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.
