Geodesic
Integrals in Involution for Groups of Linear Symplectic Transformations and Natural Mechanical Systems with Homogeneous Potential
Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 3, pp. 26-36.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that if a complex Hamiltonian system with $n$ degrees of freedom has $n$ functionally independent meromorphic first integrals in involution and the monodromy group of the corresponding variational system along some phase curve has $n$ pairwise skew-orthogonal two-dimensional invariant subspaces, then the restriction of the action of this group to each of these subspaces has a rational first integral. The result thus obtained is applied to natural mechanical systems with homogeneous potential, in particular, to the $n$-body problem. 
@article{FAA_2000_34_3_a2,
     author = {S. L. Ziglin},
     title = {Integrals in {Involution} for {Groups} of {Linear} {Symplectic} {Transformations} and {Natural} {Mechanical} {Systems} with {Homogeneous} {Potential}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {26--36},
     publisher = {mathdoc},
     volume = {34},
     number = {3},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2000_34_3_a2/}
}
TY  - JOUR
AU  - S. L. Ziglin
TI  - Integrals in Involution for Groups of Linear Symplectic Transformations and Natural Mechanical Systems with Homogeneous Potential
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2000
SP  - 26
EP  - 36
VL  - 34
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2000_34_3_a2/
LA  - ru
ID  - FAA_2000_34_3_a2
ER  - 
%0 Journal Article
%A S. L. Ziglin
%T Integrals in Involution for Groups of Linear Symplectic Transformations and Natural Mechanical Systems with Homogeneous Potential
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2000
%P 26-36
%V 34
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2000_34_3_a2/
%G ru
%F FAA_2000_34_3_a2
S. L. Ziglin. Integrals in Involution for Groups of Linear Symplectic Transformations and Natural Mechanical Systems with Homogeneous Potential. Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 3, pp. 26-36. http://geodesic.mathdoc.fr/item/FAA_2000_34_3_a2/

[1] Kozlov V. V., Simmetrii, topologiya i rezonansy v gamiltonovoi mekhanike, Izd-vo Udmurtskogo gos. un-ta, Izhevsk, 1995 | MR | Zbl

[2] Ziglin S. L., “Vetvlenie reshenii i nesuschestvovanie pervykh integralov v gamiltonovoi mekhanike, I”, Funkts. analiz i ego pril., 16:3 (1982), 30–41 | MR | Zbl

[3] Ziglin S. L., “Vetvlenie reshenii i nesuschestvovanie pervykh integralov v gamiltonovoi mekhanike, II”, Funkts. analiz i ego pril., 17:1 (1983), 8–24 | MR

[4] Yoshida H., “A criterion for the non-existence of an additional integral in Hamiltonian systems with homogeneous potential”, Phys. D, 29 (1987), 128–142 | DOI | MR | Zbl

[5] Yoshida H., “A criterion for the non-existence of an additional analitic integral in Hamiltonian systems with $n$ degrees of freedom”, Phys. Lett. A, 141 (1989), 108–112 | DOI | MR

[6] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[7] Leng S., Algebra, Mir, M., 1968

[8] Polubarinova-Kochina P. Ya., “Ob odnoznachnykh resheniyakh i algebraicheskikh integralakh zadachi o vraschenii tyazhelogo tverdogo tela okolo nepodvizhnoi tochki”, Dvizhenie tverdogo tela okolo nepodvizhnoi tochki, Sbornik, posvyaschennyi pamyati S. V. Kovalevskoi, Izd-vo AN SSSR, M.-L., 1940, 157–186

[9] Liouville R., “Sur la mouvement d'un corps solide pesant suspendu par l'un ses points”, Acta Math., 20 (1896), 239–284 | DOI | MR

[10] Alekseev V. M., “Kvazisluchainye dinamicheskie sistemy, 2: Odnomernye nelineinye kolebaniya v periodicheski vozmuschaemom pole”, Matem. sb., 77:4 (1968), 545–601 | MR | Zbl

[11] Alekseev V. M., “Kvazisluchainye dinamicheskie sistemy, 3: Kvazisluchainye kolebaniya odnomernykh ostsillyatorov”, Matem. sb., 78:1 (1969), 3–50 | MR | Zbl

[12] Alekseev V. M., “Finalnye dvizheniya v zadache trekh tel i simvolicheskaya dinamika”, UMN, 36:4 (1981), 161–176 | MR | Zbl

[13] Libre J., Simo C., “Oscillatory solutions in the planar restricted three-body problem”, Math. Ann., 248:2 (1980), 153–184 | DOI | MR

[14] Uitteker E. T., Analiticheskaya dinamika, ONTI, M.-L., 1937

[15] Ziglin S. L., “Ob otsutstvii dopolnitelnogo pervogo integrala v nekotorykh zadachakh dinamiki”, Funkts. analiz i ego pril., 31:1 (1997), 3–11 | DOI | MR | Zbl

[16] Morales-Ruiz J. J., “Differential Galois theory and nonintegrability of Hamiltonian systems”, Progress in Math., 179, Birkhäuser, 1999 | MR | Zbl