PARAMETRIC QUARTIC HAMILTONIAN MODEL. A UNIFIED TREATMENT OF CLASSIC INTEGRABLE SYSTEMS

摘要

Related to the components of the quaternionic Hopf mapping, we propose a parametric Hamiltonian function in T*R-4 which is a homogeneous quartic polynomial with six parameters, defining an integrable family of Hamiltonian systems. The key feature of the model is its nested Hamiltonian-Poisson structure, which appears as two extended Euler systems in the reduced equations. This is fully exploited in the process of integration, where we find two 1-DOF subsystems and a quadrature involving both of them. The solution is quasi-periodic, expressed by means of Jacobi elliptic functions and integrals, based on two periods. For a suitable choice of the parameters, some remarkable classical models such as the Kepler, geodesic flow, isotropic oscillator and free rigid body systems appear as particular cases.