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Why is the Hamiltonian constructed from the Lagrangian?

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I understand how to use Hamiltonian mechanics, but I never understood why you construct the Hamilitonian by first constructing the Langrangian, and then performing a Legendre transform on it.

Why can't you just construct the Hamiltonian directly? Does it have to do with the generalized coordinate [itex] p_i [/itex] being more difficult to think about than the coordinate [itex] \dot{q_i} [/itex]? This seems like the only reason to start with the Lagrangian first, if your goal is to build the Hamiltonian.

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