Generalizations of a method for constructing first integrals of a class of natural Hamiltonians and some remarks about quantization

TL;DR

This paper extends previous methods for constructing high-degree first integrals of natural Hamiltonians, generalizing to Poisson manifolds and exploring implications for quantization, with a focus on conformal symmetries and integrability.

Contribution

It generalizes the construction of first integrals to Hamiltonians on Poisson manifolds and links their existence to conformal Killing tensors, advancing the understanding of integrability and quantization.

Findings

Explicit expression for first integrals on Poisson manifolds.

Connection between conformal Killing tensors and first integrals.

Discussion on quantization of second-degree first integrals.

Abstract

In previous papers we determined necessary and sufficient conditions for the existence of a class of natural Hamiltonians with non-trivial first integrals of arbitrarily high degree in the momenta. Such Hamiltonians were characterized as (n+1)-dimensional extensions of n-dimensional Hamiltonians on constant-curvature (pseudo-)Riemannian manifolds Q. In this paper, we generalize that approach in various directions, we obtain an explicit expression for the first integrals, holding on the more general case of Hamiltonians on Poisson manifolds, and show how the construction of above is made possible by the existence on Q of particular conformal Killing tensors or, equivalently, particular conformal master symmetries of the geodesic equations. Finally, we consider the problem of Laplace-Beltrami quantization of these first integrals when they are of second-degree.