On two Abelian Groups Related to the Galois Top

ON TW O ABELIAN GR OUPS RELA TED TO THE GALOIS TOP HELMUT RUHLAND Abstract. In mathematical physics the Galois top, introduced by S. Adla j, possesses a fixed p oint on one of tw o Galois axes through its center of mass. This heavy top has tw o algebraic motion in v arian ts and an additional tran- scendental motion-inv ariant. This third inv ariant depends on an an tiderivativ e of a v ariable in the canonical phase space. In this article an ab elian semigroup and an abelian group are defined that are related to the application of the Huygens-Steiner theorem to p oints on the Galois axis of a rigid bo dy . 1. Introduction The author of [ 1 ] in tro duced the Galois top. The fixed p oin t O of this top is lo cated on one of the tw o so-called Galois axes through the cen ter of mass G of the rigid b ody . A Galois axis can b e in terpreted geometrically in this manner: it passes through G and is orthogonal to a plane that intersects the MacCullagh ellipsio d cen tered at G in a circle. Besides the t w o motion in v arian ts-energy K and angular momen tum pro jected on to the v ector of gravit y L g (ev ery hea vy top has these tw o in v ariants)-the Galois top has a third tr ansc endental constant of motion. It is widely b elieve d that suc h a transcenden tal (dep ending on an an tideriv ative of a v ariable in the canonical phase space) would not exist ; for the inv ariant see formula (3) in [ 1 ]. The application of the Huygens-Steiner theorem to p oin ts O on the Galois axis defines maps from the principal moments of inertia in the center of mass G to the principal momen ts of inertia in the p oin t O . In the next section, it is s ho wn that these maps generate an abelian semigroup. In section 3 an ab elian group is described, also generated b y these maps. 2. An abelian semigroup of iner tia maps Let M ⊂ R 3 represen t the three (ordered with resp ect to < ) principal moments of inertia of a rigid b o dy , M = { ( A, B , C ) ∈ R 3 | 0 < A < B < C } . The condition 0 < A in the definition ensures, that the 3 principal momen ts are ph ysically mean- ingful. Then the 2-v alued (2 Galois axes exist) tensor of inertia J ( d ), defined in the line b elo w form ula (2) in [ 1 ], defines maps acting on M : Definition 1. F or x ∈ R and x ≥ 0 this is a one-parameter family of maps: j ( x ) : M → M ( A, B , C ) 7→ ( λ 1 , λ 2 , λ 3 ) (2.1) Date : March 26, 2026. 2020 Mathematics Subje ct Classific ation. Primary 20M14; Secondary 20M20. Key words and phr ases. Galois top, Galois axis, MacCullagh ellipsio d, abelian semigroup. 1 2 H. R UHLAND 0 < λ 1 < λ 2 = B + x < λ 3 are the 3 ordered eigen v alues of J ( d ), d 2 replaced b y x . These maps dep end only on x , not on whic h of the tw o Galois axes is chosen. In app endix A , it is sho wn that the condition x ≥ 0 is necessary and sufficien t. Theorem 2.1. The maps j ( d 2 ) , wher e d is the distanc e of a p oint O on the Galois axis to the c enter of mass G , define an ab elian semigr oup S + = { j ( d 2 ) | d ∈ R } = { j ( x ) | x ∈ R + } . j (0) is the neutr al element. The semigr oup law is j ( x ) ◦ j ( y ) = j ( x + y ) . F or a pr o of, se e app endix B . 3. An abelian group of maps Starting with the semigroup S + defined in theorem 2.1 , w e could try get an ab elian group G = { j ( x ) | x ∈ R } adding the inv erses of all elements in G + . The in verse of j ( x ) would b e j ( − x ). But j ( x ) is only defined for x ≥ 0 and not for x ∈ R , as it w ould be necessary to get a group G . But we can define other maps with C 3 as domain/co domain. Based on these maps, w e get an ab elian group. In con trary to the semigroup S + ab o ve, this group has no longer a reference to physical tops. The tensor of inertia J ( d ), defined in the line b elow form ula (2) in [ 1 ], now defines maps acting on C 3 : Definition 2. F or x ∈ C this is a one-parameter family of maps: j ( x ) : C 3 → C 3 ( A, B , C ) 7→ ( λ 1 / 3 , λ 2 , λ 3 / 1 ) (3.1) λ 1 , λ 2 = B + x, λ 3 are the three eigen v alues of J ( d ), d 2 replaced b y x . These maps are 2-v alued/sheeted after analytic con tinuation. An inv olution i lets suc h a 2-v alued map in v ariant, it in terchanges the tw o sheets or interc hanges the comp onen ts A, C of the domain: i : C 3 → C 3 , ( A, B , C ) 7→ ( C , B , A ) (3.2) i ◦ j ( x ) = j ( x ) the 2 sheets are interc hanged, i.e. ( λ 3 , λ 2 , λ 1 ) = i ( λ 1 , λ 2 , λ 3 ). j ( x ) ◦ i = j ( x ) A, B are interc hanged, i.e. ( C , B , A ) = i ( A, B , C ). Theorem 3.1. The maps j ( x ) define an ab elian gr oup G = { j ( x ) | x ∈ C } . j (0) is the neutr al element. The inverse of j ( x ) is j ( − x ) . The gr oup law is j ( x ) ◦ j ( y ) = j ( x + y ) . 