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Geometric Generalisations of Shake and Rattle

Journal article
Authors Robert I McLachlan
Klas Modin
Olivier Verdier
Matt Wilkins
Published in Foundations of Computational Mathematics
Volume 14
Issue 2
Pages 339-370
ISSN 1615-3375
Publication year 2014
Published at Department of Mathematical Sciences, Mathematics
Pages 339-370
Language en
Subject categories Geometry


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.

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