Abstract
It is a big problem to distinguish between integrable and non-integrable Hamiltonian systems.We provide a new approach to prove the non-integrability of homogeneous Hamiltonian systemswith two degrees of freedom. The homogeneous degree can be taken from real values (not necessarily integer).The proof is based on the blowing-up theorywhich McGehee established in the collinear three-body problem.We also compare our result with Molares-Ramis theory which is the strongest theory in this field.
Highlights
Let H : D → R be a smooth function where D is an open set in R2k
The Hamiltonian system is defined by the ordinary differential equations dqj dt
A function F : D → R is called the first integral of (1) if F is conserved along each solution of (1)
Summary
Let H : D → R be a smooth function where D is an open set in R2k. The Hamiltonian system is defined by the ordinary differential equations dqj dt. Hamiltonian systems there is no analytic first integral depending analytically on a parameter By applying it to the restricted 3-body problem, he proved the non-existence of an analytic first integral depending analytically on a mass parameter. Another theory in this field was originated by Kovalevskaya [3]. By applying the Ziglin analysis, Yoshida [7] provided a criterion for the non-integrability of the homogeneous Hamiltonian systems. Our purpose is to prove the non-integrability of Hamiltonian systems from a new approach. The case of θ1 = θ−1 + 2π is allowed in assumption 2 These two critical points are essentially identical. In the final section we compare our theorem with the Morales-Ramis theorem
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