Pairs of commuting Hamiltonians quadratic in momenta

TL;DR

This paper introduces new multi-parameter families of commuting quadratic Hamiltonian pairs for two-degree systems and presents a universal method to solve the Hamilton-Jacobi equation using algebraic curves.

Contribution

It develops a universal approach to construct solutions of Hamilton-Jacobi equations via algebraic curves, expanding the class of integrable systems with quadratic Hamiltonians.

Findings

Discovered new multi-parametric families of commuting quadratic Hamiltonians.

Proposed a universal method for solving Hamilton-Jacobi equations using algebraic curves.

Applied the method to examples involving non-hyperelliptic curves.

Abstract

In the case of two degree system the pairs of quadratic in momenta Hamiltonians commuting according the standard Poisson bracket are considered. The new many-parametrical families of such pairs are founded. The universal method of constructing the full solution of Hamilton - Jacobi equation in terms of integrals on some algebraic curve is proposed. For some examples this curve is non-hyperelliptic covering over the elliptic curve.