# Proof that the Euler-Lagrange equations hold in any set of coordinates if they hold in one

This is a question about a specific proof presented in the book Introduction to Classical Mechanics by David Morin. I have highlighted the relevant portion in the picture below. In the remark, he does this: \begin{align} \sum_{k=1}^N \frac{\partial }{\partial q_k} \left( \frac{\partial x_i}{\partial q_m} \right) \dot{q_k} = \frac{\partial }{\partial q_m} \sum_{k=1}^N \frac{\partial x_i}{\partial q_k} \dot{q_k} \end{align} He has essentially changed the order of differentiation from $$\frac{\partial }{\partial q_k} \frac{\partial x_i}{\partial q_m}$$ to $$\frac{\partial }{\partial q_m} \frac{\partial x_i}{\partial q_k}$$.

However, isn't this only valid if all the $$q_k$$ are independent?

• "...isn't this a very limited proof?" How would you prefer to generalize it?
– hft
Jun 14 at 20:25

Yes, the generalized coordinates $$(q^1,\ldots, q^N)$$ are assumed to be independent, i.e. no constraints, and the cotangent vectors $$(\mathrm{d}q^1_p,\ldots,\mathrm{d}q^N_p)$$ at each point $$p$$ are linearly independent. (This is a common assumption in textbooks.) The generalized coordinates $$(q^1,\ldots, q^N)$$ constitute an arbitrary local coordinate system for the configuration manifold $$M$$; in particular they may not be orthogonal.
• Actually $x$ and $y$ are still independent since you can move one coordinate without also moving the other. Jun 14 at 20:37