  In this paper, we consider Langevin processes with mechanical constraints.
The latter are a fundamental tool in molecular dynamics simulation for sampling
purposes and for the computation of free energy differences. The results of
this paper can be divided into three parts. (i) We propose a simple
discretization of the constrained Langevin process based on a standard
splitting strategy. We show how to correct the scheme so that it samples {\em
exactly} the canonical measure restricted on a submanifold, using a Metropolis
rule in the spirit of the Generalized Hybrid Monte Carlo (GHMC) algorithm.
Moreover, we obtain, in some limiting regime, a consistent discretization of
the overdamped Langevin (Brownian) dynamics on a submanifold, also sampling
exactly the correct canonical measure with constraints. The corresponding
numerical methods can be used to sample (without any bias) a probability
measure supported by a submanifold. (ii) For free energy computation using
thermodynamic integration, we rigorously prove that the longtime average of the
Lagrange multipliers of the constrained Langevin dynamics yields the gradient
of a rigid version of the free energy associated with the constraints. A second
order time discretization using the Lagrange multipliers is proposed. (iii) The
Jarzynski-Crooks fluctuation relation is proved for Langevin processes with
mechanical constraints evolving in time. An original numerical discretization
without time-step error is proposed. Numerical illustrations are provided for
(ii) and (iii).
