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

so that

Now consider that

The first term is zero from (2). Substituting (6) we get

Hence

Putting everything together we get

Q.E.D.

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