µ B  ∧ d ˙ f A + d ˙ µ A ∧  d f A + G AB d µ B + ˙ G AB d ˙ µ B  . (177) N ow , t he form d ˙ f A onl y appears in the t hird piece, multiplied b y the I matrices. Since it is linear ly independent from all other forms, its coeﬃcients must vanish, forcing I B A = 0 and ˙ I B A = 0 . Gathering the d f A w edges from the second and t hird pieces, w e get  H B A d µ B + ˙ H B A d ˙ µ B + d ˙ µ A  ∧ d f A = 0 , forcing H B A = 0 and ˙ H B A = − δ B A . The 1-forms i ∂ ∂ ˙ x a ω L are linear combinations of d x b . Since d x b do not appear an ywhere else in the expansion, the F matrices multiplying t hem must v anish, forcing F aA = 0 and ˙ F aA = 0 . W e are left with only d ˙ µ A ∧ ( G AB d µ B + ˙ G AB d ˙ µ B ) = 0 from the third term. The linear independence of d ˙ µ A ∧ d µ B forces G AB = 0 . F or t he second part, ˙ G AB d ˙ µ A ∧ d ˙ µ B = 0 implies that t he matrix ˙ G AB must be symmetric ( ˙ G AB = ˙ G B A ). Finall y , w e im pose the s tructural condition established b y Lemma 3.24 , namel y that the extended tangent structure must lea v e the dual Lag rangian complement in v ariant: b S ( K ∗ ) ⊆ K ∗ . The subspace K ∗ is gener ated b y { ∂ ∂ µ C , ∂ ∂ ˙ µ C } . Applying our simpliﬁed tensor b S to t he basis v ector ∂ ∂ ˙ µ C , w e obtain: b S ∂ ∂ ˙ µ C ! = ˙ G C A ∂ ∂ ˙ f A − ∂ ∂ µ C . (178) F or this resulting v ector to remain within K ∗ , it cannot possess an y component along ∂ ∂ ˙ f A , which belongs to K . This geometricall y forces t he symmetric matrix ˙ G C A to be strictl y zero. Thus, all unknown coeﬃcients are strictly identically zero or uniquely ﬁx ed. Consequently , there is only one algebr aicall y permissible tensor b S on T Q . Therefore, t he map ψ identifying the tw o univ ersal splittings satisﬁes ψ ∗ e S 1 = e S 2 along T Q , concluding t he proof. 4 Regularization of non-autonomous systems 4.1 Cosym plectic Hamiltonian systems While t he natural geometric setting for autonomous Hamiltonian systems is that of symplectic geometry , for non-autonomous systems is t hat of cosym plectic geometry (see [ 33 , 34 ]). 33 Deﬁnition 4.1 ( C osymplectic manifold ) . A cosymplectic manifold is a triple ( M , ω , τ ) wher e M is a smoot h manifold of dimension 2 n + 1 , ω ∈ Ω 2 ( M ) is a closed 2-form, and τ ∈ Ω 1 ( M ) is a closed 1-form, satisfying t he non-deg eneracy condition τ ∧ ω n  = 0 . Theorem 4.2 ( D arboux’s Theorem for cosymplectic manifolds ) . The Darboux t heorem can be g eneralized to cosymplectic manifolds. Let ( M , ω , τ ) be a cosymplectic manifold of dimension 2 n + 1 . Around every point m ∈ M , ther e exist local coor dinates ( q i , p i , t ) , called Darboux coordinates , such t hat: ω = d q i ∧ d p i , τ = d t . Deﬁnition 4.3 ( Cos ymplectic Hamil tonian sy stem ) . A cosymplectic Hamiltonian system is a tuple ( M , ω , τ , H ) , where ( M , ω , τ ) is a cosymplectic manifold and H ∈ C ∞ ( M ) is a smoot h function (t he Hamiltonian). Similar to t he symplectic case, t he isomorphism  guarantees t he existence of a unique v ector ﬁeld grad H ∈ X ( M ) , t he gradient vect or ﬁeld , satisfying:  (grad H ) = d H . (179) T o obtain the dynamics deﬁned b y t he Hamiltonian (see [ 4 ]), w e use grad H to deﬁne tw o additional v ector ﬁelds. F irst, t he Hamiltonian vector ﬁeld : X H = grad H − R ( H ) R , (180) and second, t he evolution vect or ﬁeld : E H = X H + R . (181) In Darboux coordinates ( q i , p i , t ) , using t he local expression for grad H , t he ev olution v ector ﬁeld takes the form: E H = ∂ H ∂ p i ∂ ∂ q i − ∂ H ∂ q i ∂ ∂ p i + ∂ ∂ t . (182) The solutions of t he non-autonomous Hamiltonian system are the integral cur v es ( q i ( ε ) , p i ( ε ) , t ( ε )) of E H , satisfying t he time-dependent Hamilton ’s equations: d q i d ε = ∂ H ∂ p i , d p i d ε = − ∂ H ∂ q i , d t d ε = 1 . (183) Since d t d ε = 1 , w e ha v e t = ε + const , allowing us to identify t he cur v e parameter wit h t he time coordinate t and recov er the standard non-autonomous Hamilton ’s equations. A particular case of high relev ance is obtained by taking t he product T ∗ Q × R , where T ∗ Q denotes the co tangent bundle of the conﬁgur ation manifold Q . Indeed, if w e deno te b y ω = − d θ Q , where θ Q is t he Liouville 1-form on T ∗ Q , t hen t he pair ( ω , d t ) deﬁnes a cosymplectic structure on T ∗ Q × R , where here ω and d t are t he ob vious extensions, t being t he standar d coordinate in R . A direct calculation show s t hat the natural bundle coordinates ( t, q i , p i ) are Darboux coordinates for this cosymplectic manifold. Remark 4.4 ( Cos ymplectic d ynamics are Reeb d ynamics ) . Assume that H is a Hamilt onian function on a cosymplectic manifold ( M , ω , τ ) . Then, we can construct an additional cosymplectic structur e depending on H , say ( ω H = ω + d H ∧ τ , τ ) . A simple comput ation shows that t he evolution vector ﬁeld for H wit h respect to ( ω , τ ) coincides with t he Reeb vect or ﬁeld for ( ω H , τ ) , E H = R H , so that autonomous Hamiltonian dynamics may be studied as Reeb dynamics. This point of view will be particular l y useful in the Lagrangian setting. 34 The point of view of R emark 4.4 is the picture that w e will adhere to on war ds, as it is t he most natural setting to study t he non-autonomous Lagrangian side (see [ 18 , 32 ] for a compr hensiv e account on t he Lagrangian description of time-dependent mechanics in ter ms of jet bundles, and [ 11 , 12 ] for t he singular case). 