Geometry of Lax pairs: particle motion and Killing-Yano tensors
arXiv:1210.3079 · doi:10.1103/PhysRevD.87.024002
Abstract
A geometric formulation of the Lax pair equation on a curved manifold is studied using phase space formalism. The corresponding (covariantly conserved) Lax tensor is defined and the method of generation of constants of motion from it is discussed. It is shown that when the Hamilton equations of motion are used, the conservation of the Lax tensor translates directly to the well known Lax pair equation, with one matrix identified with components of the Lax tensor and the other matrix constructed from the (metric) connection. A generalization to Clifford objects is also discussed. Nontrivial examples of Lax tensors for geodesic and charged particle motion are found in spacetimes admitting hidden symmetry of Killing--Yano tensors.
arXiv comment13 pages, 1 figure
References in corpus (27)
- Rotating Black Holes in Higher Dimensions with a Cosmological Constant
- General Kerr-NUT-AdS Metrics in All Dimensions
- Gravitational Perturbations of Higher Dimensional Rotating Black Holes: Tensor Perturbations
- Complete Integrability of Geodesic Motion in General Kerr-NUT-AdS Spacetimes
- Separability of Hamilton-Jacobi and Klein-Gordon Equations in General Kerr-NUT-AdS Spacetimes
- Higher-Dimensional Black Holes: Hidden Symmetries and Separation of Variables
- Killing-Yano Tensors, Rank-2 Killing Tensors, and Conserved Quantities in Higher Dimensions
- Hidden Symmetries of Higher Dimensional Black Holes and Uniqueness of the Kerr-NUT-(A)dS spacetime
- Closed conformal Killing-Yano tensor and Kerr-NUT-de Sitter spacetime uniqueness
- Separability of Dirac equation in higher dimensional Kerr-NUT-de Sitter spacetime
- Constants of Geodesic Motion in Higher-Dimensional Black-Hole Spacetimes
- Charged particle in higher dimensional weakly charged rotating black hole spacetime
- Dirac Equation in Kerr-NUT-(A)dS Spacetimes: Intrinsic Characterization of Separability in All Dimensions
- Complete Set of Commuting Symmetry Operators for the Klein-Gordon Equation in Generalized Higher-Dimensional Kerr-NUT-(A)dS Spacetimes
- Closed conformal Killing-Yano tensor and geodesic integrability
- Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets
- Hidden Symmetry and Exact Solutions in Einstein Gravity
- Killing-Yano tensors and some applications
- Generalized Kerr-NUT-de Sitter metrics in all dimensions
- Killing-Yano equations and G-structures
- Separability of Gravitational Perturbation in Generalized Kerr-NUT-de Sitter Spacetime
- Higher rank Killing tensors and Calogero model
- Lax pair tensors and integrable spacetimes
- Hidden Symmetries of Euclideanised Kerr-NUT-(A)dS Metrics in Certain Scaling Limits
- Third rank Killing tensors in general relativity. The (1+1)-dimensional case
- Lax Tensors, Killing Tensors and Geometric Duality
- Geometrization of the Lax Pair Tensors
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- On Integrability of the Geodesic Deviation Equation
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