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I was about to answer this question:

Testing the second type of equations of Lagrangian Mechanics

when I refreshed the page and saw it was put on hold as off-topic. Now, it might be a duplicate (although I can't find one) but I don't think it is directly off-topic and is asking a reasonable question: why do the E-L equations not work if you substitute in on-shell values for the velocity?

(OP has clearly recently discovered Lagrangian mechanics and is a bit confused about the whole thing, so they don't make this point clearly but it is the heart of their question. Perhaps this counts as insufficient prior research, although that seems a quite high bar to me.)

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    $\begingroup$ Can you be more specific what about the closing notice is unclear to you? It says: ""Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" $\endgroup$ – ACuriousMind Apr 6 at 18:39
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    $\begingroup$ Note that you can try editing to improve the question if you feel that you can boil it down to the "heart of their question". It's a homework-type question currently but with a bit of work, it can be beaten into a good conceptual question (of course, only if the OP gives their consent). $\endgroup$ – S.D. Apr 6 at 18:53
  • $\begingroup$ @Blue I'll submit an edit. $\endgroup$ – jacob1729 Apr 6 at 19:21
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    $\begingroup$ @ACuriousMind I think it does show some effort (clearly defines coordinate system and shows working for the kinetic and potential parts of Lagrangian) and comes to the conclusion that something is wrong. I think an answer explaining what the issue is (which as I commented, is that they've actually already solved the equations of motion and are substituting the results in to get some sort of 'on-shell' Lagrangian which clearly won't work) would be worthwhile being on the site. Which should count as a conceptual question imo. $\endgroup$ – jacob1729 Apr 6 at 19:23
  • $\begingroup$ Whatever I could have shown by my efforts have already been done. I'm grateful for the editors that they also beautified the contents of the question by understanding that I'm a beginner. That's all I have. If you think that I've missed anything, please tell me so that others can help me knowing the mistake ;) $\endgroup$ – Abhas Kumar Sinha Apr 7 at 2:32
  • $\begingroup$ @AbhasKumarSinha Your question needs to focus on the concepts that are giving you trouble. The last 2 paragraphs kind of do that, except for the very last sentence asking for a review of your work - that turns it into a "check my work" question, which is off-topic. $\endgroup$ – PM 2Ring Apr 7 at 8:15
  • $\begingroup$ IMO from the original question it was hard to get any more clue from the math about how to answer it than "the OP doesn't understand anything about what all this means," which was too broad to be useful. From the revised question, I agree it is possible to see what the OP's conceptual mistake was. $\endgroup$ – alephzero Apr 8 at 10:06
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After various improvements, I agree with OP that his question contains an underlying conceptional issue, and have therefore reopened it.

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