So let's say we have a mechanical system described by some Lagrangian [itex]L=L(q_i,\dot{q}_i)[/itex], where the q_{i}'s are the generalized coordinates of the system. Does the condition [tex]\frac{\partial L}{\partial q_i}=0[/tex] give the equilibrium configurations of the system? Intuitively it seems so, but I can't prove it.
[tex]\frac{\partial V}{\partial x}=0./tex]. I'm having trouble connecting this to Lagrangians though....