Some inequalities satisfied by periodical solutions of multi-time Hamilton equations

The objective of this paper is to find some inequalities satisfied by periodical solutions of multi-time Hamilton systems, when the Hamiltonian is convex. To our knowledge, this subject of first-order field theory is still open. Section 1 recall well-known facts regarding the equivalence between Euler-Lagrange equations and Hamilton equations and analyses the action that produces multi-time Hamilton equations, emphasizing the role of the polysymplectic structure. Section 2 extends two inequalities of [21] from a cube to parallelipiped and proves two inequalityes concerning multiple periodical solutions of multi-time Hamilton equations.