Curl force dynamics: symmetries, chaos and constants of motion

This is a theoretical study of Newtonian trajectories governed by curl forces, i.e. position-dependent but not derivable from a potential, investigating in particular the possible existence of conserved quantities. Although nonconservative and nonhamiltonian, curl forces are not dissipative because volume in the position-velocity state space is preserved. A physical example is the effective forces exerted on small particles by light. When the force has rotational symmetry, for example when generated by an isolated optical vortex, particles spiral outwards and escape, even with an attractive gradient force, however strong. Without rotational symmetry, and for dynamics in the plane, the state space is four-dimensional, and to search for possible constants of motion we introduce the Volume of section: a numerical procedure, in which orbits are plotted as dots in a three-dimensional subspace. For some curl forces, e.g. optical fields with two opposite-strength vortices, the dots lie on a surface, indicating a hidden constant of motion. For other curl forces, e.g. those from four vortices, the dots explore clouds, in an unfamiliar kind of chaos, suggesting that no constant of motion exists. The curl force dynamics generated by optical vortices could be studied experimentally.