REMARKS ON PERIODS OF PLANAR HAMILTONIAN-SYSTEMS

Criteria for monotonicity of the periods of planar Hamiltonian systems from various authors are logically related. From the more restrictive criteria, not only monotonicity is proved, but even a further convexity property of the energy-period function. This implies estimates for the canonical partition function-which is the Laplace transform of the energy-period function-and some averages from the canonical ensemble of thermodynamics. For the standard system of classical mechanics with quadratic kinetic energy and symmetric potential energy, the Laplace transform technique allows one to solve the inverse problem to determine the potential energy function from given periods. Several examples are discussed.