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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2000, Volume 34, Issue 3, Pages 26–36 (Mi faa309)

Integrals in Involution for Groups of Linear Symplectic Transformations and Natural Mechanical Systems with Homogeneous Potential

S. L. Ziglin

Kotel'nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences

Abstract: 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.

DOI: https://doi.org/10.4213/faa309

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English version:
Functional Analysis and Its Applications, 2000, 34:3, 179–187

Bibliographic databases:

UDC: 517.913

Citation: S. L. Ziglin, “Integrals in Involution for Groups of Linear Symplectic Transformations and Natural Mechanical Systems with Homogeneous Potential”, Funktsional. Anal. i Prilozhen., 34:3 (2000), 26–36; Funct. Anal. Appl., 34:3 (2000), 179–187

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa309
• https://doi.org/10.4213/faa309
• http://mi.mathnet.ru/eng/faa/v34/i3/p26

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. L. Ziglin, “First Integrals of Groups of Complex Linear Transformations and of Natural Mechanical Systems with Homogeneous Potential”, Math. Notes, 70:6 (2001), 765–770
2. Tsygvintsev, A, “On the absence of an additional meromorphic first integral in the planar three-body problem”, Comptes Rendus de l Academie Des Sciences Serie i-Mathematique, 333:2 (2001), 125
3. Maciejewski, AJ, “Non-integrability of Gross-Neveu systems”, Physica D-Nonlinear Phenomena, 201:3–4 (2005), 249
4. Morales-Ruiz, JJ, “ON THE MEROMORPHIC NON-INTEGRABILITY OF SOME N-BODY PROBLEMS”, Discrete and Continuous Dynamical Systems, 24:4 (2009), 1225
5. Maciejewski A.J., Przybylska M., “Partial Integrability of Hamiltonian Systems with Homogeneous Potential”, Regular & Chaotic Dynamics, 15:4–5 (2010), 551–563
6. Morales-Ruiz J.J., Ramis J.-P., “Integrability of Dynamical Systems through Differential Galois theory: a practical guide”, Differential Algebra, Complex Analysis and Orthogonal Polynomials, Contemporary Mathematics, 509, 2010, 143–220
7. Combot T., “Non-Integrability of the Equal MASS N-Body Problem with Non-Zero Angular Momentum”, Celest. Mech. Dyn. Astron., 114:4 (2012), 319–340
8. Combot T., “A Note on Algebraic Potentials and Morales-Ramis Theory”, Celest. Mech. Dyn. Astron., 115:4 (2013), 397–404
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