This post is a derivation requested by some students in a course on classical mechanics at my university. Consider the Lagrangian which satisfies the Euler-Lagrange equation
Let be new coordinates such that we can express the as functions of the as
Such changes of variables are known as point transformations. It is relatively easy to show, by variational calculus, that the Euler-Lagrange equation is invariant under point transformations. Here we show this invariance explicitly via the chain rule.
First note that
Since is an independent variable that depends on nothing else, and since the partial derivative is taken with other coordinates fixed, the last term is thus zero so that
Next, consider that
Now consider that
Putting everything together we get