I understand how the legendre transform of the Lagrangian with respect to $\dot{q}$ yields the hamiltonian, but I do not understand why one would think to do this in the first place? what is the deeper reason for taking the legendre transform of the lagrangian?
While reading David Morin's Introduction to classical mechanics he says:
Motivation:
Let’s now take a step back and ask why we should consider applying a Legendre
transform to the Lagrangian in the first place. There are some deeper reasons for
considering a Legendre transform, but a practical motivation is the following...
He then makes an argument that we can make the Euler Lagrange equations look more symmetric through the Hamiltonian, but what are the other deeper reasons that he might have been hinting at?
This answer does not touch on the fundamental motivation for using the Legendre transform, also this answer again does not argue generally what the legendre transform helps us obtain and why it is a useful tool in this case.
