Deriving the canonical equations of Hamilton

[kesgab]

As i know there are several diffrent way to derive the canonical equations.

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

 

[Lojzek]

In my opinion the Hamiltonian equations don't have any physical meaning until you find the Hamiltonian which corresponds to your system. That derivation only proves that a mathematical system which is defined by least action principle with a certain Lagrangian L can also be defined with corresponding Hamiltonian H and Hamiltonian equations. L and H(L) is not specified yet.
The equations get physical meaning only when we prove experimentally that a certain physical system respects Lagrange/Hamiltonian equations and we define L/H which correctly describes the system. Experiment is not necessary for classical mechanics, since we already know that it respects Newton's laws: it is enough to find L for which the Lagrange equations are equivalen to second Newton's law.