# Lagrangian with non-holonomic constraints

Given a smooth manifold $M$ and a smooth Lagrangian $\mathcal{L}(x,\dot{x})$ on $M$, the curves which make stationary the corresponding action are those which solve the Euler-Lagrange equations.

If a non-holonomic constraint is added $F(\dot{x})=0$ for all times, what then are the EL equations?

Strictly speaking there are no "correct" equations. The Euler--Lagrange equations correspond to a vector field $X_\textrm{EL}$ on $TM$. The constraints define a subbundle (or a distribution) $\mathcal{D} \subset TM$ to which the dynamics should be restricted. However, in general, the Euler--Lagrange vector field $X_\textrm{EL}$ is not tangent to $\mathcal{D}$, so does not leave it invariant. Thus one needs to project in some way $X_\textrm{EL}|_\mathcal{D}$ onto $T\mathcal{D}$. There are in principle many ways to do this and d'Alembert's principle is one of these; for example, another way is the vakonomic principle, which means enforcing the constraints before doing variations.