## Discrete Routh reduction

S. M. Jalnapurkar, M. Leok, J. E. Marsden, and M. West

*Journal of Physics A: Mathematical and General* **39**(19), 5521-5544, 2006.

This paper develops the theory of Abelian Routh reduction
for discrete mechanical systems and applies it to the variational
integration of mechanical systems with Abelian symmetry. The reduction
of variational Runge-Kutta discretizations is considered, as well as
the extent to which symmetry reduction and discretization
commute. These reduced methods allow the direct simulation of
dynamical features such as relative equilibria and relative periodic
orbits that can be obscured or difficult to identify in the unreduced
dynamics. The methods are demonstrated for the dynamics of an Earth
orbiting satellite with a non-spherical *J*_{2}
correction, as well as the double spherical pendulum. The
*J*_{2} problem is interesting because in the unreduced
picture, geometric phases inherent in the model and those due to
numerical discretization can be hard to distinguish, but this issue
does not appear in the reduced algorithm, where one can directly
observe interesting dynamical structures in the reduced phase space
(the cotangent bundle of shape space), in which the geometric phases
have been removed. The main feature of the double spherical pendulum
example is that it has a non-trivial magnetic term in its reduced
symplectic form. Our method is still efficient as it can directly
handle the essential non-canonical nature of the symplectic
structure. In contrast, a traditional symplectic method for canonical
systems could require repeated coordinate changes if one is evoking
Darboux' theorem to transform the symplectic structure into canonical
form, thereby incurring additional computational cost. Our method
allows one to design reduced symplectic integrators in a natural way,
despite the non-canonical nature of the symplectic structure.

DOI: 10.1088/0305-4470/39/19/S12

Full text: JaLeMaWe2006.pdf