(1+1)-dimensional separation of variables

Nonlinear Sciences – Exactly Solvable and Integrable Systems

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

To appear on Journal of Mathematical Physics

Scientific paper

10.1063/1.2811706

In this paper we explore general conditions which guarantee that the geodesic flow on a 2-dimensional manifold with indefinite signature is locally separable. This is equivalent to showing that a 2-dimensional natural Hamiltonian system on the hyperbolic plane possesses a second integral of motion which is a quadratic polynomial in the momenta associated with a 2nd-rank Killing tensor. We examine the possibility that the integral is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have conservation at arbitrary values of the Hamiltonian (strong integrability). Using null coordinates, we show that the leading-order coefficients of the invariant are arbitrary functions of one variable in the case of weak integrability. These functions are quadratic polynomials in the coordinates in the case of strong integrability. We show that for $(1+1)$-dimensional systems there are three possible types of conformal Killing tensors, and therefore, three distinct separability structures in contrast to the single standard Hamilton-Jacobi type separation in the positive definite case. One of the new separability structures is the complex/harmonic type which is characterized by complex separation variables. The other new type is the linear/null separation which occurs when the conformal Killing tensor has a null eigenvector.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

(1+1)-dimensional separation of variables does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with (1+1)-dimensional separation of variables, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and (1+1)-dimensional separation of variables will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-456142

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.