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This question was recently closed as Too broad. While I see how that would be the case if the question is interpreted as

Explain the entire theories of Newtonian, Lagrangian and Hamiltonian mechanics,

I think the question that's being asked is

How do the differences in these three methods manifest themselves the solving of a problem?

and that seems to me like a decent question which could probably be addressed in a reasonably sized answer. In fact, I was half-way through writing an answer when the question was closed. I must say it was slightly disappointing to think my time on that answer was wasted, so perhaps I am prejudiced. :)

How do you see this? Is the question too broad? If so, can it be rephrased (I've already made a humble attempt myself) or is it unsalvageable without significantly altering the core question?

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The key statement of the original question (v1) is

Can you show me a problem and solved with basic characteristics of Newtonian, Lagrangian and Hamiltonian methods. like
Problem : ........................
Solution with Newtonian : ....................
Solution with Lagrangian : ........................
Solution with Hamiltonian : ......................

There are several reasons I think this is a bad question:

  • Any time someone says effectively "answer the question in this format," it's a strong (though not definite) indicator of an asker who wants to be spoon-fed, and is not going to put in the effort to understand a reasonable answer. This is supported by the lack of any reference to an attempt by the OP to answer the question themselves, or even to look up a suitable problem and solutions in a textbook or problem book. So I think it's very clear that the OP has not put in anywhere close to sufficient effort to craft a good question for this site. As far as I'm concerned, this warrants a downvote, and one could argue also closure under the homework-like reason (though that could be a bit of a stretch). If we were to have an "insufficient effort" close reason then this would certainly qualify.

  • This question, and in particular the way in which the question is asked, suggests it's being asked by someone who has not studied Lagrangian and Hamiltonian mechanics. And in fact the poster says as much earlier in the question. That means they won't have the expertise required to make sense of the kind of answer we'd give them. What they need for this particular misunderstanding is a textbook or a course in classical mechanics, not a question on this site, and thus any useful answer to their question is likely to be very long (in an attempt to incorporate much of the material that a class would cover). That's part of the description of the "too broad" close reason.

    When I say this is a site for expert-level questions, this is the kind of thing I mean to exclude, and thus for this reason as well I'm inclined to downvote the question.

  • Beyond that, the question is very much non-specific. It doesn't give us any indication of what the asker is really confused about, and in particular it doesn't address any sort of conceptual misunderstanding. Without that information, it will be a shot in the dark for us to provide a problem and solutions that are actually helpful. This qualifies it to be closed as "unclear what you're asking" in my opinion.

  • And finally, there's an endless list of possible answers, which is one of the criteria mentioned explicitly in the description for the "too broad" close reason. Now, of course any question can have a large number of possible answers, if you go by word-for-word comparison, but usually there is something in the question that will cause all possible correct answers to basically be saying the same thing. In this case, if the OP had identified some aspect of the various formalisms that they were confused about (as I mentioned in the previous bullet point), that would give the answers something to address and perhaps excuse the question from being too broad, but in this revision there is nothing of the sort.

As for the rephrased version (v3): the key statement is

Can you use an example to outline the basic characteristics (and key differences) of Newtonian, Lagrangian and Hamiltonian methods in their approach to classical mechanics problems?

This is certainly an improvement over v1, as it addresses the formatting issue from the first bullet point above, and partially the third, in the sense that it starts to give a clue as to why the asker is confused. But the second and fourth, and part of the third, still stand. In particular, the question is still being asked by someone who is not prepared to receive a reasonable answer, and there's still nothing sufficiently specific identifying a particular concept that confuses the OP. Any edit we could make to remedy that would involve a significant amount of guessing on the editors' part as to what the OP really wanted to know. So I don't think it's possible to fix this question through editing without intervention by the OP, and thus I think this should remain on hold unless and until the OP does edit it in a way that addresses these issues.

