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: they admit geodesically equivalent metrics; one can use them to construct a large family of natural systems admitting integrals quadratic in momenta; the integrability of such systems can be generalized to the quantum setting; these natural systems are integrable by quadratures. Crown Copyright (C) 2009 Published by Elsevier B.V. All rights reserved.
Bolsinov, A., Matveev, V., Pucacco, G. (2009). Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta. JOURNAL OF GEOMETRY AND PHYSICS, 59(7), 1048-1062 [10.1016/j.geomphys.2009.04.010].
Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta
PUCACCO, GIUSEPPE
2009-01-01
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: they admit geodesically equivalent metrics; one can use them to construct a large family of natural systems admitting integrals quadratic in momenta; the integrability of such systems can be generalized to the quantum setting; these natural systems are integrable by quadratures. Crown Copyright (C) 2009 Published by Elsevier B.V. All rights reserved.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.