Feedback Integrators

TL;DR

This paper introduces feedback integrators that ensure numerical solutions of dynamical systems on manifolds stay on the manifold and preserve integrals, compatible with standard integrators and demonstrated on classical problems.

Contribution

The paper presents a novel feedback integrator method that maintains manifold constraints and first integrals during numerical integration, compatible with common numerical schemes.

Findings

Method effectively keeps trajectories on the manifold.

Preserves first integrals during numerical integration.

Outperforms standard projection and splitting methods.

Abstract

A new method is proposed to numerically integrate a dynamical system on a manifold such that the trajectory stably remains on the manifold and preserves first integrals of the system. The idea is that given an initial point in the manifold we extend the dynamics from the manifold to its ambient Euclidean space and then modify the dynamics outside the intersection of the manifold and the level sets of the first integrals containing the initial point such that the intersection becomes a unique local attractor of the resultant dynamics. While the modified dynamics theoretically produces the same trajectory as the original dynamics, it yields a numerical trajectory that stably remains on the manifold and preserves the first integrals. The big merit of our method is that the modified dynamics can be integrated with any ordinary numerical integrator such as Euler or Runge-Kutta. We illustrate this method by applying it to three famous problems: the free rigid body, the Kepler problem and a perturbed Kepler problem with rotational symmetry. We also carry out simulation studies to demonstrate the excellence of our method and make comparisons with the standard projection method, a splitting method and St\"ormer-Verlet schemes.