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    $\begingroup$ I'm inclined to agree. OP likely needs a textbook or a course, not an answer on this site. Also, I agree on the level-argument - I hadn't really thought about that. I think it's a question worth asking, but it shouldn't (can't) be addressed here and OP doesn't seem to have sufficient background knowledge. My answer was going to be an overview of each method for the very simple (but important) example of a harmonic oscillator, where the differences aren't too big but I could point them out nonetheless. However, I cannot be sure this would answer OP's question without them rephrasing it. $\endgroup$ – Wouter Apr 29 '14 at 19:42
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If I were to answer that question my answer would look something like what appears below. The problem I see with that kind of answer is that it is only useful if you already have experience with these methods so that you can relate to the amount of work (and level of creativity/cleverness) each step demands.

An alternate approach is to claim that Newtonian physics is concerned with forces, Lagrangian with energies and Hamiltonian with momenta, but again that is not helpful unless the reader has a framework to hang the explanation on and beginning students may incorrectly feel that I mean the treatment of energy and momenta in the Newtonian mechanics.

In both cases, the problem is that the reader has to understand before they can understand and that calls for an entire chapter with exercises in a textbook.


Problem solving steps in each formalism

Newtonian

  1. Draw picture, choose coordinate system
  2. Identify forces (i.e. free body diagrams go here)
  3. Write the algebraic relationships between various quantities in the system
  4. Bash the algebra and calculus until you get to the answer

Lagrangian

  1. Draw picture, choose generalized coordinates
  2. Write down the kinetic and potential energies in terms of the generalized coordinate (and thereby get the Lagrangian). Identify what form (1st, 2nd, undetermined multiplies) you want to use
  3. Extract derivatives and write down Equations of Motion
  4. Bash the system of PDEs until you have solved the system

Hamiltonian

  1. Draw picture, choose generalized coordinates
  2. Write down the kinetic and potential energies in terms of the generalized coordinate (and thereby get the Lagrangian).
  3. Extract the generalized momenta from the Lagrangian
  4. Invert the momenta and write down the Hamiltonian
  5. Extract the derivatives of the Hamiltonian thus arriving at the remaining Equations of Motion
  6. Bash the system of PDEs until you have solved the system
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  • $\begingroup$ I would probably add what are they "usually" useful for: newtonian when there is friction, hamiltonian for perturbation and Hamilton-Jacobi (difficult problems), lagrangian for anything in between. $\endgroup$ – Davidmh May 12 '14 at 18:51
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I think it is a little broad, but mostly because the OP wants examples of solving the same problem done in the three methods which can take up quite some space (especially if you chose a problem that is rather complicated in one method to show the power of the others).

If we neglected that aspect, the question would then be,

What are the key differences between Newtonian, Lagrangian, and Hamiltonian dynamics?

which probably could be answered easily and quickly enough, but it might be an "insufficient effort" type question and remain closed.

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    $\begingroup$ "because the OP wants examples of solving the same problem done in the three methods" Surely every source that explain Lagrangian and Hamiltonian mechanics does a example or two of a system that the reader already knows how to do in the Newtonian formalism, anyway. At least I'm not aware of one that doesn't. $\endgroup$ – dmckee Apr 29 '14 at 14:56
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    $\begingroup$ I wouldn't consider your rephrasing an "insufficient effort" type question. It might still be too broad, but certainly much closer to being okay than it is now. $\endgroup$ – David Z Apr 29 '14 at 17:41
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    $\begingroup$ @DavidZ: I think that if they spent any time looking at the three methods, they'd easily see the key differences and wouldn't need to ask such a question. $\endgroup$ – Kyle Kanos Apr 29 '14 at 17:45
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    $\begingroup$ @KyleKanos perhaps. But I think a person who has spent a considerable amount of time looking at the three methods would see the key differences; not necessarily so with someone who is still learning the subject. $\endgroup$ – David Z Apr 29 '14 at 17:46
  • $\begingroup$ @DavidZ: That is possibly true, though I recall wondering why we were using energy instead of forces when I first encountered LM back in undergrad. Maybe I should also point out that I did state might in my post and I think in my comment above, meaning these are my opinions and not set in stone answers. $\endgroup$ – Kyle Kanos Apr 29 '14 at 17:49

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