Equations of motion for dynamical systems with angular momentum on Finsler geometries

Abstract

This article aims to derive equations of motion for dynamical systems with angular momentum on Finsler geometries. To this end, we apply Souriau's Principle of General Covariance, which is a geometrical framework to derive diffeomorphism invariant equations of motion. The equations we obtain are the generalization of that of Mathisson-Papapetrou-Dixon (MPD) on Finsler geometries, and we give their conserved quantities which turn out to be formally identical to the Riemannian case. These equations share the same properties as the MPD equations, and we mention different choices possible for supplementary conditions to close this system of equations. These equations have an additional requirement, which is the choice of what defines the tangent direction of the Finsler manifold, since the velocity and momentum are not parallel in general. After choosing the momentum as the Finsler direction and to define the center of mass, we give the complete equations of motion in 3 spatial dimensions, which we find coincide to previously known equations for Finsler spinoptics, and novel equations in 4 dimensions for massive and massless dynamical systems.

Figures (1)

• Figure 1: The geometry of the Principle of General Covariance. The dotted area represents the orbit $\mathcal{O}_g$ of all metrics connected to the metric $g$ by a diffeomorphism. This orbit is projected as a point $[g]$ on the quotient space $\mathop{\mathrm{Geom}}\nolimits(M)$.