A Dirichlet criterion for the stability of periodic and relative periodic orbits in Hamiltonian systems
The problem of nonlinear stability of relative equilibria of the symmetric Hamiltonian systems when the dimension of the space are bigger than two is considered. It is known that Dirichlet's theorem has been adapted already to the study of the stability of relative equilibria (periodic orbits or relative periodic orbits). In that situations it received the names of Arnold, energy-Casimir, enery-momentum method or others. All these techniques are based on the definiteness of the second variation of an augmented energy function (which in infinite dimensions needs to be replaced with certain convexity estimates) and, as in the case of Dirichlet's theorem, they provide only sufficient stability conditions. The classical Dirichlet criterion is generalized and Dirichlet's like sufficient conditions for the stability of periodic orbits and relative periodic orbits are formulated in the article. For terminological convenience and to differentiate these results from the situations dealing with equilibria and relative equilibria the authors call their conditions the energy-integrals and symmetric energy-integrals methods, respectively. Some application of the results to several elementary examples are given. A short review on the known results is given.
scientific article; zbMATH DE number 1396337
scientific article