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.
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.