A Riemannian geometrical description for Lie-Poisson systems and its application to idealized magnetohydrodynamics

Abstract

This paper presents a general theorem which enables description of Lie-Poisson systems for semi-direct product groups in terms of Riemannian geometry. The method employed is unique in that it changes the non-quadratic right- or left-invariant Hamiltonian to the quadratic form in a right- or left-invariant I-form on the corresponding group by removing the respective invariance, which obtains the Riemannian metric and its induced Riemannian (Levi-Civita) connection. The resultant geodesic equation proves to be equivalent to the equation of motion, while the corresponding Jacobi equation determines its instability. In addition, this method is applied to idealized magnetohydrodynamics having isentropic flow, with a simple example being provided that considers the motion of an isentropic gas with no magnetic field present.