4. An open question Ma yb e it can b e shown that arbitrary axes through the center of mass G , except the t wo Galois axes, do not allo w the definition of such abelian semi(groups)? Besides the characterization of the t wo Galois axes using sections of the Mac- Cullagh ellipsiod, this would allow us to characterize the Galois axes as the only axes with assigned (semi)groups. App endices ON TWO ABELIAN GROUPS RELA TED TO THE GALOIS TOP 3 Appendix A. Proof tha t the ima ge of j ( x ) is a subset of the codomain It m ust to b e pro v en: { j ( x ) m | m ∈ M } ⊆ M for x ≥ 0. Let K ( x ) b e the symmetric 2 × 2 submatrix of J ( d ) with rows/columns 1 and 3, d 2 is replaced b y x ; then: T r x = tr ace ( K ( x )) = A + B + x D et x = det ( K ( x )) = A C (1 + x/B ) (A.1) The c haracteristic equation of K ( x ) is then: λ 2 − T r x λ + D et x = 0 The 2 eigen v alues of K ( x ), and thus the low est and highest eigen v alues of J ( d ), are: λ 1 / 3 = ( T r x ∓ p ∆ x ) / 2 ∆ x = T r 2 x − 4 Det x (A.2) F or the square ro ot √ ∆ x , the principal v alue has to b e taken. Map 2.1 no w has this form: j ( x ) : M → M ( A, B , C ) 7→ ( A x = ( T r x − p ∆ x ) / 2 , B x = B + x, C x = ( T r x + p ∆ x ) / 2) (A.3) Because A x and C x are eigen v alues of K ( x ), see the c haracteristic equation abov e, w e hav e: A x + C x = T r x , A x C x = D et x . W e pro of the following inequalities for x ≥ 0: (1) The square ro ot √ ∆ x defined ab o v e is real, i.e., ∆ x is p ositiv e: ∆ x = x 2 + ... is a quadratic p olynomial in x . Solving d ∆ x /dx = 0 gives x min ; the minim um is 4 A C ( B − A ) ( C − B ) /B 2 and is p ositiv e 1 , here for x ∈ R . (2) A x < B x : ( T r x − √ ∆ x ) / 2 − B x < 0. With D = A + B + x − 2 ( B + x ) we hav e D < + √ ∆ x . D 2 − ∆ x = − 4 ( B + x ) ( B − A ) ( C − B ) /B < 0, hence | D | < √ ∆ x . Therefore, for eac h of the t wo p ossible signs of D , D < + √ ∆ x , whic h implies A x < B x . (3) B x < C x : ( T r x + √ ∆ x ) / 2 − B x > 0. With D as in (2), we ha v e D > − √ ∆ x F rom (2), it was already sho wn that | D | < √ ∆ x , so for eac h of the tw o p ossible signs of D , we ha v e D > − √ ∆ x , whic h implies B x < C x . (4) 0 < A x : In (3), it is already prov en that 0 < B x < C x . Moreov er, A x C x = D et x = A C (1 + x/B ) > 0, whic h implies 0 < A x . Th us, 0 < A x < B x < C x , so the image lies in the co domain □ Appendix B. Proof of the (semi)group la w It has to be prov en: the comp osition of 2 maps j ( x ) , j ( y ) satisfies the group law j ( x ) ◦ j ( y ) = j ( x + y ) or j ( x ) j ( y ) m = j ( x + y ) m for m ∈ M . 1 a ”physical pro of” of ∆ x is p ositive for x ≥ 0: b ecause the principal moments of inertia A, B , C > 0 in the center of mass are ph ysical, the principal moments in all p oints are ph ysical, i.e. real. This implies ∆ x is positive 4 H. R UHLAND Using the form ula A.3 for j ( x ), the composition of t w o maps j ( x ) j ( y ) ( A, B , C ) yields the v ector: ( A xy , B xy , C xy ) A xy = ( A y + B y + x − p ∆ xy ) , / 2 , B xy = B y + x C xy = ( A y + B y + x + p ∆ xy ) / 2 A xy = ( T r y + x − p ∆ xy ) / 2 , B xy = B + x + y C xy = ( T r y + x + p ∆ xy ) / 2 A xy = ( T r x + y − p ∆ xy ) / 2 , B xy = B + x + y C xy = ( T r x + y + p ∆ xy ) / 2 (B.1) The first indications of commutativit y and additivit y are already visible; it re- mains to calculate ∆ xy : ∆ xy = ( A + C + x + y ) 2 − 4 A y C y (1 + x/B y ) A y C y = D et y = T r 2 x + y − 4 A C (1 + y /B ) (1 + x/ ( B + y )) = T r 2 x + y − 4 A C (1 + ( x + y ) /B ) = T r 2 x + y − 4 Det x + y = ∆ x + y (B.2) Hence, j ( x ) j ( y ) ( A, B , C ) = j ( x + y ) ( A, B , C ) □ References 1. S. Adla j. The Galois top and its motion-invariant . Avialable_at_the_author’ s_website Sant a F ´ e, La Habana, Cuba Email address : helmut.ruhland50@web.de