4.2 Coisotropic regular ization of pre-cosym plectic Hamiltonian systems In general, when w or king with singular time-dependent t heories (such as time-dependent Hamiltonian gaug e t heories and time-dependent singular Lag rangian t heories), w e w or k on a pre-cosymplectic manifold rather t han on a cosymplectic one. Deﬁnition 4.5 ( Pre-c osymplectic Hamil tonian sy stem ) . A pr e-cosymplectic Hamiltonian system can be fundamentall y underst ood t hrough its Reeb dynamics (as per Remark 4.4 ). Let ( M , ω , τ ) be a pre-cosym plectic manifold. The dynamics ar e formall y governed by the equations: i X ω = 0 , and i X τ = 1 . (184) As in the autonomous case, t hese equations pose two distinct problems: 1. Existence: A vect or ﬁeld X satisfying bot h equations may not exis t. 2. Uniqueness: If a solution X exis ts, it is not unique, as it is deﬁned onl y up to the addition of any vect or ﬁeld Y ∈ V := k er ω ∩ ker τ . These tw o problems identify , again, tw o classes of pre-cosymplectic systems: Inconsistent Hamiltonian Systems. A system is inconsist ent if t he existence condition fails. In t his case, a cosymplectic generalization of t he pre-symplectic constr aint algorithm [ 5 ] can be used to ﬁnd t he submanifold where consistent dynamics can be deﬁned. The algorit hm proceeds iterativ el y too. W e deﬁne M 0 := M and deﬁne t he ﬁrst constraint manifold M 1 as the locus of points where the equations are com patible with tangency conditions: M 1 := { m ∈ M 0 | ∃ X ∈ T m M 0 with ( i X ω ) m = 0 and τ m ( X ) = 1 } . (185) Assuming M 1 is a smooth submanifold, the algorit hm imposes solutions of ( 184 ) on M 1 to be tangent to M 1 . Ev entually , at each step k ≥ 1 , one ﬁnds the submanifold M k +1 ⊂ M k : M k := { m ∈ M k − 1 | ∃ X ∈ T m M k − 1 with ( i X ω ) m = 0 and τ m ( X ) = 1 } . (186) As for t he pre-sym plectic case, if the algorit hm s tabilizes, w e denote by ω f and τ f the restrictions of ω and τ to M f . Then, ( M f , ω f , τ f ) is a consistent pre-cosymplectic manifold which has Reeb dynamics deﬁned. Consistent Hamiltonian Systems. A system is consistent if it admits global Reeb dynamics. This corresponds to a sys tem t hat eit her started consistent or is the result ( M f , ω f , τ f ) of appl ying the constraint algorithm. As for t he pre-symplectic case, t he dynamics can still be modiﬁed b y an y v ector ﬁeld Y taking v alues in the characteristic distribution V = k er ω f ∩ ker τ f and one can regularize t he system using a cosymplectic v ersion of t he coisotropic embedding theorem [ 30 ]. Remark 4.6 ( C onstr uction of the cos ymplectic thickening ) . Let V := ker ω ∩ k er τ . Since it is t he intersection of t he kernels of two closed forms, it is integr able in the sense of F robenius. Let ( x a , f A ) 35 denot e adapted coor dinates to the foliation. Since, by construction, ther e is a Reeb vect or ﬁeld, we may furt her specify coor dinates adapt ed to the pair ( ω , τ ) obtaining coordinat es ( q a , p a , f A , t ) such that ω = d q a ∧ d p a , τ = d t . (187) Then, t he r egularized structur e is deﬁned on the bundle V ∗ → M , and is deﬁned in g eneral using an almost-pr oduct structur e complementing t he distribution V . In gener al, this will be deﬁned by a project or P = P A ⊗ ∂ ∂ f A =  d f A − P A a d q a − P Aa d p a − Q A d t  ⊗ ∂ ∂ f A . (188) This project or , as in the symplectic case, induces an embedding j P : V ∗ → T ∗ M , θ 7→ P ∗ θ . (189) Then, if θ M denot es the taut ological 1 -form on T ∗ M , we deﬁne ϑ P := ( j P ) ∗ θ M and let e ω = π ∗ ω + d ϑ P , e τ = π ∗ τ , (190) wher e π : V ∗ → M denot es the projection. Locall y , employing coordinat es ( q a , p a , t, f A , µ A ) on V ∗ , t his r eads as e ω = d q a ∧ d p a + d µ A ∧ P A + µ A d P A , e τ = d t . (191) Ho w ev er , unlike t he symplectic case, t his embedding is not unique. R ecently , t he aut hors studied all possible coisotropic embeddings, and uniqueness is guaranteed b y a choice of Reeb vect or ﬁeld tangent to the ﬁnal constr aint and t he orbits of t he Reeb v ector ﬁeld on t he extension. More particularl y: Theorem 4.7 ( Uniqueness of pre-c osymplectic c oisotr opic embedding ) . [ 30 ] Let ( M , ω , τ ) be a pr e-cosymplectic manifold of const ant rank and i j : M  → f M j be coisotr opic embeddings into cosymplectic manifolds ( f M j , e ω j , e τ j ) , for j = 1 , 2 . Then: • Ther e ar e neighbor hoods U 1 and U 2 of M in f M 1 and f M 2 and a diﬀeomorphism ψ : U 1 → U 2 . • F urthermor e, if the Reeb vect or ﬁelds R 1 and R 2 of f M 1 and f M 2 (which are tang ent to M ) coincide on M , we have t hat t he diﬀeomorphism ψ can be chosen such that its pushforw ar d ψ ∗ : T f M 1 | M − → T f M 2 | M (192) deﬁnes an isomor phism of cosymplectic vect or bundles. • F inally , if t her e is a diﬀ eomorphism ψ 0 : U 1 → U 2 t hat is the identity on M and satisﬁes ( ψ 0 ) ∗ R 1 = R 2 (namel y , both Reeb vector ﬁelds have t he same orbits), ψ can be chosen to be a cosymplect omorphism. Remark 4.8. In short, coisotr opic embeddings of a pr e-cosymplectic manifold ( M , ω , τ ) (in particular , ( M f , ω f , τ f ) ) are unique topologicall y . Furt hermore, if in advance we ﬁx a Reeb vect or ﬁeld on M f , t hen t he coisotropic embedding is unique "on M ". F inall y , if the Reeb dynamics of bot h t hickenings are conjug ate, the embeddings are neighborhood equiv alent. 4.3 Coisotropic regular ization of non-autonomous Lagrangian systems Let π : Q → R be a ﬁber bundle wit h standard ﬁber Q , which will denote t he conﬁguration manifold. As in t he autonomous case, w e ﬁrs t f ormally deﬁne the g eometric structures associated wit h an y non-autonomous Lagrangian L ∈ C ∞ ( J 1 π ) . 