Geodesic
Integral invariants of the Hamilton equations
Matematičeskie zametki, Tome 58 (1995) no. 3, pp. 379-393.

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Conditions are found for the existence of integral invariants of Hamiltonian systems. For two-degrees-of-freedom systems these conditions are intimately related to the existence of nontrivial symmetry fields and multivalued integrals. Any integral invariant of a geodesic flow on an analytic surface of genus greater than 1 is shown to be a constant multiple of the Poincaré–Cartan invariant. Poincaré's conjecture that there are no additional integral invariants in the restricted three-body problem is proved. 
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V. V. Kozlov. Integral invariants of the Hamilton equations. Matematičeskie zametki, Tome 58 (1995) no. 3, pp. 379-393. http://geodesic.mathdoc.fr/item/MZM_1995_58_3_a6/

[1] Puankare A., “Novye metody nebesnoi mekhaniki, III”, Izbr. trudy, T. II, Nauka, M., 1972

[2] Kartan E., Integralnye invarianty, Gostekhizdat, M.–L., 1940

[3] Godbiion K., Differentsialnaya geometriya i analiticheskaya mekhanika, Mir, M., 1973

[4] Hwa-Chung-Lee, “The universal integral invariants of Hamoltonian systems and application to the theory of canonical transformations”, Proc. R. Soc. of Edinburgh. Sect. A. Part 3, LXII (1946–1948), 237–246

[5] Kozlov V. V., “Liuvillevost invariantnykh mer vpolne integriruemykh sistem i uravnenie Monzha–Ampera”, Matem. zametki, 53:4 (1993), 45–52 | MR | Zbl

[6] Kozlov V. V., “Dynamical systems determined by the Navier–Stokes equations”, Rus. J. Math. Phys., 1:1 (1993), 57–69 | MR | Zbl

[7] Bogolyubov N. N., Mitropolskii Yu. L., Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974

[8] Kozlov V. V., Metody kachestvennogo analiza v dinamike tverdogo tela, Izd-vo MGU, M., 1980 | Zbl

[9] Sharle K. L., Nebesnaya mekhanika, Nauka, M., 1966

[10] Anosov D. V., Sinai Ya. G., “Nekotorye gladkie ergodicheskie sistemy”, UMN, 22:5 (1967), 107–172 | MR | Zbl

[11] Kozlov V. V., “O gruppakh simmetrii dinamicheskikh sistem”, PMM, 52:4 (1988), 531–541 | MR | Zbl

[12] Kozlov V. V., “O gruppakh simmetrii geodezicheskikh potokov na zamknutykh poverkhnostyakh”, Matem. zametki, 48:5 (1990), 62–67 | MR

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

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

[15] Bolotin S. V., Kozlov V. V., “Symmetry fields of geodesic flows”, Rus. J. Math. Phys., 3:3 (1995), 279–296 | MR