A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential

doi:10.1016/0167-2789(87)90050-9

Physica D: Nonlinear Phenomena

Volume 29, Issues 1–2, November–December 1987, Pages 128-142

https://doi.org/10.1016/0167-2789(87)90050-9 Get rights and content

Abstract

A non-integrability criterion on the basis of Ziglin's theorem is given for two degrees of freedom Hamiltonians with a homogeneous potential of integer degree. As simple examples the anisotropic Kepler problem and the one-dimensional Newtonian three-body problem are proved to be non-integrable. A self-contained proof of Ziglin's theorem (n-dimensional homogeneous potential version) is also given.

References (11)

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Existence of exponentially unstable periodic solutions and the non-integrability of homogeneous Hamiltonian systems
Physica
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H. Yoshida
A type of second order linear ordinary differential equations with periodic coefficients for which the characteristic exponents have exact expressions
Celestial Mech.
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V.V. Kozlov
Integrability and non-integrability in Hamiltonian mechanics
Russian Math. Surveys
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S.L. Ziglin
Branching of solutions and non-existence of first integrals in Hamiltonian mechanics
Functional Anal. Appl.
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S.L. Ziglin
Branching of solutions and non-existence of first integrals in Hamiltonian mechanics
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Non-integrability of Hénon-Heiles system and a theorem of Ziglin
Kodai Math. J.
(1985)

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A criterion for the non-existence of an additional integral in Hamiltonian systems with a homogeneous potential

Abstract

Physica

Celestial Mech.

Integrability and non-integrability in Hamiltonian mechanics

Russian Math. Surveys

Branching of solutions and non-existence of first integrals in Hamiltonian mechanics

Functional Anal. Appl.

Branching of solutions and non-existence of first integrals in Hamiltonian mechanics

Functional Anal. Appl.

Non-integrability of Hénon-Heiles system and a theorem of Ziglin

Kodai Math. J.