Non-integrability criteria for Hamiltonians in the case of Lamé normal variational equations
Within the frame of the theorem of non-existence of the first integrals in a vicinity of a given solution obtained by S. L. Ziglin, it arises a question on solution of linear differential equations with time-dependent coefficients, called normal variational equations. In different applications to systems in two degrees of freedom, these equations are of Lamé type. In the paper under review the authors study a set of potentials of the classical Hamiltonian functions with an invariant plane such that an appropriate normal variational equation is of Lamé type. For these equations the conditions of non-integrability are obtained. The results are illustrated by examples from dynamics of Toda lattices of systems of two particles with cubic, quartic and rational potentials. The case of the homogeneous Henon-Heiles potential is studied in detail, including the dynamical meaning of the non-integrability.
normal variational equations of Lamé type
nonintegrability differential Galois approach
scientific article; zbMATH DE number 933271