Geometric integration on symmetric spaces

Abstract

We consider geometric numerical integration algorithms for differential equations evolving on symmetric spaces. The integrators are constructed from canonical operations on the symmetric space, its Lie triple system (LTS), and the exponential from the LTS to the symmetric space. Examples of symmetric spaces are n-spheres and Grassmann manifolds, the space of positive definite symmetric matrices, Lie groups with a symmetric product, and elliptic and hyperbolic spaces with constant sectional curvatures. We illustrate the abstract algorithm with concrete examples. In particular for the n-sphere and the n-dimensional hyperbolic space the resulting algorithms are very simple and cost only O(n) operations per step.