Existence of multiple normal mode trajectories on convex energy surfaces of even, classical Hamiltonian systems

Hamiltonian systems of n degrees of freedom for which the Hamiltonian is a function that is even both in its joint n coordinate variables as well as in its joint n momentum variables are discussed. For such systems the number of distinct trajectories which correspond to particular periodic solutions (normal modes) with the same energy, is investigated. To that end a constrained dual action principle is introduced. Applying min-max methods to this variational problem, several results are obtained, among which the existence of at least n distinct trajectories if specific conditions are satisfied.