Non-Symplectic Geometry of First Order Time-Dependent Mechanics

TL;DR

This paper develops a frame-covariant formulation of time-dependent mechanics that remains consistent under transformations, using a geometric approach on a bundle without a fixed fibration, extending to relativistic mechanics.

Contribution

It introduces a frame-covariant geometric framework for time-dependent mechanics that does not rely on a fixed splitting of the event space, applicable to relativistic contexts.

Findings

Provides a geometric phase space structure using the vertical cotangent bundle.

Establishes a canonical Poisson structure compatible with frame transformations.

Extends the formulation to relativistic mechanics with non-fixed fibrations.

Abstract

The usual formulation of time-dependent mechanics implies a given splitting $Y=R\times M$ of an event space $Y$. This splitting, however, is broken by any time-dependent transformation, including transformations between inertial frames. The goal is the frame-covariant formulation of time-dependent mechanics on a bundle $Y\to R$ whose fibration $Y\to M$ is not fixed. Its phase space is the vertical cotangent bundle $V^*Y$ provided with the canonical 3-form and the corresponding canonical Poisson structure. An event space of relativistic mechanics is a manifold $Y$ whose fibration $Y\to R$ is not fixed.