Topics in Magnetic Geometry: Interpolation, Intersections and Integrability

TL;DR

This paper explores the interplay between contact geometry, magnetic dynamics, and symmetry, establishing interpolation properties, integrals of motion, and structural results for magnetic submanifolds.

Contribution

It introduces new links between contact geometry and magnetic flows, including an interpolation property, Poisson-commuting integrals, and structural insights into magnetic submanifolds.

Findings

Magnetic geodesic flow interpolates between sub-Riemannian and magnetic vector flows.

Hamiltonian group actions yield Poisson-commuting integrals of motion.

Fixed-point sets and intersections of magnetic submanifolds are themselves totally magnetic.

Abstract

This paper develops new links between contact geometry, magnetic dynamics, and symmetry in exact magnetic systems. First, we establish an interpolation property for Killing magnetic systems on contact manifolds under an additional condition. Specifically, we show that the corresponding magnetic geodesic flow interpolates smoothly between the sub-Riemannian geodesic flow on the contact distribution and the flow of the vector field associated with a primitive of the magnetic field. Second, we show that Hamiltonian group actions associated with the magnetomorphism group produce Poisson-commuting integrals of motion for the magnetic flow. Finally, we obtain new structural results on totally magnetic submanifolds, showing that fixed-point sets of magnetomorphisms and intersections of totally magnetic submanifolds are again totally magnetic. The latter two results may be viewed as extensions of classical phenomena from Riemannian geometry to magnetic geometry.