On a unified formulation of completely integrable systems
The purpose of this article is to show that a $\mathcal{C}^1$ differential system on $\R^n$ which admits a set of $n-1$ independent $\mathcal{C}^2$ conservation laws defined on an open subset $\Omega\subseteq \R^n$, is essentially $\mathcal{C}^1$ equ…
Authors: Ru{a}zvan M. Tudoran
On a unified form ulation of completel y in te- grable systems R˘ azv an M. T udoran Abstract. The purp ose of this article is to show that a C 1 differential system on R n whic h admits a set of n − 1 in depend en t C 2 conserv ation la ws defined on an open subset Ω ⊆ R n , is essen tially C 1 equiv alent on an open and dense subset of Ω, with the linear differen tial system u ′ 1 = u 1 , u ′ 2 = u 2 , . . . , u ′ n = u n . The main results are illustrated in th e case of tw o concrete dynamical systems, namely the three dimensional Lotk a-V olterra system, and respectively the Euler equations from the free rigid b ody dynamics. Mathematics Sub ject Classification (2000). 37J35; 37K10; 70H05; 70H06. Keywords. integrable systems; Hamiltonian dynamics; linear normal forms. 1. Intro duction Recently , in [ 6 ] it is pr o ved that an in tegra ble C 1 planar diff erential system is roughly sp eaking C 1 equiv ale nt to the linea r differential sy s tem u ′ 1 = u 1 , u ′ 2 = u 2 . The pur p ose of this article is to g eneralize this result in the n dimensional case for a C 1 differential sy stem that admits a set of n − 1 indep enden t conser v ation laws. In the seco nd section we show that such a sys tem can alwa ys b e realized as a Hamilton-Poisson dynamical s ystem on a full measure op en subset of R n with re- sp ect to a r ank 2 Poisson structure. In the third sectio n a new time tr ansformation will b e explicitly constructed in order to bring the system to a linear different ial system o f the type u ′ 1 = u 1 , u ′ 2 = u 2 , . . . , u ′ n = u n . In the last section w e illustr ate the main results in the case of tw o concrete dynamical systems, namely the three dimensional Lotk a-V olterr a system, and res p ectively the Euler equations from the free rigid b ody dynamics. F or details on Poisson geometry and Hamiltonian dynamics, se e , e.g. [ 1 ], [ 2 ], [ 11 ], [ 8 ], [ 9 ], [ 10 ], [ 1 2 ]. 2 R˘ azv a n M. T udoran 2. Hamilto nian div ergence free v ector fields naturally asso ciated t o in tegrable systems In this s ection we give a metho d to co nstruct a Hamilton-Poisson divergence free vector field, naturally a ssoc iated with a given Hamilton-Poisson realiza tio n of a n dimensional differential system a dmitting n − 1 indep enden t conserv ation laws. First step in this approach is to construct a Hamilton-Poisson realizatio n of a given n dimensiona l differen tial system admitting n − 1 independent in tegra ls o f motion. Let us consider a C 1 differential system on R n : ˙ x 1 = X 1 ( x 1 , . . . , x n ) ˙ x 2 = X 2 ( x 1 , . . . , x n ) · · · ˙ x n = X n ( x 1 , . . . , x n ) , (2.1) where X 1 , X 2 , . . . , X n ∈ C 1 ( R n , R ) are arbitrary real functions. Suppose tha t C 1 , . . . , C n − 2 , C n − 1 : Ω ⊆ R n → R a re n − 1 indep endent C 2 int egr als of mo- tion of ( 2.1 ) defined on a nonempty o p en subset Ω ⊆ R n . Since C 1 , . . . , C n − 2 , C n − 1 : Ω ⊆ R n → R ar