Lagrangian formalism and classical statistical ensemble

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.