(D+1)-Dimensional Formulation for D-Dimensional Constrained Systems

TL;DR

This paper develops a (D+1)-dimensional canonical framework for D-dimensional constrained systems, revealing stochastic consistency as second class constraints and deriving Langevin equations as Hamiltonian dynamics.

Contribution

It introduces a novel (D+1)-dimensional formulation that explicitly incorporates stochastic constraints and derives Langevin equations from Hamiltonian principles.

Findings

Stochastic consistency conditions are identified as second class constraints.

Lagrange multiplier fields are determined within the (D+1)-dimensional framework.

Langevin equations are derived as Hamilton's equations with conjugate momenta acting as noise fields.

Abstract

D-dimensional constrained systems are studied with stochastic Lagrangian and\break Hamiltonian. It is shown that stochastic consistency conditions are second class constraints and Lagrange multiplier fields can be determined in (D+1)-dimensional canonical formulation. The Langevin equations for the constrained system are obtained as Hamilton's equations of motion where conjugate momenta play a part of noise fields.