Hamiltonians and variational principles for Alfvén simple waves

Abstract

The evolution equations for the magnetic field induction B with the wave phase for Alfvén simple waves are expressed as variational principles and in the Hamiltonian form. The evolution of B with the phase (which is a function of the space and time variables) depends on the generalized Frenet–Serret equations, in which the wave normal n (which is a function of the phase) is taken to be tangent to a curve X, in a 3D Cartesian geometry vector space. The physical variables (the gas density, fluid velocity, gas pressure and magnetic field induction) in the wave depend only on the phase. Three approaches are developed. One approach exploits the fact that the Frenet equations may be written as a 3D Hamiltonian system, which can be described using the Nambu bracket. It is shown that B as a function of the phase satisfies a modified version of the Frenet equations, and hence the magnetic field evolution equations can be expressed in the Hamiltonian form. A second approach develops an Euler–Poincaré variational formulation. A third approach uses the Frenet frame formulation, in which the hodograph of B moves on a sphere of constant radius and uses a stereographic projection transformation due to Darboux. The equations for the projected field components reduce to a complex Riccati equation. By using a Cole–Hopf transformation, the Riccati equation reduces to a linear second order differential equation for the new variable. A Hamiltonian formulation of the second order differential equation then allows the system to be written in the Hamiltonian form. Alignment dynamics equations for Alfvén simple waves give rise to a complex Riccati equation or, equivalently, to a quaternionic Riccati equation, which can be mapped onto the Riccati equation obtained by stereographic projection.