Energy and time as conjugate dynamical variables

TL;DR

This paper explores the geometric structure of energy and time as conjugate variables in classical mechanics, revealing their relation to Lorentz transformations and the speed of light through symplectic geometry.

Contribution

It introduces a geometric framework for energy and time as conjugate variables, connecting Galilei and Lorentz transformations via symplectic geometry and extended phase-space.

Findings

Galilei action is canonical but not Hamiltonian equivariant

Lorentz transformations emerge as symplectic transformations

Speed of light is identified through electromagnetic potentials

Abstract

The energy and time variables of the elementary classical dynamical systems are described geometrically, as canonically conjugate coordinates of an extended phase-space. It is shown that the Galilei action of the inertial equivalence group on this space is canonical, but not Hamiltonian equivariant. Although it has no effect at classical level, the lack of equivariance makes the Galilei action inconsistent with the canonical quantization. A Hamiltonian equivariant action can be obtained by assuming that the inertial parameter in the extended phase-space is quasi-isotropic. This condition leads naturally to the Lorentz transformations between moving frames as a particular case of symplectic transformations. The limit speed appears as a constant factor relating the two additional canonical coordinates to the energy and time. Its value is identified with the speed of light by using the relationship between the electromagnetic potentials and the symplectic form of the extended phase-space.