My question is what is the theoretical origin of Hamilton's principle. I mean is there any rigorous mathematical proof of this principle from some more basic principles?
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yes, you derive the Lagrangian and apply Legendre transforms. – Courage Nov 19 '15 at 03:37
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1In what way does How Hamilton's Principle was found? fail to answer your question? Or Why the Principle of Least Action? – dmckee --- ex-moderator kitten Nov 19 '15 at 04:04
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1More possible duplicates: http://physics.stackexchange.com/q/9/2451 and links therein. Specifically, the derivation of Hamilton's principle from Newton's laws is discussed in http://physics.stackexchange.com/q/78138/2451 – Qmechanic Nov 19 '15 at 06:53