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

This article is cited in 8 scientific papers (total in 8 papers)

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
Received: 19.03.1999

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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\paper Integrals in Involution for Groups of Linear Symplectic Transformations and Natural Mechanical Systems with Homogeneous Potential
\jour Funktsional. Anal. i Prilozhen.
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\pages 26--36
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\yr 2000
\vol 34
\issue 3
\pages 179--187
\crossref{https://doi.org/10.1007/BF02482407}
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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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Maciejewski, AJ, “Non-integrability of Gross-Neveu systems”, Physica D-Nonlinear Phenomena, 201:3–4 (2005), 249  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Morales-Ruiz, JJ, “ON THE MEROMORPHIC NON-INTEGRABILITY OF SOME N-BODY PROBLEMS”, Discrete and Continuous Dynamical Systems, 24:4 (2009), 1225  crossref  mathscinet  zmath  isi  scopus
    5. Maciejewski A.J., Przybylska M., “Partial Integrability of Hamiltonian Systems with Homogeneous Potential”, Regular & Chaotic Dynamics, 15:4–5 (2010), 551–563  crossref  mathscinet  zmath  adsnasa  isi  scopus
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Combot T., “A Note on Algebraic Potentials and Morales-Ramis Theory”, Celest. Mech. Dyn. Astron., 115:4 (2013), 397–404  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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