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.(adsbygoogle = window.adsbygoogle || []).push({});

Consider some system which can be described by N generalized coordinates [itex]q_1,...,q_N[/itex] and a Lagrangian [itex]L(q_i,\dot{q}_i,t)[/itex]. (I'll just use [itex]q_i[/itex] as a stand in for [itex]q_1,...,q_N[/itex]). Let [itex]q_i(t)[/itex] 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 [itex]q_i(t) \rightarrow Q_i(t)[/itex] such that the Lagrangian doesn't change. I want to prove that [itex]Q_i(t)[/itex] 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.

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

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