The usual way to derive Hamilton's equations is to perform Legendre transformation of the Lagrangian and then use the stationarity principle. However, this procedure seems a little artificial to me from a "physical" perspective, because in this case the integrand of the action is basically a rewritten Lagrangian, and it appears to be impossible to conceive the former without considering the latter first. Lagrange's approach, on the other hand, is fully self-sufficient. My question is, is there some way to derive Hamilton's equations from a number of general ideas completely independently?
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By "from a number of general ideas," do you mean that you are allowing different sets of assumptions than the usual principle of least action? And what do you mean by "completely independently"? If we use something that is equivalent to the Lagrangian but we don't call it the Lagrangian, does that count? – probably_someone Jan 23 '19 at 13:58
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Derive Hamilton's equations completely independently from what? Related: https://physics.stackexchange.com/q/105912/2451 – Qmechanic Jan 23 '19 at 13:59
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@probably_someone It would be ideal to see the usual principle of least action from another perspective, but yes, considering different sets of assumptions is equally interesting. Something that is literally equivalent to the Lagrangian doesn't count. – Dmst Jan 23 '19 at 14:22