Evolution to symmetry

TL;DR

This paper studies how a time-dependent Hamiltonian system with cubic potential evolves towards mirror symmetry, revealing drastic dynamical changes influenced by frequency ratios and timescales, inspired by galaxy evolution.

Contribution

It analyzes the slow evolution to symmetry in a Hamiltonian system with cubic potential, highlighting the role of frequency ratios and phase-space dynamics, inspired by galactic evolution models.

Findings

Identification of adiabatic invariants during evolution

Normal modes vanish and re-emerge, affecting stability

Significant changes in phase-space velocity distribution

Abstract

A natural example of evolution can be described by a time-dependent two degrees-of-freedom Hamiltonian. We choose the case where initially the Hamiltonian derives from a general cubic potential, the linearised system has frequencies 1 and $\omega >0$. The time-dependence produces slow evolution to discrete (mirror) symmetry in one of the degrees-of-freedom. This changes the dynamics drastically depending on the frequency ratio $\omega$ and the timescale of evolution. We analyse the cases $\omega = 1, 2, 3$ where the ratio's 1,2 turn out to be the most interesting. In an initial phase we find 2 adiabatic invariants with changes near the end of evolution. A remarkable feature is the vanishing and emergence of normal modes, stability changes and strong changes of the velocity distribution in phase-space. The problem is inspired by the dynamics of axisymmetric, rotating galaxies that evolve slowly to mirror symmetry with respect to the galactic plane, the model formulation is quite general.