Integrals in involution for groups of linear symplectic transformations and natural mechanical systems. (English. Russian original) Zbl 0995.37042
Funct. Anal. Appl. 34, No. 3, 179-187 (2000); translation from Funkts. Anal. Prilozh. 34, No. 3, 26-36 (2000).
Author’s summary: 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.
Reviewer: Iskander A.Taimanov (Novosibirsk)
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 70H05 | Hamilton’s equations |
| 70F10 | \(n\)-body problems |
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