Nonlocal criteria for the existence and stability of periodic oscillations in autonomous Hamiltonian systems (Q1089148)

The conditions under which single-parameter families of periodic solutions (the existence in a sufficiently small neighbourhood of the origin of coordinates follows from the Lyapunov theorem can be continued in a parameter to the boundary of the given domain, in particular to a certain isoenergetic surface, are found. These conditions, which can be verified by the use of the Hessian of a Hamilton function, also ensure the orbital stability of solutions to a first approximation. Bilateral estimates of the oscillation periods are obtained, and it is established that any solution with a period which satisfies such an estimate belongs to the corresponding family. As an example, the nonlinear oscillations of a string with lumped asses are examined.