Solving 2-Body Problem w/ Lagrangian: What Substitutions?

[EduardoToledo]

Hi, I was trying to solve the classical two body problem with Lagrangian Principle. I replaced the angular velocity before taking the partial derivatives (which respect to the distance to the virtual particle) and the result was completely different. I would like to ask, therefore, which substitutions can I do before taking these partial derivatives. I think the answer may be "the ones with holonomic constraints", but I really would like the demonstration for that

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[wrobel]

I would like to ask, therefore, which substitutions can I do before taking these partial derivatives

$$q\mapsto f(t,Q),\quad \dot q\mapsto f_t+f_Q\dot Q$$