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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.

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  • $\begingroup$ Possible duplicate: What's the point of Hamiltonian mechanics? $\endgroup$
    – Qmechanic
    Commented Aug 7, 2022 at 19:03
  • $\begingroup$ Related : A mathematically illogical argument in the derivation of Hamilton's equation in Goldstein. $\endgroup$
    – Frobenius
    Commented Aug 7, 2022 at 20:27
  • $\begingroup$ Some additional motivation. If you start by looking at $\frac{dL}{dt}$ in the case when $\frac{\partial L}{\partial t}=0$ you will see that the quantity $\dot q \frac{\partial L}{\partial \dot q}-L$ is constant. This quantity is also called the Hamiltonian. This at least provides some general motivation for why you might want to work with the Hamiltonian. $\endgroup$
    – hft
    Commented Aug 8, 2022 at 21:40
  • $\begingroup$ (The question is still a duplicate. There are at least a few different questions/answer already on this website that are duplicative still.) $\endgroup$
    – hft
    Commented Aug 8, 2022 at 21:41
  • $\begingroup$ It's really not a motivation at all. My question Is rather on the nature of legendre transforms and why one when formulating a theory might motivate one to apply it. It would be great to not be dismissive of the question and see what discussions can arise from it. $\endgroup$ Commented Aug 9, 2022 at 2:30

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