36 Deﬁnition 4.9 ( Second Order Differential Equ a tion (SODE) ) . A vect or ﬁeld X ∈ X ( J 1 π ) is a Second Order Diﬀerential Eq uation (SODE) ﬁeld if it correctl y relat es the position, velocity , and time coordinat es. In natur al local coordinat es ( q i , ˙ q i , t ) , a gener al SODE vect or ﬁeld takes t he form: X = ∂ ∂ t + ˙ q i ∂ ∂ q i + X i ( q , ˙ q , t ) ∂ ∂ ˙ q i . (193) Intrinsicall y , this condition is expressed by the two equations: (d t )( X ) = 1 , and S ( X ) = 0 , (194) wher e S is the vertical endomorphism on J 1 π (see Deﬁnition 2.24 ). Deﬁnition 4.10 ( Regular Non- a uton omous La grangian sy stem ) . A non-autonomous La- g rangian syst em is a pair ( Q , L ) , wher e L : J 1 π → R . W e deﬁne: • The Poincaré-Car tan 1-for m θ L := L d t + d S L = L d t + i S d L , locally reading: θ L = L − ∂ L ∂ ˙ q i ˙ q i ! d t + ∂ L ∂ ˙ q i d q i . (195) • The Lag rangian 2-for m ω L := − d θ L . A syst em is regular if the pair ( ω L , d t ) deﬁnes a cosymplectic structur e on J 1 π . This happens if and onl y if t he Hessian matrix  W ij = ∂ 2 L ∂ ˙ q i ∂ ˙ q j  is non-singular . In this case, the Euler -Lagr ange equations corr espond pr ecisely to the unique Reeb dynamics of the cosymplectic manifold ( J 1 π , ω L , d t ) . The follo wing t heorem extends Theorem 3.11 to t he cosymplectic context. Theorem 4.11 ( Chara cteriza tion of La grangian 2-forms ) . Let π : Q − → R be a ﬁber bundle. Then, a 2 -form ω ∈ Ω 2 ( J 1 π ) is locally a Lagrangian 2 -form if and onl y if t he following conditions hold: • It is closed: d ω = 0 . • It satisﬁes i S ω = 0 , where S is t he vertical endomorphism (equiv alently , ω ( S ( X ) , Y ) = ω ( S ( Y ) , X ) ). Proof. Let us introduce ﬁbered coordinates on Q , which w e denote by ( q i , t ) . Since ω is closed, it is locally exact so t hat ω = d  A j d q j + B j d ˙ q j + C d t  . (196) By computing i S ω L , and emplo ying t he same argument as in Theorem 3.11 , w e ma y reduce it to t he case where B j = 0 , in which case t he contraction reads as ∂ A j ∂ ˙ q i  d q i − ˙ q i d t  ∧ d q j + ∂ C ∂ ˙ q i  d q i − ˙ q i d t  ∧ d t . (197) F or this to v anish, two conditions need to hold: ∂ A j ∂ ˙ q i = ∂ A i ∂ ˙ q j and ∂ A j ∂ ˙ q i ˙ q i + ∂ C ∂ ˙ q j = 0 . (198) The ﬁrst condition implies t he existence of L such that A j = ∂ L ∂ ˙ q i . Rearr anging the second ter m, this implies that t here is a function F = F ( q , t ) such t hat C = L − ∂ L ∂ ˙ q i ˙ q i + F . By making t he change e L = L + F , w e hav e t hat ω = d ∂ e L ∂ ˙ q i d q i + e L − ∂ e L ∂ ˙ q i ˙ q i ! d t ! , (199) so t hat it is Lag rangian. N ecessity follo ws trivially . 37 Deﬁnition 4.12 ( Degenera te Non- a utonomous La grangian s ys tems ) . A non-aut onomous Lagr angian syst em ( Q , L ) is degenerate (or singular ) if the pair ( ω L , d t ) is pre-cosym plectic (i.e., t he Hessian matrix W ij is singular). A degener ate non-autonomous Lagrangian sys tem is precisel y a pre-cosymplectic Hamiltonian system ( J 1 π , ω L , d t ) , but one which carries t he "extr a" kinematic constr aint that its physical dynamics must be a SODE ﬁeld. As in the autonomous case, t his pre-cosymplectic system ( J 1 π , ω L , d t ) can be either inconsistent or consistent. Inconsistent non-autonomous Lagrangian systems. In the presence of an inconsistent system, w e ma y initiate t he analogue of t he constraint algorit hm in t he non-autonomous Lagrangian setting. This is obtained b y requiring, at each step, t he follo wing conditions. Set F 0 = J 1 π . Then, w e iterativ ely deﬁne F k +1 to t he set of points m ∈ M k (where F k is assumed to be a submanifold) such t hat there exists a tangent v ector X m ∈ T m F k satisfying • Reeb condition: i X m ω L = 0 and (d t )( X m ) = 1 . • SODE condition: S ( X m ) = 0 . As in t he autonomous case, if t he algorit hm con v erg es at a ﬁnite step, it ﬁnds a ﬁnal constr aint manifold F f where t he Lagrangian system becomes consistent, and which is, in g eneral, a subset of t he Hamiltonian one, F f ⊆ M f . No w , ( F f , ω L , d t ) ma y hav e gauge ambiguity , in which case w e can select a product structure adapted to V = ker ω L ∩ k er d t and apply the cosymplectic coisotropic embedding t heorem to remo v e the ambiguity in the equations. In general F f fails to be a jet manifold (or , more generall y , to ha v e a jet structure), so t hat there is little to no hope t hat t he regularization e F − → F f inherits a jet structure making t he regularized system Lagrangian. Remark 4.13. A similar discussion applies to that of Remark 3.16 . The geometric version of the constr aint algorithm for time-dependent Lagrangians was developed in [ 5 ], g eneralizing the procedur e pr esented, tog ether with t he SODE construction. The same relation applies: In gener al, S f , namely t he ﬁnal submanifold of the non-autonomous version of t he contraint algorit hm is a submanifold of F f . Consistent non-autonomous Lag rangian systems. When in t he presence of a consistent Lagrangian L ∈ C ∞ ( J 1 π ) , where π : Q − → R denotes the conﬁguration bundle, w e ha v e t he existence of global Reeb dynamics which are of second-order , namely w e hav e that t here exists X ∈ X ( J 1 π ) satisfying i X ω L = 0 , (d t )( X ) = 1 , and S ( X ) = 0 . (200) Ho w ev er , if { 0 }  = ker ω L ∩ ker d t , t he dynamics fails to be uniquely deﬁned, yielding gaug e ambiguities. After applying t he pre-cosymplectic coisotropic embedding t heorem, w e obtain a cosymplectic manifold ( f M , e ω , e τ ) in which ( J 1 π , ω , τ ) embeds as a coisotropic submanifold. Here w e ma y w onder whether f M admits a jet structure in such a wa y that t he cosymplectic manifold ( f M , e ω , e τ ) arises from an extended Lag rangian e L : f M − → R . (201) More particularl y , w e are looking for a • A ﬁber bundle e π : f Q − → R , tog ether wit h a ﬁber bundle embedding Q  → f Q . Denote by e S t he v ertical endomorphism on J 1 e π . 