Constants of Motion for Constrained Hamiltonian Systems: A Particle around a Charged Rotating Black Hole
arXiv:1005.1815 · doi:10.1103/PhysRevD.83.065027
Abstract
We discuss constants of motion of a particle under an external field in a curved spacetime, taking into account the Hamiltonian constraint which arises from reparametrization invariance of the particle orbit. As the necessary and sufficient condition for the existence of a constant of motion, we obtain a set of equations with a hierarchical structure, which is understood as a generalization of the Killing tensor equation. It is also a generalization of the conventional argument in that it includes the case when the conservation condition holds only on the constraint surface in the phase space. In that case, it is shown that the constant of motion is associated with a conformal Killing tensor. We apply the hierarchical equations and find constants of motion in the case of a charged particle in an electro-magnetic field in black hole spacetimes. We also demonstrate that gravitational and electro-magnetic fields exist in which a charged particle has a constant of motion associated with a conformal Killing tensor.
20 pages
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- Conformal Killing Tensors and covariant Hamiltonian Dynamics
- Non-commutative oscillator with Kepler-type dynamical symmetry
- Integrability of geodesic motions in curved manifolds through non-local conserved charges
- Scale invariance and constants of motion
- Covariant Approach of the Dynamics of Particles in External Gauge Fields, Killing Tensors and Quantum Gravitational Anomalies
- Integrability of Particle System around a Ring Source as the Newtonian Limit of a Black Ring
- Hidden symmetries in a gauge covariant approach, Hamiltonian reduction and oxidation
- Chaotic particle motion around a homogeneous circular ring
- Properties of Particle Trajectory Around a Weakly Magnetized Black Hole
- Construction of test Maxwell fields with scale symmetry