Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems
Related to the components of the quaternionic Hopf mapping, we propose a parametric Hamiltonian function in \mathbb{T}\mathbb{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.