38 • A Lag rangian e L : f M − → R such t hat it restricts to the original Lagrangian L under t he previous inclusion. • A diﬀeomorphism α : f M − → J 1 e π in such a wa y that t he cosymplectic structure ( e ω , e τ ) is precisel y the cosymplectic structure structure obtained as follo ws e ω = d  e L d t + i α ∗ e S d e L  , τ = d t . (202) 4.4 Existence and uniqueness of Lag rangian regular ization The objectiv e of this section is to prov e the existence of a non-autonomous Lagrangian regularization, under some conditions on t he gauge ambiguities of L . W e also discuss the matter of uniqueness, and although global uniqueness is no t guar anteed, as a plethora of Lagrangians ma y be considered, w e prov e that any tangent structure on a particular cosymplectic regularization f M must be ‘isomor phic on M’ to t he one we build, giv en t hat t he Reeb v ector ﬁelds coincide. N amel y , as in the autonomous case, t he ﬁrst order germ of t he extension is unique. The main assumption that w e will make to endow the regularization wit h a Lagrangian structure is that t he characteristic distribution V = ker ω L ∩ ker d t on J 1 π is t he complete lif t of a v ertical (completel y integ rable) distribution on π : Q → R . This ma y be t hought of as a time-dependent generalization of t he case presented in Section 3 . That is, V = K f C . Let Q ha v e local ﬁbered coordinates ( t, x a , f A ) , where x a are coordinates on t he lea v es of t he regular foliation F K e induced by K f , and f A parameterize the ﬁbers (t he distribution K f ). The ﬁrst jet bundle J 1 π has natural coordinates ( t, x a , f A , ˙ x a , ˙ f A ) . U nder this h ypothesis, t he characteristic distribution V is locally spanned by : span    ∂ ∂ f A ! C = ∂ ∂ f A , ∂ ∂ f A ! V = ∂ ∂ ˙ f A    . (203) The cosymplectic t hickening ( f M , e ω , e τ ) is constructed as a neighborhood of t he zero section in t he dual bundle V ∗ → J 1 π , as described in Remar k 4.6 . As in t he autonomous case, w e assume t his thickening coincides with t he whole V ∗ b y assuming that an almost product structure P with a vanishing Nĳenhuis tensor can be chosen. On t he other hand, t he t hickened space f M = V ∗ can be identiﬁed as the cotang ent bundle of t he foliation F V gener ated by V : Proposition 4.14. The following canonical isomor phism exis ts: f M = V ∗ ≃ T ∗ F V := G F ∈F V T ∗ F , (204) wher e F deno tes a leaf of F V . Proof. The proof is completel y analogous to the symplectic case presented in Section 3.3 , taking into account that t he lea v es F are exactl y t he maximal integ ral manif olds of t he characteris tic distribution V . The coordinates of f M are ( t, x a , f A , ˙ x a , ˙ f A , µ f A , µ ˙ f A ) , where ( µ f A , µ ˙ f A ) are t he ﬁber coordinates dual to the kernel gener ators { ∂ ∂ f A , ∂ ∂ ˙ f A } . W e now deﬁne a new extended conﬁguration bundle e π : f Q → R . W e identify f Q as the cotang ent bundle of the f oliation F K e , denoted 39 f Q := T ∗ F K e ≡ K f ∗ . The manifold f Q has local coordinates ( t, x a , f A , µ A ) . The ﬁrst jet bundle of t his new space is J 1 e π , with local coordinates ( t, x a , f A , µ A , ˙ x a , ˙ f A , ˙ µ A ) . Remark 4.15. A gain, as pointed out in Remar k 3.19 , we can wor k with a almost product structur e wit hout vanishing Nĳenhuis tensor , simpl y r estricting to an open subset to obtain regularity . Proposition 4.16 ( Tulczyjew isomorphism for jets ) . Ther e exis ts a canonical isomor phism α t hat r elates J 1 e π to the t hickened space f M : α : J 1 e π → f M . (205) Proof. The proof is an immediate generalization of the t heory presented in Section 2.2 . Let j 1 s µ ∈ J 1 e π denote a jet, where µ : R → K f ∗ is a section, and X C ∈ X ( J 1 π ) denotes a v ector ﬁeld tangent to V = K f C , where X is a v ertical v ector ﬁeld taking v alues in K f . Then, it is enough to deﬁne t he pairing: ⟨ j 1 s µ, X C ⟩ := d d t      t = s ⟨ µ, X ⟩ . (206) In local coordinates, t he isomor phism reads exactly as in t he autonomous case ( 42 ) , simpl y carrying t he time coordinate t : α ( t, x a , f A , µ A , ˙ x a , ˙ f A , ˙ µ A ) = ( t, x a , f A , ˙ x a , ˙ f A , µ f A = ˙ µ A , µ ˙ f A = µ A ) . (207) Remark 4.17. As thor oughl y discussed in Section 3.3 for the autonomous scenario, the construction of t he r egularized 2-form e ω based solel y on t he choice of an almost-pr oduct structur e P on J 1 π cannot be adapted so that e ω is a Lagr angian form with respect to the canonical jet structur e on J 1 e π . There is a fundamental incompatibility between the s tandar d coisotr opic approac h and the SODE g eometr y , which carries over identically to this time-dependent setting. Therefore, to regularize t he system, w e will proceed exactl y as introduced in the pre-symplectic case. W e construct t he regularized cosymplectic 2-form e ω on J 1 e π b y adding a correction term that is Lag rangian b y construction. This term takes the form − dd e S F , where F ∈ C ∞ ( J 1 e π ) is a globall y deﬁned smooth function. W e t hen deﬁne t he extended Lag rangian as: e L = L + F , (208) so t hat w e obtain a global Lag rangian cosymplectic structure ( e ω = − dd e S e L, τ = d t ) . The deﬁnition of t he function F , exactl y as in the autonomous case, is not canonical and depends on t he choice of tw o speciﬁc ingredients: • An Ehresmann connection ∇ on the bundle K f ∗ − → Q (again giv en b y a splitting of t he tangent bundle in v ertical and horizontal v ectors T K f ∗ = V ⊕ H ∇ ) in such a w a y that the splitting at Q is t he canonical splitting T K f ∗ | Q = V ⊕ T Q . (209) • An almost product structure P on Q , which complements t he distribution K f . Remark 4.18 ( C oordin a te expressions ) . Locally , we expr ess t he components of t he connection as H ∇ = span ( ∂ ∂ t + Γ A ∂ ∂ µ A , ∂ ∂ q i + Γ iA ∂ ∂ µ A , ∂ ∂ f B + Γ B A ∂ ∂ µ A ) . (210) 40 The condition on ∇ inducing the canonical splitting at the zero section is reﬂect ed on the Γ ’ s vanishing at Q (ag ain identiﬁed via the zero section). And we express the components of the almost product structur e as P = P A ⊗ ∂ ∂ f A =  d f A − Q A d t − P A i d q i  ⊗ ∂ ∂ f A . (211) Generalizing the construction made in Section 3.3 , consider the follo wing diag ram ( K f C ) ∗ J 1 e π V ⊕ H ∇ = T K f ∗ J 1 π T Q V K f K f ∗ τ α i e π p V i π P , (212) where t he map τ : ( K f C ) ∗ − → J 1 π is t he canonical projection and the arrow V − → K f ∗ is t he usual identiﬁcation of t he v ertical bundle with t he ﬁber of a v ector bundle. Deﬁne the map F P, ∇ : J 1 e π − → R (213) b y F P, ∇ ( ξ ) = ⟨ ( P ◦ i π ◦ τ ◦ α )( ξ ) , ( p V ◦ i e π )( ξ ) ⟩ , (214) where ξ ∈ J 1 e π , and ⟨· , ·⟩ denotes t he natural pairing betw een K f and K f ∗ . Remark 4.19 ( Local expression of F P, ∇ ) . Using the coor dinate components of ∇ and P from Remar k 4.18 , we have t hat F P, ∇ =  ˙ µ A − Γ A − ˙ q i Γ iA − ˙ f B Γ B A  ·  ˙ f A − Q A − ˙ q j P A j  . (215) W e t hen hav e t he follo wing: Theorem 4.20 ( La grangian coisotr opic embedding ) . Let L : J 1 π − → R be a singular Lagrangian, wher e π : Q − → R denot es a conﬁgur ation bundle. Suppose t hat L is consis tent and that the char acteris tic distribution K = k er ω L ∩ ker d t is the complet e lift of a vertical distribution K f on Q . Then given an Ehresmann connection ∇ and an almost product structur e P as above t he embedding J 1 π  → ( K f C ) ∗ ∼ = J 1 e π (216) is a coisotr opic embedding on a neighborhood of J 1 π for t he cosymplectic structur e ( ω e L , d t ) , where e L = L + F P, ∇ . Proof. F ollow s wit h a similar discussion as in t he symplectic ase. W e will ﬁrst sho w t hat T ( J 1 e π )    J 1 π is a cosymplectic v ector bundle, so t hat t he pair ( e ω , d t ) deﬁnes a cosymplectic structure on some neighborhood of J 1 π . Indeed d e S e L =d S L + ˙ f A d µ A + ˙ µ A d f A (217) −  Γ iA ( ˙ f A − Q A − ˙ q j P A j ) + P A i ( ˙ µ A − Γ A − ˙ q i Γ iA − ˙ f B Γ B A )  d q i (218) −  Γ B A ( ˙ f A − Q A − ˙ q j P A j ) + Γ A + ˙ q i Γ iA + ˙ f B Γ B A  d f A (219) − ( Q A + ˙ q j P A j )d µ A . (220) 41 So t hat dd e S e L = ω L + d ˙ f A ∧ d µ A + d ˙ µ A ∧ d f A + (semi-basic terms) . (221) N otice t hat t he ﬁrst three terms on t he right hand side, tog ether wit h t he 1 -form d t , deﬁne a cosymplectic structure. Since adding semi-basic terms (wit h respect to t he projection onto Q ) does not change regularity , w e hav e t hat T ( J 1 e π )    J 1 π is a cosymplectic v ector bundle. Finall y , notice that it is a coisotropic embedding as, again, d e S L    J 1 π = d S L . (222) Hence, w e conclude t he discussion of t he existence of Lagrangian regularization in the cosymplectic scenario. Ag ain, w e ma y ask about its uniqueness. A similar discussion applies. Ho w ev er , w e encounter one natural obstruction to uniqueness, which is t he arbitrariness of the Reeb v ector ﬁeld on J 1 π . Indeed, recall ( Theorem 4.7 ) that uniqueness ’ on J 1 π ’ for t he coisotropic embedding is guaranteed provided the Reeb v ector ﬁeld is ﬁxed. N ev ert heless, once t his is taken into account, w e hav e t hat t he Lag rangian regularization is rigid to ﬁrst order : Theorem 4.21 ( Uniqueness of La grangian c oisotr opic embeddin g to first order ) . Let ( f M i , e ω i , τ i ) , be cosym plectic r egularizations of ( J 1 π , ω L , d t ) , with i = 1 , 2 . Suppose that f M i is endowed wit h a jet structur e ( e S i , τ i ) in such a way that i e S i e ω i = 0 , and the embedding J 1 π  → f M i (223) pr eserves the jet structur e. If t he induced Reeb vector ﬁelds (which ar e tang ent t o J 1 π ) coincide, ther e exis ts neighbor hoods U i of J 1 π in f M i and a diﬀeomorphism ψ : U 1 − → U 2 (224) t hat is t he identity on J 1 π such that t he induced map ψ ∗ : T f M 1    J 1 π − → T f M 2    J 1 π (225) pr eserves all tensor s, namely ψ ∗ ( e ω 1 , e τ 1 , e S 1 ) = ( e ω 2 , e τ 2 , e S 2 ) on J 1 π . Proof. Denote b y V = k er ω L ∩ ker d t the characteristic distribution. Let R denote t he induced Reeb v ector ﬁeld on J 1 π (b y an y of the embeddings), and let W be a distribution on J 1 π such that T J 1 π = V ⊕ span { R } ⊕ W and such t hat S ( W ) ⊆ W (this can be achiev ed b y taking a complete lif t). Then, ( W , ω L ) is a symplectic v ector bundle, and its ( e ω i , e τ i ) -orthogonal W ⊥ , e ω i , e τ i := { v ∈ T f M i : i v e τ i = 0 and e ω i ( v , W ) = 0 } (226) is as w ell and satisﬁes e S ( W ⊥ , e ω i , e τ i ) ⊆ W ⊥ . Indeed, the ﬁrst property is an elementary consequence of symplectic linear algebr a and t he second follo ws by the compatibility of e S with e ω i , as w e ha v e e ω i  e S ( W ⊥ , e ω i , e τ i ) , W  = e ω i ( e S ( W ) , W ⊥ , e ω i , e τ i ) = e ω i ( S ( W ) , W ⊥ , e ω i , e τ i ) = 0 , (227) since w e are requiring S ( W ) ⊆ W . 