In this paper we explore general conditions which guarantee that the geodesic flow on a two-dimensional manifold with indefinite signature is locally separable. This is equivalent to showing that a two-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 secind 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. (c) 2007 American Institute of Physics.

Pucacco G , & Rosquist, K. (2007). (1+1)-dimensional separation of variables. JOURNAL OF MATHEMATICAL PHYSICS, 48(11), 2903-2925 [10.1063/1.2811706].

(1+1)-dimensional separation of variables

PUCACCO, GIUSEPPE;
2007

Abstract

In this paper we explore general conditions which guarantee that the geodesic flow on a two-dimensional manifold with indefinite signature is locally separable. This is equivalent to showing that a two-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 secind 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. (c) 2007 American Institute of Physics.
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore FIS/02 - Fisica Teorica, Modelli e Metodi Matematici
Settore FIS/05 - Astronomia e Astrofisica
Settore MAT/07 - Fisica Matematica
English
Con Impact Factor ISI
INTEGRALS OF HAMILTONIAN-SYSTEMS; UNIFIED TREATMENT OF ARBITRARY ENERGY INVARIANTS;
Pucacco G , & Rosquist, K. (2007). (1+1)-dimensional separation of variables. JOURNAL OF MATHEMATICAL PHYSICS, 48(11), 2903-2925 [10.1063/1.2811706].
Pucacco, G; Rosquist, K
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/35224
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