On Killing tensors and constants of motion

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Abstract
Some general properties belonging to constants of motion for geodesics and charged particle orbits are derived. A constant of motion for geodesics is seen to be a function on the cotangent bundle which has vanishing Poisson bracket with the ``energy function'' determined by the metric tensor. The resulting algebraic structure on the set of constants of motion is closely related to the Lie algebra of Killing tensors. Each constant of motion is shown to provide a family of mappings of geodesics into geodesics. Constants of motion for charged particles also possess a Lie algebra structure. The relationship of Killing tensors to charged particle constants of motion is derived. The linear and quadratic constants of motion for charged particle orbits in the charged Kerr metric illustrate the results. Examples of valence 2 Killing tensors are given in an appendix.

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