The representation of periodic solutions of Newtonian systems

The integration of Hamiltonian systems with one degree of freedom x’’ = f(x), (sometimes called Newtonian systems), with f(x) analytical in a neighborhood of the origin, f(0) = 0, if f(0) < 0, can be notoriously reduced to quadratures by considering the Hamiltonian function H(x, y). This reduction, however, presents the disadvantage of introducing, already in the polynomial case, elliptic and hyperelliptic integrals, which in addition must be inverted in order to express x(t). The purpose of the present note is to show that, for sufficiently small initial conditions, the (periodic) solution of the given Hamiltonian system can be obtained in closed form. More precisely, we prove that it is possible to represent this solution on a fundamental interval [0, l] as the sum of a uniformly convergent series of particular maps (semitrigonometric polynomials), each of them can be explicitly determined from the given data. In the second section we apply these results to Jacobian elliptic functions and compare the rate of convergence of our representation with that one of other known developments of these functions. Some numerical experiments, referring to classic problems, are solved by a symbolic program and results are reported in details; they show that, the convergence rate of the expansion looks quite good.