Computing Invariant Tori for Energetic Particles

Collisionless energetic charged particles in magnetic confinement devices can exhibit two kinds of motion: chaotic and integrable. When the motion is integrable, the trajectories lie on geometric objects known as invariant tori. For full-orbit dynamics, these tori are 3D submanifolds in the full 6D phase space, whereas for guiding-center dynamics, they are 2D surfaces in 4D phase space. Charged particle dynamics on invariant tori can be conjugated to the flow by a single rotation vector, which is analogous to the rotational transform for magnetic flux surfaces. Here, we will show how we can compute parameterizations of these tori from individual trajectories. We will demonstrate this capability for many charged particles in a stellarator, allowing for a comprehensive visualization of the geometry of charged particle dynamics. Through the rotation vector, this also allows for accurate computations of the transit and bounce frequencies of the charged particles.