Hi, A fundamental aspect in the Hamiltonian framework of mechanics is that the q's and p's are independent. I feel like I understand the steps in the Legendre transform from Lagrangian to Hamiltonian mechanics, but I don't see how you can go from a system where only the q's are independent (the qdots are just time derivatives of the q's) to one where both q and p are independent. In thermodynamics Legendre transforms you always go from from sets of 2 independent quantities to other sets of 2 independent quantities. It seems to be saying that in Lagrangian mechanics q and qdot are independent, but I can't seem to make sense of that. Any help would be appreciated!