42 N otice t hat, b y construction, V ⊆ W ⊥ , e ω i and V is isotropic. Let us sho w that it is actuall y Lagragian, b y computing its dimension. Suppose dim J 1 π = 2( n + r ) + 1 , where 2 r is the rank of V . Then, since J 1 π  → f M i is coisotropic, 2 r = rank ( T ( J 1 π ) ⊥ , e ω 1 , e τ i ) = dim f M i − dim J 1 π , (228) w e conclude dim f M i = dim J 1 π + 2 r . N o w , rank W = 2 n , so that rank W ⊥ , e ω i , e τ i = 2 n + 4 r − 2 n = 4 r . Finall y , since rank V = 2 r , w e conclude t hat it is Lag rangian. By usual techniques (see Lemma 3.24 ), w e can build a v ector bundle symplect omor phism Φ : W ⊥ , e ω i , e τ i − → V ⊕ V ∗ , (229) so t hat w e obtain t he isomor phism T f M i    J 1 π = span { R } ⊕ W ⊕ W ⊥ , e ω i , e τ i ∼ = span { R } ⊕ W ⊕ V ⊕ V ∗ , (230) and such t hat, by deﬁning b S i = Φ ◦ e S i ◦ Φ , w e ha v e b S i ( V ∗ ) ⊆ V ∗ . The previous isomorphism of v ector bundles is clearl y t he identity on T J 1 π = span { R } ⊕ W ⊕ V . N ow , since span { R } ⊕ W ⊕ V ⊕ V ∗ = T V ∗ | J 1 π , (231) w e ma y choose a neighborhood V of J 1 π in V ∗ (identiﬁed as the zero section) and tw o diﬀeomorphisms ψ i : U i − → V , where U i are neighborhoods of J 1 π in f M i , and in such a w a y t hat it induces t he isomor phism abo v e. Clear ly , t he abo v e identiﬁcation makes ( ψ i ) ∗ an isomorphism of cosymplectic v ector bundles on ov er J 1 π . It only remains to show t hat it induces an isomor phism of jet structures as w ell. This, emplo ying the same technique as in Theorem 3.27 , namely w e will show that under t he previous construction, t here exists a unique tensor b S that satisﬁes t he following properties (all consequence of the construction presented): • It is a jet structur e : b S 2 = 0. • It makes ω Lagrangian : i S ω = 0 . • It extends the v ertical endomorphism of J 1 π , which w e denote by S . • It satiﬁes b S ( V ∗ ) ⊆ V ∗ . In particular ( ψ i ) ∗ e S i = b S , and the diﬀeomorphism ψ − 1 2 ◦ ψ 1 necessaril y preserv es the jet structure on J 1 π . Indeed, by taking adapted coordinates ( t, x a , f A , ˙ x a , ˙ f A ) on J 1 π and coordinates ( µ A , ˙ µ A ) in the ﬁbers of V ∗ , w e ha v e t hat the canonical form on V ∗ | J 1 π reads as ω = ω L + d µ A ∧ d f A + d ˙ µ A ∧ d ˙ f A − ( ˙ f A d µ A + Q A d ˙ µ A ) ∧ d t , (232) where R = ∂ ∂ t + ˙ x a ∂ ∂ x a + ˙ f A ∂ ∂ f A + R a ∂ ∂ ˙ x a + Q A ∂ ∂ ˙ f A is the chosen Reeb v ector ﬁeld. A gener al (1 , 1) tensor b S extending S = (d x a − ˙ x a d t ) ⊗ ∂ ∂ ˙ x a + (d f A − ˙ f A d t ) ⊗ ∂ ∂ ˙ f A takes t he follo wing 43 expression b S =  d x a − ˙ x a d t + F iA d µ A + ˙ F iA d ˙ µ A  ⊗ ∂ ∂ ˙ x a (233) +  d f A − ˙ f A d t + G AB d µ B + ˙ G AB d ˙ µ B  ⊗ ∂ ∂ ˙ f A (234) +  H B A d µ B + ˙ H B A d ˙ µ B + H A d t  ⊗ ∂ ∂ µ A (235) +  I B A d µ B + ˙ I B A d ˙ µ B + I A d t  ⊗ ∂ ∂ ˙ µ A . (236) It follo ws by similar com putations t hat t he three properties abo v e impl y b S = (d x a − ˙ x a d t ) ⊗ ∂ ∂ q i + (d f A − ˙ f A d t ) ⊗ ∂ ∂ ˙ f A − d ˙ µ A ⊗ ∂ ∂ µ A , (237) Finall y , deﬁning ψ := ψ − 1 2 ◦ ψ 1 , w e ha v e t hat it preserv es t he cosymplectic s tr ucture b y construction and t hat ψ ∗ ( e S 1 ) = ( ψ − 1 2 ) ∗  ( ψ 1 ) ∗ e S 1  = ( ψ − 1 2 ) ∗ b S = e S 2 , (238) which show s t hat it preser v es t he jet structure as w ell. 4.5 Exam ples 4.5.1 T r ivialized bundles Here w e deal with conﬁgurations bundles π : Q − → R which are trivialized, namely that w e choose a bundle isomor phism Q ∼ = Q × R − → R . Then, t his splitting induces a diﬀeomorphism J 1 π ∼ = T Q × R and, in particular , any Lag rangian L : J 1 π − → R simpl y reads as a time- dependent Lag rangian L : T Q × R − → R . (239) W e s tudy in t his section how the (Lag rangian) regularization procedure beha v es in the trivialized case. Firs t notice that, b y deﬁnition, v ertical and complete lifts correspond to v ertical and complete lifts on T Q after trivializing. The P oincaré–Cartan form still reads (in natural coordinates ( q i , ˙ q i , t ) ) as θ L = L − ∂ L ∂ ˙ q i ˙ q i ! d t + ∂ L ∂ ˙ q i d q . (240) Remark 4.22 ( Hypothesis on the chara cteristic distribution ) . Recall that to obatain a Lagrangian r egularization, we imposed the condition on the char acteris tic distribution K := ker ω L ∩ ker d t (241) to be the complet e lif t of a distribution vertical distribution on Q − → R . This may be stat ed using t he trivialized bundle as follows. F irst notice that t he char acteris tic distribution K on the trivialized bundle T Q × R − → R may be t hought of as time-dependent complet ely integr able distribution on T Q . Denot e by K t t he distribution at time t . Then, the condition on K to be the complete lif t of a vertical distribution K f t on Q − → R tr anslates to K t being a complet e lif t of a complet ely integr able distribution K f t , for every t . Incidentall y , K f is simpl y the g luing of all K f t . 44 N ow , if w e are in t he case abov e, w e ma y w onder whet her on t he bundle K f ∗ − → R w e ha v e a natural trivialization. This w ould be t he case if the ‘time dependent’ distribution K f t is constant but, other wise, w ould fail to hold. W e could also ask whether , although K f t is not constant, t hey are all isomor phic, in t he sense that t here is a smooth famil y of diﬀeomorphisms ψ t : Q − → Q , ψ 0 = id Q (242) such that K f t = ( ψ t ) ∗ ( K 0 ) . This, again, does not hold in gener al, as t he follo wing example sho ws: Exam ple 4.23 ( La grangian on trivial bundle with non-conts t ant chara cteristic distribu- tion ) . The following example, albeit artiﬁcal, shows the exis tence of time-dependent Lagr angians on a trivialized bundle L : T Q × R − → R (243) such that V = ker ω L ∩ ker d t = ( K f ) C , for certain distribution K f on Q × R which is not constant . Her e, “not constant ” means t hat ther e is not a time dependent famil y of diﬀeomorphisms ψ t : Q − → Q such that ψ ∗ t ( K f | Q ×{ t } ) = K | Q ×{ 0 } , essentially for cing the jet bundle point of view present ed. On S 1 , let exp : R − → S 1 , t 7→ (cos t, sin t ) (244) denot e the exponential and deﬁne I t to be the image of the interv al h − t 2 1+ t 2 , t 2 1+ t 2 i under exp , namely I t := exp " − t 2 1 + t 2 , t 2 1 + t 2 #! . (245) W e clear ly hav e that I t is diﬀeomorphic to the closed interv al [0 , 1] for t  = 0 and onl y a point for t = 0 . Deﬁne Q t := ( S 1 × S 1 ) \ ( I t × { p } ) , (246) for a ﬁxed point p ∈ S 1 . S tandar d tec hniques of diﬀer