> I've never been able to understand if there is something "truly fundamental" about H compared to L, or if H is more of a mathematical convenience for making the equations first-order.
Actually, Hamiltonian formulation, being equivalent, offers more room for finding solutions. Lagrangian formulation of the Least Action principle allows you to search for a solution employing arbitrary smooth re-parameterizations of the configuration variables `q`. The Hamiltonian formulation, on the other hand, allows you to re-parametrize the entire phase space (q,p) and find solutions that are much harder to get in Lagrangian formulation.
That sounds interesting. Do you know of a particular example where that helps?
I guess maybe it's 'all of them'. The pedagogy on Hamiltonian mechanics had been strangely hard for me to learn beyond the elementary level, like it doesn't make enough sense for my brain to organize it in a memorable way.
Actually, Hamiltonian formulation, being equivalent, offers more room for finding solutions. Lagrangian formulation of the Least Action principle allows you to search for a solution employing arbitrary smooth re-parameterizations of the configuration variables `q`. The Hamiltonian formulation, on the other hand, allows you to re-parametrize the entire phase space (q,p) and find solutions that are much harder to get in Lagrangian formulation.