On Hamiltonian flows whose orbits are straight lines

TL;DR

This paper investigates Hamiltonian systems with flows that are linear in time, demonstrating the limitations of simplifying polynomial Hamiltonians of degree 4 or higher through linear symplectic transformations.

Contribution

It proves that polynomial Hamiltonians of degree 4 or higher generally cannot be reduced to simpler forms via linear symplectic changes, unlike degree 3 cases, and provides conditions for such reductions.

Findings

Degree 3 polynomial Hamiltonians can be simplified via linear symplectic transformations.

Degree 4 or higher polynomial Hamiltonians generally cannot be reduced to simpler forms.

Nondegenerate degree 4 homogeneous Hamiltonians satisfy the reduction condition.

Abstract

We consider real analytic Hamiltonians whose flow depends linearly on time. Trivial examples are Hamiltonians $H(q,p)$ that do not depend on the coordinate $q$. By a theorem of Moser, every polynomial Hamiltonian of degree 3 reduces to such a $q$-independent Hamiltonian via a linear symplectic change of variables. We show that such a reduction is impossible, in general, for polynomials of degree 4 or higher. But we give a condition that implies linear-symplectic conjugacy to another simple class of Hamiltonians. The condition is shown to hold for all nondegenerate Hamiltonians that are homogeneous of degree 4.