ential topology [ 35 ] show that ther e is a famil y of embeddings ψ t : ( S 1 × S 1 ) \ ( { p } × { p } ) − → S 1 × S 1 , (247) t hat varies smoot hl y wit h t ∈ R and such that it deﬁnes a diﬀeomorphism with Q t , for each t . Deﬁne t he following Lagrangian e L : T  S 1 × S 1  − → R , e L ( θ 1 , θ 2 , ˙ θ 1 , ˙ θ 2 ) = ( ˙ θ 1 ) 2 2 , (248) wher e ( θ 1 , θ 2 ) denot e angular (local) coordinat es on the torus, and ( ˙ θ 1 , ˙ θ 2 ) denot e the induced global coor dinates on T ( S 1 × S 1 ) . Let Q := ( S 1 × S 1 ) \ ( { p } × { p } ) and deﬁne t he following time-dependent Lagr angian: L : T Q × R − → R , L ( v , t ) := e L (( ψ t ) ∗ v ) . (249) Locall y , after a chang e of coordinat es, the previous Lagrangian is pr ecisely e L , but not globall y. Indeed, in t = 0 , the char acteris tic distribution of L has precisel y one non-compact leaf. This no long er holds for t  = 0 , which shows that the char acteris tic distribution K f cannot be made const ant after a g lobal time dependent famil y of diﬀeomorphisms. 45 4.5.2 Degenerate metr ics Let us deal wit h t he example of degener ate metrics. A canonical exam ple of an autonomous degener ate Lagrangian is that of a degenerate metric, namel y a symmetric and positiv e semideﬁnite tensor g on a manifold M . Giv en such a tensor , we ma y study its kinetic energy L : T M − → R , L ( v ) := 1 2 g ( v , v ) . (250) In t he autonomous realm w e can gener alize this in tw o w a ys: • A degener ate metric on a manifold M , which is time dependent, sa y g t for t ∈ R and study its time-dependent energy L : T M × R − → R , L ( v , t ) := 1 2 g t ( v , v ) . (251) • Or , giv en a ﬁber bundle π : Q − → R , to w ork with a (possibl y degener ate) metric g on Q , together with t he energy L : J 1 π − → R , L ( j 1 t γ ) := g γ ∗ ∂ ∂ t , γ ∗ ∂ ∂ t ! . (252) The latter has t he advantag e of including t he ﬁrst as a particular case and, also, allowing for potentials. Indeed, in g eneral, and for ﬁbered coordinates ( q i , t ) on Q , w e hav e g := g ij d q i d q j + 2 A i d t d q i − 2 V d t 2 . (253) Then, wit h natural coordinates ( q i , ˙ q i , t ) on J 1 π , the Lag rangian reads as L = 1 2 g ij ˙ q i ˙ q j + A i ˙ q i − V . (254) This Lag rangian yields the equations for t he mov ement of a charg ed (with charg e 1 ) particle on Q (the standard ﬁber) under an electric potential V and magnetic potential A i d q i in a cur v ed (b y t he metric g ij ) space. Let us study its regularity . Firs t, notice that its P oincaré–Cartan form is θ L = L d t + d S L =  1 2 g ij ˙ q i ˙ q j + V  d t + ( g ij ˙ q i + A i )d q i , (255) so t hat ω L = − d θ L (256) = g ij d q i ∧ d ˙ q j − ∂ g ij ∂ q k ˙ q j d q k ∧ d q i − ∂ A i ∂ q j d q j ∧ d q i (257) − d  1 2 g ij ˙ q j ˙ q j + V  + ∂ g ij ∂ t ˙ q i + ∂ A i ∂ t ! d q i ! ∧ d t (258) Hence, it is immediate to see that Proposition 4.24. Let Q t = π − 1 ( t ) denot e the ﬁber of π : Q − → R and let g t denot e the res triction of g to Q t . Then, the Lagrangian L is regular if and onl y if eac h metric g t is non-degener ate. 46 N ow , let us study t he consistency conditions when g t is not deﬁnite positiv e (so that ( ω L , d t ) is no longer a cosymplectic manifold). Here, letting X denote a SODE ﬁeld X = ∂ ∂ t + ˙ q i ∂ ∂ q i + X i ∂ ∂ ˙ q i (259) and imposing i X ω L = 0 w e g et t he equations − g ij X j = 1 2 ∂ g ij ∂ q k + ∂ g kj ∂ q i − ∂ g ki ∂ q j ! ˙ q j ˙ q k + ∂ A i ∂ q j − ∂ A j ∂ q i + ∂ g ij ∂ t ! ˙ q j (260) + ∂ V ∂ q i + ∂ A i ∂ t . (261) By applying t he constraint algorit hm, if the Lagrangian is degenerate (hence t he metric), w e ma y contract on both sides b y a v ector W = W i ∂ ∂ q i taking values in t he characteris tic distribution C t = { w ∈ T Q t : i w g t = 0 } (262) to get t he follo wing consistency conditions: 0 = w i 2 ∂ g ij ∂ q k + ∂ g kj ∂ q i − ∂ g ki ∂ q j ! ˙ q j ˙ q k + w i ∂ A i ∂ q j − ∂ A j ∂ q i + ∂ g ij ∂ t ! ˙ q j (263) + w i ∂ V ∂ q i + ∂ A i ∂ t ! . (264) This, in gener al, imposes new conditions that w e need to in v estig ate furt her . N ev ertheless, it giv es us suﬃcient and necessary conditions for L to be consistent. Indeed, if t he abov e equations are trivially satisﬁed, b y taking derivativ es with respect to ˙ q i tw o times w e obtain the follo wing conditions on t he metric: w i ∂ g ij ∂ q k + ∂ g kj ∂ q i − ∂ g ki ∂ q j ! = 0 (265) w i ∂ A i ∂ q j − ∂ A j ∂ q i + ∂ g ij ∂ t ! = 0 (266) w i ∂ V ∂ q i + ∂ A i ∂ t ! = 0 , (267) for ev er y W = w i ∂ ∂ q i taking values in the characteristic distribution. Remark 4.25 ( Geometric conditions for L to be consis tent ) . Suppose that we choose a trivializa- tion Q = Q × R . Then, the metric g is speciﬁed by a choice of • A time dependent metric on Q , which we denot e by g . • A time dependent 1 -form on Q , which we denote by A . • A time dependent potential V on Q , which we denote by V . Denot e by C t t he char acteris tic distribution on Q , for every t . Then, the previous conditions read as £ W g = 0 , (268) i W ∂ g ∂ t − d A ! = 0 , (269) i W d V + ∂ A ∂ t ! = 0 , (270) for all W t aking values in the char acteris tic distribution. 