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.