A method for obtaining Hamiltonians with simple periodic solutions

The authors present a method for obtaining Hamiltonian systems with accurate periodic solutions. The general solution of a degenerate integrable Hamiltonian is expressed in a polynomial form when the ratio of the frequencies q = ω₁/ω₂ is an integer, or as a power series when q is a rational number. Adding this expression as a perturbation in the above Hamiltonian, a new nontrivial Hamiltonian with one exact periodic solution is obtained. Some possible applications of these findings are also discussed.