Periodical Solutions of Multi-Time Hamilton Equations

Abstract

To our knowledge, there are two main references [9], [12] regarding the periodical solutions of multi-time Euler-Lagrange systems, even if the multi-time equations appeared in 1935, being introduced by de Donder. That is why, the central objective of this paper is to solve an open problem raised in [12]: what we can say about periodical solutions of multi-time Hamilton systems when the Hamiltonian is convex? Section 1 recall well-known facts regarding the equivalence between Euler-Lagrange equations and Hamilton equations. Section 2 analyzes the action that produces multi-time Hamilton equations, and introduces the Legendre transform of a Hamiltonian together a new dual action. Section 3 proves the existence of periodical solutions of multi-time Hamilton equations via periodical extremals of the dual action, when the Hamiltonian is convex.