Periodic Solutions of Hamiltonian Systems with Symmetries

After a brief historical account, starting with the celebrated Poincaré–Birkhoff Theorem, we provide a multiplicity result for periodic solutions of some Hamiltonian systems whose Hamiltonian function \(H(t,x,y)\) is periodic in the space variables x, and even in the variables \((t,y)\) . Our result is based on a recent theorem by R. Ortega and the author, and it does not require any twist condition on the solutions of the system.