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

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The discussion focuses on solving the classical two-body problem using the Lagrangian principle, specifically addressing the issue of substitutions before taking partial derivatives. The original poster encountered differing results after substituting angular velocity and seeks clarification on appropriate substitutions. They suggest that substitutions involving holonomic constraints may be relevant but request a demonstration to support this. The conversation emphasizes the importance of understanding the implications of these substitutions in the context of Lagrangian mechanics. Overall, the thread aims to clarify the correct approach to substitutions in solving the two-body problem.
EduardoToledo
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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
substitution in euler lagrangian equation.JPG
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EduardoToledo said:
I would like to ask, therefore, which substitutions can I do before taking these partial derivatives
$$q\mapsto f(t,Q),\quad \dot q\mapsto f_t+f_Q\dot Q$$
 

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