Complex variables for separation of the Hamilton-Jacobi equation on real pseudo-Riemannian manifolds

Abstract

In this paper the geometric theory of separation of variables for time-independent Hamilton-Jacobi equation is extended to include the case of complex eigenvalues of a Killing tensor on pseudo-Riemannian manifolds. This task is performed without to complexify the manifold but just considering complex-valued functions on it. The simple formalism introduced allows to extend in a very natural way the classical results on separation of variables (including Levi-Civita criterion and Stäckel-Eisenhart theory) to the complex case. Orthogonal variables only are considered.

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

“…In (real) pseudo-Riemannian manifolds, KTs (and CKTs) may have complex conjugated eigenvalues, in this case it is not possible to define real separable coordinates. However, it is possible to introduce separated complex variables allowing the Jacobi integration (see [13]). The application of our eigenvalue method to the complex case is in progress [14].…”

“…However, it is possible to introduce separated complex variables allowing the Jacobi integration (see [13]). The application of our eigenvalue method to the complex case is in progress [14]. For manifolds of constant curvature the whole spaces of Killing and conformal-Killing tensors are well known, then it is possible to apply our method to get computer-graphical representations of the webs.…”

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

“…The corresponding natural Hamiltonian problem on the hyperbolic plane has recently been treated in [39] following an approach used by Rosquist and Uggla [40]. Systems with indefinite signature have been investigated before in the classical works by Kalnins and Miller on separation of variables [20,38], see also [13,16,36]. Other possible approaches are based on Killing tensor theory [5], r-matrix theory [24] and algebraic methods [15].…”

Nonstandard Separability on the Minkowski Plane

“…Separability of free motion on the hyperbolic plane has already been investigated [7][8][9]. In particular, [9] describes a general theory of complex variable separation on pseudoRiemannian manifolds.…”

“…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.…”