The structure of invariants in conformal mechanics

We investigate the integrals of motion of general conformal mechanical systems with and without confining harmonic potential as well as of the related angular subsystems, by employing the sl(2,R)sl(2,R) algebra and its representations. In particular, via the tensor product of two representations we construct new integrals of motion from old ones, both in the classical and in the quantum case. Furthermore, the temporally periodic observables (including the integrals) of the angular subsystem are explicitly related to those of the full system in a confining harmonic potential. The techniques are illustrated for the rational Calogero models and their angular subsystems, where they generalize known methods for obtaining conserved charges beyond the Liouville ones.