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As far as I understand it:

Hamiltons principle:

δI = δ∫Ldt = 0

is only satisfied for systems, where the generalized forces can be put in a form such that the equation:

d/dt([itex]\partial[/itex]L/[itex]\partial[/itex]q') - [itex]\partial[/itex]L/[itex]\partial[/itex]q = 0

is satisfied. I have seen that this can be done for electromagnetic systems and system in which the force can be written as the gradient of a potential with respect to the position coordinates.

My question is: Are all the fundamental forces observed in nature in a form like this? And is that then the deep content in Hamiltons principle?