A criterion for non-existence of invariant tori for Hamiltonian systems (Q913337)

A criterion for the non-existence of a specific class of invariant tori for Hamiltonian systems is derived. It is based on a classical result of the calculus of variations which goes back to Weierstrass. It can be applied to certain Hamiltonian systems that satisfy the Legendre condition (i.e., \(\det (\partial^ 2H/\partial p_ i\partial p_ j)\) is positive definite): If an orbit has a variation which decreases the action or if this orbit possesses conjugate points, then there is no invariant Lagrangian graph (i.e., a Mayer field that fulfils the Hamilton-Jacobi equation) containing it. The motion of a particle in the field of two waves is investigated as an example.