47 T o deal with t he issue of Lagrangian regularization, as discussed, w e need to f ocus on consistent Lag rangians, so that henceforth w e assume t he previous conditions to hold. Then, w e need to study when t he characteristic distribution V = k er ω L ∩ ker d t is t he complete lif t of a v ertical integ rable distribution on Q . If it w ere t he case, it is clear that t he choice of t he v ertical distribution on Q w ould be t he disjoint union of the characteristic distributions of each ( Q t , g t ) K f = G t ∈ R C t , (271) as w e would need K f v to be t he v ertical elements in V . Hence, w e onl y need to ﬁnd conditions on C t that, together wit h t he abo v e consistency conditions, guarantee that V = K f C . Proposition 4.26. If the Lagr angian deﬁned by the metric g is consist ent, we hav e t hat V = K f C if and only if C t is integr able and the 1 -form A satisﬁes i C t d A = 0 . Proof. If each distribution C t (and hence K f ) is integ rable, w e can ﬁnd adapted coordinates ( x a , f A , t ) on Q in such a wa y that C t = span ( ∂ ∂ f A ) (272) By deﬁnition, since C t is t he characteris tic distribution, the metric g reads as g = g ab d x a d x b + 2 A d t − 2 V d t 2 , (273) where A denotes a 1 -form on each ﬁber . Since L is consistent (b y appl ying Eq. (265) ), w e hav e that g ab onl y depends on x a , so t hat its P oincaré–Cartan form reads as θ L =  1 2 g ab ˙ x a ˙ x b + V  d t + g ab ˙ x a d x b + A . (274) Hence, ω L = { terms t hat only depend on x a } − d A − ∂ A ∂ t ∧ d t − d V ∧ d t (275) Contracting b y an arbitrary element in K f C = span ( ∂ ∂ f A , ∂ ∂ ˙ f A ) (276) and taking into account t he compatibility condition of Eq. Eq. (267) w e conclude t he result. Remark 4.27. A diﬀer ent, but equivalent way of st ating that C t is an integr able distribution and t hat £ W g t = 0 , if W ∈ Γ( C t ) , is to requir e the exist ence of an adapt ed, tor sionless connection ∇ t to the metric g t , for every t . If such a connection can be chosen for every t , it is not so complicat ed to show t hat it can be chosen so t hat ∇ t varies smoot hl y . N ow , under t he conditions of the abov e result, as w e sho w ed, the regularization procedure reco v ers a Lagrangian system. Here, w e ma y emplo y the existence of an adapted connection to build the connection on the bundle K f ∗ = span ( ∂ ∂ µ A ) − → Q , (277) but not so much wit h the product structure. In local, adapted coordinates to t he characteristic distribution C , an arbitrary connection ∇ (identiﬁed wit h its linear splitting of the tangent bundle T M ) takes the follo wing local expression T M = span ( ∂ ∂ ˙ x a , ∂ ∂ ˙ f A ) ⊕ H ∇ , (278) 48 where H ∇ = span ( ∂ ∂ x a + Γ c ab ˙ x b ∂ ∂ ˙ x c +  Γ A ab ˙ x b + Γ A aB ˙ f B  ∂ ∂ ˙ f A , ∂ ∂ f A +  Γ C AB ˙ f B + Γ C Aa ˙ x a  ∂ ∂ ˙ f C ) . (279) Since ∇ t ( g t ) = 0 , w e also ha v e ∇ t (Γ( C t )) ⊆ Γ( C t ) , so that, in particular , there is an induced linear connection on the bundle K f − → Q . In particular , it induces a dual linear connection on K f ∗ , which we denote by ∇ ∗ . Identiﬁed as a spitting of the tangent bundle, it reads as follo ws: T K f ∗ = V ⊕ span ( ∂ ∂ x a − Γ C aB µ C ∂ ∂ µ B , ∂ ∂ f A − Γ C AB µ A ∂ ∂ µ C ) . (280) Finall y , giv en an almost product structure on Q , sa y T Q = span ( ∂ ∂ f A ) ⊕ span ( ∂ ∂ t + Q A ∂ ∂ f A , ∂ ∂ x a + P A a ∂ ∂ f A ) , (281) w e obtain t he regularized Lag rangian e L = 1 2 g ab ˙ x a ˙ x b + A a ˙ x a + A A ˙ f A − V +  ˙ f A − Q A − P A a ˙ x a   ˙ µ A + ˙ x a µ B Γ B aA + ˙ f B µ C Γ C B A  (282) 5 Conclusions and fur t her w ork In this paper , w e ha v e dev eloped a met hod for regularizing singular time-dependent La- grangian systems. T o do so, w e ﬁrst anal ysed in detail the met hod dev eloped b y A. Ibort and J. Marín-Solano [ 27 ] for singular time-independent Lagrangian systems, impro ving some of their results, namely explicitly constructing a global regular Lag rangian with t he auxiliar y help of a connection rather t han a Riemannian metric in Theorem 3.23 and proving that t he embedding is unique to ﬁrst order in Theorem 3.27 . Since the basis of t he construction was the coisotropic embedding theorem in pre-symplectic manifolds, our method has been based on the coisotropic embedding for pre-cosymplectic manifolds, which yields t he analogue of t he previous results, namely Theorem 3.23 and Theorem 4.7 , respectiv ely . The other key ingredients ha v e been the use of almost-product structures adapted to t he singularity of t he Lagrangian, which facilitates t he use of t he constr aints algorit hm for singular Lag rangians; and the use of T ulczyjew triples adapted to a foliation, which allow s for t he regularization to inherit a natural tangent (or jet) structure. This paper opens some new and interesting research lines that w e aim to discuss in coming papers: • Extend t he regularization problem for singular contact systems; a constraint algorithm has been recently dev eloped in [ 10 ]. • Extend the geometric approach to the inv erse problem for implicit equations of [ 40 , 41 ] to time-dependent implicit diﬀerential equations. • A main issue is t he extension of the regularization problem for singular classical ﬁeld theories, due to t he complexity of multisymplectic geometry . In such a case, w e should dev elop a co v ariant t heory on the space of solutions [ 6 , 7 , 14 ]. • Other interesting research pur poses are t he follo wing ones: the case when w e ha v e symmetries for t he Lagrangian function, or the regularization of t he Hamilton–Jacobi equation [ 13 ], and furt hermore, the discretization of the original singular Lag rangian and its relation wit h t he regularized one. 49 A cknow ledgements W e ackno w ledg e ﬁnancial support of the Minist erio de Ciencia, Innovación y Univ ersidades (Spain), grant PID2022-125515NB-C21; w e also ackno w ledge ﬁnancial support from t he Sev ero Ochoa Prog ramme for Centers of Excellence in R&D and Grant CEX2023-001347-S funded by MICIU/AEI/10.13039/501100011033. P ablo Soto also acknow ledges a J AE-Intro scholarship for undergraduate students. R ubén Izquierdo-López wishes to thank the Spanish Ministry of Science, Innov ation and U niv ersities for t he contract FPU/02636. Ref erences [1] R. Abraham and J. E. 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