An Expansion Term In Hamilton's Equations

TL;DR

This paper introduces an additional expansion term in Hamilton's equations for systems with a preferred time coordinate, impacting fluid dynamics and cosmology, and explores the implications for symplectic structure and Finsler curvature.

Contribution

It reveals a previously unnoticed expansion term in Hamilton's equations when using absolute time, and proposes a method to recover symplectic form by introducing particle number variables.

Findings

The expansion term appears in Hamilton's equations for fluid and cosmological systems.

Replacing expansion with particle number N restores the symplectic form.

The study suggests possible non-standard symplectic structures and Finsler curvature implications.

Abstract

For any given spacetime the choice of time coordinate is undetermined. A particular choice is the absolute time associated with a preferred vector field. Using the absolute time Hamilton's equations are $- (\delta H_{c})/(\delta q)=\dot{\pi}+\Theta\pi, $+ (\delta H_{c})/(\delta \pi)=\dot{q}$, where $\Theta = V^{a}_{.;a}$ is the expansion of the vector field. Thus there is a hitherto unnoticed term in the expansion of the preferred vector field. Hamilton's equations can be used to describe fluid motion. In this case the absolute time is the time associated with the fluid's co-moving vector. As measured by this absolute time the expansion term is present. Similarly in cosmology, each observer has a co-moving vector and Hamilton's equations again have an expansion term. It is necessary to include the expansion term to quantize systems such as the above by the canonical method of replacing Dirac brackets by commutators. Hamilton's equations in this form do not have a corresponding sympletic form. Replacing the expansion by a particle number $N\equiv exp(-\int\Theta d \ta)$ and introducing the particle numbers conjugate momentum $\pi^{N}$ the standard sympletic form can be recovered with two extra fields N and $\pi^N$. Briefly the possibility of a non-standard sympletic form and the further possibility of there being a non-zero Finsler curvature corresponding to this are looked at.