Non-commutative integrability, exact solvability and the Hamilton–Jacobi theory

Abstract

The non-commutative integrability (NCI) is a property fulfilled by some Hamiltonian systems that ensures, among other things, the exact solvability of their corresponding equations of motion. The latter means that an “explicit formula” for the trajectories of these systems can be constructed. Such a construction rests mainly on the so-called Lie theorem on integrability by quadratures. It is worth mentioning that, in the context of Hamiltonian systems, the NCI has been for around 40 years, essentially, the unique criterium for exact solvability expressed in the terms of first integrals (containing the usual Liouville–Arnold integrability criterium as a particular case). Concretely, a Hamiltonian system with n degrees of freedom is said to be non-commutative integrable if we know a set of independent first integrals \(F_{1},\ldots ,F_{l}\) such that: the kernel of the \(l\times l\) matrix with coefficients \(\left\{ F_{i},F_{j}\right\} \), where \(\left\{ \cdot ,\cdot \right\} \) denotes the canonical Poisson bracket, has dimension \(2n-l\) (isotropy); and each bracket \(\left\{ F_{i},F_{j}\right\} \) is functionally dependent on \(F_{1},\ldots ,F_{l}\) (closure). In this paper, we develop two procedures for constructing the trajectories of a Hamiltonian system which only require isotropic first integrals (closure condition is not needed). One of them is based on an extended version of the geometric Hamilton–Jacobi theory, and does not rely on the above mentioned Lie’s theorem. We do all that in the language of functions of several variables.