Separable systems with quadratic in momenta first integrals

Abstract

Geometric separability theory of Gel'fand-Zakharevich bi-Hamilto-nian systems on Riemannian manifolds is reviewed and developed. Particular attention is paid to the separability of systems generated by the so-called special conformal Killing tensors, i.e. Benenti systems. Then, infinitely many new classes of separable systems are constructed by appropriate deformations of Benenti class systems. All such systems can be lifted to the Gel'fand-Zakharevich bi-Hamiltonian form.

A class of recursion operators on a tangent bundle

“…An equivalent definition of the sequence (2.5) is given by [4]). When we know the eigenvalues λ i of L, we can also write…”

Natural coordinates for a class of Benenti systems

“…The scalar functions V (k) r are basic separable potentials, see below for details. The contravariant metric tensors G m have the form [6] …”

“…[7,8] and references therein. More general separable metrics can be obtained from G m via the so-called k-hole deformations [6,9]. For m = 0, .…”

“…The potentials V (k) r in the Hamiltonians (2) can be obtained from the following recursion relation [6]:…”

Eigenvalues of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds

“…Each section is completed by illustrative examples: the spherical coordinates in R 3 (Section 3), the L-systems [5], also known as Benenti systems [8,9,15] (Section 4), two non-orthogonal 4-dimensional coordinate systems (one of them with null coordinates) in Section 5, and the conformal separable coordinate system, known as tangent-spheres coordinates [20] (Section 6). Moreover, by applying our analysis to L-systems, we prove an interesting geometrical property of these systems: for n > 2, none of the common eigenvectors of the associated Killing tensors is a proper conformal Killing vector.…”