Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems

Sebastián Ferrer; Francisco Crespo

doi:10.3934/jgm.2014.6.479

Article Contents

2014, Volume 6, Issue 4: 479-502. Doi: 10.3934/jgm.2014.6.479

This issue Previous Article Higher-order variational problems on lie groups and optimal control applications Next Article Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings

Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems

Sebastián Ferrer^1, and
Francisco Crespo^1,

1.
Departamento de Matemática Aplicada, Universidad de Murcia, 30100 Espinardo

Received: April 2014

Revised: August 2014

Published: December 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 37N05, 53D20, 70E40; Secondary: 37J35, 70H06, 70H15.

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