Hamiltonian mechanics is an elegant way of formulating problems is classical mechanics. Also, it provides insight into the world of quantum mechanics, as is evident with the Schrodinger equation.

The basic equations:

$$\dot{q}=\frac{\partial }{\partial p}H$$

$$\dot{p}=-\frac{\partial }{\partial q}H$$

Where $$q=q(t)$$ are the *generalized coordinates* and $$p=p(t)$$ are the *generalized momenta*. H is the *Hamiltonian* and represents the total energy of the closed system (i.e. conservative) under consideration. $$H=T+V$$ where T is the *kinetic* energy and V is the *potential* energy. Also note that $$\dot{q}$$ represents the time derivative of the position, or the *velocity*, and that $$\dot{p}$$ represents the time derivative of the momentum, or the *force*.

These two equations yield the evolution of the mechanical system. Aesthetically they are quite pleasing, owing to their nearly perfect symmetry.