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:
Where 
are the generalized coordinates and 
are the generalized momenta. H is the Hamiltonian and represents the total energy of the closed system (i.e. conservative) under consideration. 
where T is the kinetic energy and V is the potential energy. Also note that 
represents the time derivative of the position, or the velocity, and that 
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.
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