## Does a Lagrangian preserving transformation obey the equations of motion?

This seems like such a simple question that I fully expect its solution to be embarrassingly easy, but try as I might I can't get the answer.

Consider some system which can be described by N generalized coordinates $q_1,...,q_N$ and a Lagrangian $L(q_i,\dot{q}_i,t)$. (I'll just use $q_i$ as a stand in for $q_1,...,q_N$). Let $q_i(t)$ be a solution to Lagrange's equations ie an actual possible trajectory through phase space that the system can follow.

Now we make the transformation $q_i(t) \rightarrow Q_i(t)$ such that the Lagrangian doesn't change. I want to prove that $Q_i(t)$ also satisfies Lagrange's equations.

This seems like it'd be so trivial to prove, and it probably is, but I can't brain today (or yesterday, apparently) and would appreciate your help.
 it's not trivial to prove, in fact it's quite the opposite. this is the beggining of the proof of Noether's theorem. I personally don't remember the proof, but you can google it easily.
 Unfortunately, I was motivated to ask this question because the proof of Noether's theorem in my textbook asserted this without proof! And all other proofs that I've seen are constructed to avoid the problem.