Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

Abstract

We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct (Theorem 1) 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 (Theorem 2);• one can use them to construct a large family of natural systems admitting integrals quadratic in momenta (Theorem 4);• the integrability of such systems can be generalized to the quantum setting (Theorem 5);• these natural systems are integrable by quadratures (Section 2.2.2).

Two-dimensional metrics admitting precisely one projective vector field

“…The projective class of the metric g from the case 3d of Theorem 1 coincides with G(g,ḡ,g), whereḡ is the canonical projectively equivalent metric, andg is the metric given by (8).…”

Nonstandard Separability on the Minkowski Plane

“…It can be proven [3,4] that, even if the metric does not take an explicitly separated form, the Hamilton-Jacobi equation is indeed separable.…”

“…A general approach based on conformal coordinate transformations to solve Killing tensor equations has been given in [2]. It has been applied [3][4][5] to get a complete classification of separating coordinate systems of the Hamilton-Jacobi equation with the corresponding separated potentials and second integral of motion.…”

“…In particular, [9] describes a general theory of complex variable separation on pseudoRiemannian manifolds. In [3,4] the general picture of separability is extended to indefinite natural systems, showing how different kinds of separation structures are needed for different regions in configuration space. In particular, it is in general necessary to use different separating variables, even for the integration of a single orbit.…”

“…3 we resume the classification of the coordinate systems and the corresponding separation structures; in Sec. 4 we provide a detailed analysis of four natural Hamiltonian systems that give an overview of the possible cases; Sec. 5 contains concluding remarks.…”