Beyond Quartic Hamiltonians

TL;DR

This paper explores Hamiltonian systems derived from high-order homogeneous polynomials, focusing on conditions where the Hessian relates to Weierstrass functions, revealing new covariant-based criteria.

Contribution

It identifies specific covariants whose vanishing characterizes when the Hessian of such Hamiltonians is a Weierstrass  function, advancing understanding of these complex systems.

Findings

Hessian of the polynomial can be a Weierstrass  function under certain conditions.

Vanishing of two covariants of the polynomial is key to this property.

Provides new criteria for analyzing high-order polynomial Hamiltonian systems.

Abstract

We investigate the Hamiltonian system generated by a homogeneous binary polynomial $U$ of order greater than four. In particular, we study the circumstances under which the associated Hessian is a Weierstrass $\wp$ function and find that the vanishing of two covariants of $U$ is involved.