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Now when we are given a constraint, that [tex]x \in M[/tex], where M is a differentiable manifold and when the coordinates on this manifold are [tex]q_i[/tex], then the Lagrange equations look "almost" the same, only the coordinate x is "replaced" by q:

[tex]\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0[/tex]

So now to the question:

I understand how we obtain the Lagrange equations without constraints, but I can't find any proof of the equations with constraints. How is this done? Is the proof difficult (assuming only some basic knowledge of differential geometry).