On averaging, reduction, and symmetry in hamiltonian systems

摘要: Abstract The existence of periodic orbits for Hamiltonian systems at low positive energies can be deduced from the nondegenerate critical points an averaged on associated “reduced space.” Alternatively, in classical (kinetic plus potential energy) Hamiltonians such often established by elementary geometrical arguments. present paper unifies two approaches exploiting discrete symmetries, including reversing diffeomorphisms, that occur a given system. symmetries are used to locate Hamiltonian, and thence original when continued under perturbations admitting same symmetries. In applications Henon-Heiles it is illustrated how “higher order” averaging sometimes overcome degeneracies encountered first order.