First integrals from conformal symmetries: Darboux-Koenigs metrics and beyond. (English) Zbl 1430.37063

Summary: On spaces of constant curvature, the geodesic equations automatically have higher order integrals, which are just built out of first order integrals, corresponding to the abundance of Killing vectors. This is no longer true for general conformally flat spaces, but in this case there is a large algebra of conformal symmetries. In this paper we use these conformal symmetries to build higher order integrals for the geodesic equations. We use this approach to give a new derivation of the Darboux-Koenigs metrics, which have only one Killing vector, but two quadratic integrals. We also consider the case of possessing one Killing vector and two cubic integrals. The approach allows the quantum analogue to be constructed in a simpler manner.


37J37 Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
70H20 Hamilton-Jacobi equations in mechanics
17B63 Poisson algebras
81S10 Geometry and quantization, symplectic methods
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