Symmetry is fundamental to formulating and understanding our physical models. Many of the models which have been most influential to our understanding of mechanics often posses an abundance of conserved quantities. The Kepler two body problem and its quantisation, the hydrogen atom, have an additional “hidden” symmetry, encoded in the Laplace-Runge-Lenz vector. For classical Hamiltonian systems, hidden symmetries correspond to higher order Killing tensors, which are also known to be required for separability of the Hamilton-Jacobi and Schrödinger equations.
In the presence of a non-zero potential, constants of motion are naturally described by conformal Killing tensors with multiple Killing scales, that is, metrics in the conformal class for which these tensors become genuine Killing tensors. Both conformal Killing tensors and Killing scales admit a natural treatment via prolongation, as developed in recent work of Gover, Kress, and Leistner. In this talk, I will explain how tractor prolongations of conformal Killing tensors provide an effective framework for analysing constants of motion, and how this perspective can be used to address problems in Hamiltonian mechanics and superintegrable systems.
This is joint work with R. Gover and J. Kress.