  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.
