Ehrenfest regularization of Hamiltonian systems

TL;DR

This paper introduces Ehrenfest regularization, a method to add irreversible, dissipative effects to Hamiltonian systems, demonstrated on various physical models, and discusses discretization techniques that preserve key properties.

Contribution

It proposes a novel Ehrenfest regularization method for Hamiltonian equations, incorporating irreversibility and dissipation while maintaining core physical properties.

Findings

Regularization stabilizes rotations in rigid body dynamics.

Method successfully applied to fluid mechanics and kinetic theory.

Discretization preserves energy and entropy behavior.

Abstract

Imagine a freely rotating rigid body. The body has three principal axes of rotation. It follows from mathematical analysis of the evolution equations that pure rotations around the major and minor axes are stable while rotation around the middle axis is unstable. However, only rotation around the major axis (with highest moment of inertia) is stable in physical reality (as demonstrated by the unexpected change of rotation of the Explorer 1 probe). We propose a general method of Ehrenfest regularization of Hamiltonian equations by which the reversible Hamiltonian equations are equipped with irreversible terms constructed from the Hamiltonian dynamics itself. The method is demonstrated on harmonic oscillator, rigid body motion (solving the problem of stable minor axis rotation), ideal fluid mechanics and kinetic theory. In particular, the regularization can be seen as a birth of irreversibility and dissipation. In addition, we discuss and propose discretizations of the Ehrenfest regularized evolution equations such that key model characteristics (behavior of energy and entropy) are valid in the numerical scheme as well.