Lagrangian formalism and classical statistical ensemble

TL;DR

This paper introduces a Lagrangian-based formulation of classical statistical mechanics using tangent bundle dynamics, employing Wick rotation and symplectic geometry to define statistical ensembles directly in velocity variables.

Contribution

It develops a novel Lagrangian formalism for classical statistical mechanics that preserves natural measures and allows ensemble construction in velocity space.

Findings

Establishes a Hamiltonian structure in velocity variables.

Demonstrates measure-preserving dynamics on tangent bundle.

Enables direct definition of statistical ensembles in Lagrangian variables.

Abstract

We present a formulation of classical statistical mechanics based on a Lagrangian description on the tangent bundle. In this approach, a Wick rotation from real time to imaginary time is employed as a technical device that facilitates the construction of a Hamiltonian structure expressed in velocity variables. The resulting dynamics preserves a natural measure induced by the associated symplectic form on the tangent bundle. This measure-preserving property enables the consistent definition of classical statistical ensembles directly in terms of Lagrangian variables.