Equivalent to Lagrangian mechanics but formulated in a different way.
Motivation: Lagrangian vs Hamiltonian.
TODO understand original historical motivation, www.youtube.com/watch?v=SZXHoWwBcDc says it is from optics.
Intuitively, the Hamiltonian is the total energy of the system in terms of arbitrary parameters, a bit like Lagrangian mechanics.
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The key difference from Lagrangian mechanics is that the Hamiltonian approach groups variables into pairs of coordinates called the phase space coordinates:This leads to having two times more unknown functions than in the Lagrangian. However, it also leads to a system of partial differential equations with only first order derivatives, which is nicer. Notably, it can be more clearly seen in phase space.
- generalized coordinates, generally positions or angles
- their corresponding conjugate momenta, generally velocities, or angular velocities
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Analogous to what the Euler-Lagrange equation is to Lagrangian mechanics, Hamilton's equations give the equations of motion from a given input Hamiltonian:
So once you have the Hamiltonian, you can write down this system of partial differential equations which can then be numerically solved.
