Integration of geodesic flows on homogeneous spaces: the case of a wild lie group

TL;DR

This paper develops a method for integrating geodesic flows on homogeneous spaces of wild Lie groups using canonical transformations and coadjoint orbits, providing necessary and sufficient conditions for integrability in quadratures.

Contribution

It introduces a new integration algorithm based on canonical transformations and coadjoint orbits, applicable to wild Lie group homogeneous spaces.

Findings

Derived necessary and sufficient conditions for integrability in quadratures.

Developed a canonical transformation-based integration algorithm.

Applicable to geodesic flows on homogeneous spaces of wild Lie groups.

Abstract

We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces $M$ with invariant and central metrics. The proposed integration algorithm consists in using a special canonical transformation in the space $T^*M$ based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions. This algorithm is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie group.