  This paper is concerned with the numerical approximation of stochastic
mechanical systems with nonlinear holonomic constraints. Such systems are
described by second order stochastic differential-algebraic equations involving
an implicitly given Lagrange multiplier process. The explicit representation of
the Lagrange multiplier leads to an underlying stochastic ordinary differential
equation, the drift coefficient of which is typically not globally one-sided
Lipschitz continuous. We investigate a half-explicit drift-truncated Euler
scheme which fulfills the constraint exactly. Pathwise uniform
$L_p$-convergence is established. The proof is based on a suitable
decomposition of the discrete Lagrange multipliers and on norm estimates for
the single components, enabling the verification of consistency, semi-stability
and moment growth properties of the scheme. To the best of our knowledge, the
presented result is the first strong convergence result for a
constraint-preserving scheme in the considered setting.
