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Some of them starts from a physical principle like Hamilton's principle or the Lagrange equations.

But it can be derived also by simply make a Legendre transormation on the Lagrange function and then make derivatives on it. I don't understand how can we end up having equations that have physical meaning when we don't start from a physical law or principle and don't use any during the derivation. We actually don't state anything during this derivation.

(On Wikipedia - Hamiltonian mechanics, you can see this two kind of derivation, one uses Lagrange equations and the other is just mathematical.)

If somebody has any thought about this, i would be glad to hear, because i has been thinking on this for a couple of days now and haven't been able to come up with a solution.

Thanks,

kesgab