Hamiltonian Mechanics: Another Alternative to the Traditional Newtonian Formulation
Exploring beyond Newtonian and Lagrangian mechanics
Apart from the Newtonian and Lagrangian formulation of classical mechanics, there also exist yet another formulation known as Hamiltonian mechanics which, although similar to the Lagrangian formalism, can be even more versatile in certain situations.
For this post, I would like to briefly introduce Hamiltonian mechanics by deriving the necessary equations and elaborating on various advantages it has over the Lagrangian formalism (which you can review in this post).
Brief Review of Lagrangian Mechanics
As Hamiltonian mechanics contains virtually the same information as Newtonian and Lagrangian mechanics, we are able to derive the equations for Hamiltonian mechanics from them. In fact, due to how similar Hamiltonian and Lagrangian mechanics is, deriving from the Lagrangian equations happens to be the most natural approach.
Therefore, as a quick review of Lagrangian mechanics, we can recall that this formalism hinges on a certain mathematical object known as the ‘Lagrangian’ which, in most cases, is just defined as T - U, the kinetic energy minus the potential energy. This will result in the…