e integrals of motion of the v ector field X = X 1 ∂ x 1 + · · · + X n ∂ x n ∈ X ( R n ), w e obtain tha t for each i ∈ { 1 , . . . , n − 1 } h∇ C i ( x ) , X ( x ) i = n X j =1 ∂ x j C i · ˙ x j = 0 , for ev ery x = ( x 1 , . . . , x n ) ∈ Ω, where h· , ·i stand for the canonical inner pro duct on R n , and res pectively ∇ stand for the gradient with r espect to h· , ·i . Hence, by a standa rd m ultilinear algebra argument, the C 1 vector field X is given as the C 1 vector field ⋆ ( ∇ C 1 ∧ · · · ∧ ∇ C n − 1 ) multiplied b y a C 1 real function (rescaling function), where ⋆ sta nd for the Ho dge star op erato r for multiv ector fields (see fo r details e.g. [ 5 ]). It may happen that the domain of definition for the rescaling function to be a pr oper subset of Ω. In the following w e will consider the generic case when the r escaling function is defined on an op en a nd dense subset of Ω. In order to simplify the no tations, we will a lso deno te this set by Ω. Consequently , the vector field X can b e realized on the op en set Ω ⊆ R n as the Hamilton-Poisson vector field X H ∈ X (Ω) with r espect to the Hamiltonia n function H := C n − 1 and r espectively the Poisson brack et o f class C 1 defined by: { f , g } ν ; C 1 ,...,C n − 2 dx 1 ∧ · · · ∧ dx n = ν dC 1 ∧ . . . dC n − 2 ∧ d f ∧ dg , where ν ∈ C 1 (Ω , R ) is a given real function (rescaling). F or ν ≡ 1, the asso ciated Poisson br a c ket in the smo oth category , it is exactly the Flaschk a-Rat ¸ iu bracket . F or similar Hamilton-Poisson formulations of completely integrable systems see also [ 3 ], [ 7 ]. In co ordinates, the brack et { f , g } ν ; C 1 ,...,C n − 2 is given by: { f , g } ν ; C 1 ,...,C n − 2 = ν · ∂ ( C 1 , . . . , C n − 2 , f , g ) ∂ ( x 1 , . . . , x n ) . A unified formulation of completely integrable systems 3 Note that { C 1 , . . . , C n − 2 } is a complete set of Ca s imirs for the Poisson br a c ket {· , ·} ν ; C 1 ,...,C n − 2 . Recall tha t the Hamiltonia n vector field X H ∈ X (Ω) is acting on an arbitrary real function f ∈ C k (Ω , R ), k ≥ 2 as: X H ( f ) = { f , H } ν ; C 1 ,...,C n − 2 ∈ C 1 (Ω , R ) . Hence, the differential system ( 2.1 ) can be locally wr itten in Ω a s a Hamilton- Poisson dynamical sys tem of the type: ˙ x 1 = { x 1 , H } ν ; C 1 ,...,C n − 2 ˙ x 2 = { x 2 , H } ν ; C 1 ,...,C n − 2 · · · ˙ x n = { x n , H } ν ; C 1 ,...,C n − 2 , or equiv a len tly ˙ x 1 = ν · ∂ ( C 1 , . . . , C n − 2 , x 1 , H ) ∂ ( x 1 , . . . , x n ) ˙ x 2 = ν · ∂ ( C 1 , . . . , C n − 2 , x 2 , H ) ∂ ( x 1 , . . . , x n ) · · · ˙ x n = ν · ∂ ( C 1 , . . . , C n − 2 , x n , H ) ∂ ( x 1 , . . . , x n ) . (2.2) Consequently , the compo nen ts of the vector field X = X 1 ∂ x 1 + · · · + X n ∂ x n which genera tes the differential system ( 2.1 ), ar e given in Ω as follows: X i = ν · ∂ ( C 1 , . . . , C n − 2 , x i , H ) ∂ ( x 1 , . . . , x n ) , for i ∈ { 1 , . . . , n } . Next r esult gives a metho d to co nstruct a divergence free vector field out of the v ector field X . The divergence op erator w e will use in this approach is the div ergence a s socia ted with the standard Lebesgue measure on R n , na mely L X dx 1 ∧ · · · ∧ dx n = div X dx 1 ∧ · · · ∧ dx n , wher e L X stand fo r the Lie deriv ative along the v ector fie ld X . Theorem 2 .1 . The ve ctor field e X := 1 ν · X is a diver genc e f r e e ve ctor field on Ω \ Z ( ν ) , wher e Z ( ν ) = { ( x 1 , . . . , x n ) ∈ Ω | ν ( x 1 , . . . , x n ) = 0 } . Pr o of. Note that the comp onents of the vector field e X = e X 1 ∂ x 1 + · · · + e X n ∂ x n are given by: e X i = ∂ ( C 1 , . . . , C n − 2 , x i , H ) ∂ ( x 1 , . . . , x n ) , for i ∈ { 1 , . . . , n } . By definition, the vector field e X is a Hamilton-Poisson vector field with resp ect to the Flas chk a- Ra t ¸iu brac ket, and ha ving the same Hamiltonian H as the vector field X . 4 R˘ azv a n M. T udoran Hence, the divergence of e X is given by: div( e X ) = n X i =1 ∂ x i e X i = n X i =1 ∂ x i ∂ ( C 1 , . . . , C n − 2 , x i , H ) ∂ ( x 1 , . . . , x n ) = n X i =1 ( − 1) i + n − 1 ∂ x i ∂ ( C 1 , . . . , C n − 2 , H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) , where the notation ” b x i ” means that ” x i ” is o mitted. Let us now analyze the gener al term in the ab o ve sum. By us ing the deriv ative of a determinant we obtain the following: ∂ x i ∂ ( C 1 , . . . , C n − 2 , H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) = ∂ ( ∂ x i C 1 , . . . , C n − 2 , H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) + · · · + ∂ ( C 1 , . . . , ∂ x i C n − 2 , H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) + ∂ ( C 1 , . . . , C n − 2 , ∂ x i H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) . Hence, div( e X ) = n X i =1 ( − 1) i + n − 1 ( ∂ ( ∂ x i C 1 , . . . , C n − 2 , H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) + · · · + ∂ ( C 1 , . . . , ∂ x i C n − 2 , H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) + ∂ ( C 1 , . . . , C n − 2 , ∂ x i H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) ) = n X i =1 ( − 1) i + n − 1 ∂ ( ∂ x i C 1 , . . . , C n − 2 , H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) + · · · + n X i =1 ( − 1) i + n − 1 ∂ ( C 1 , . . . , ∂ x i C n − 2 , H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) + n X i =1 ( − 1) i + n − 1 ∂ ( C 1 , . . . , C n − 2 , ∂ x i H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) . Next we pr ove that each of the ab ov e sums v a nishes. In or de r to do that, it is enough to show that the ge ne r al sum S k v anishes , wher e S k := n X i =1 ( − 1) i + n − 1 ∂ ( C 1 , . . . , C k − 1 , ∂ x i C k , C k +1 , . . . , C n − 2 , H ) ∂ ( x 1 , . . . , b x i , . . . , x n ) . A unified formulation of completely integrable systems 5 Indeed, we obtain that: S k = n X i =1 ( − 1) i + n − 1 i − 1 X j =1 ( − 1) j + k ∂ 2 x j x i C k · ∂ ( C 1 , . . . , b C k , . . . , C n − 2 , H ) ∂ ( x 1 , . . . , b x j , . . . , b x i , . . . , x n ) + n X i =1 ( − 1) i + n − 1 n X j = i +1 ( − 1) j + k − 1 ∂ 2 x j x i C k · ∂ ( C 1 , . . . , b C k , . . . , C n − 2 , H ) ∂ ( x 1 , . . . , b x i , . . . , b x j , . . . , x n ) = n X i =1 ( − 1) i + n − 1 i − 1 X j =1 ( − 1) j + k ∂ 2 x j x i C k · ∂ ( C 1 , . . . , b C k , . . . , C n − 2 , H ) ∂ ( x 1 , . . . , b x j , . . . , b x i , . . . , x n ) + n X j =1 ( − 1) j + n − 1 n X i = j + 1 ( − 1) i + k − 1 ∂ 2 x i x j C k · ∂ ( C 1 , . . . , b C k , . . . , C n − 2 , H ) ∂ ( x 1 , . . . , b x j , . . . , b x i , . . . , x n ) = ( − 1 ) n + k − 1 X 1 ≤ j
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