Geometric Generalisations of SHAKE and RATTLE

TL;DR

This paper provides a geometric analysis of Shake and Rattle methods for constrained Hamiltonian systems, revealing their geometric foundations, relations, and generalizations to broader classes of systems.

Contribution

It introduces a geometric framework for Shake and Rattle, generalizing these methods beyond classical constraints and clarifying their underlying assumptions.

Findings

Identifies geometric foundations of Shake and Rattle methods.

Establishes the relation between Shake and Rattle.

Generalizes methods to wider class of systems.

Abstract

A geometric analysis of the Shake and Rattle methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises Shake and Rattle to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting. In order for Shake and Rattle to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given.