Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta
Mathematical Physics
2015-05-13 v2 math.MP
Exactly Solvable and Integrable Systems
Abstract
We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely: 1) they admit geodesically equivalent metrics; 2) one can use them to construct a big family of natural systems admitting integrals quadratic in momenta; 3) the integrability of such systems can be generalized to the quantum setting; 4) these natural systems are integrable by quadratures.
Cite
@article{arxiv.0803.0289,
title = {Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta},
author = {Alexey V. Bolsinov and Vladimir S. Matveev and Giuseppe Pucacco},
journal= {arXiv preprint arXiv:0803.0289},
year = {2015}
}