## The Hamiltonian

The Hamiltonianrepresents the energy of the system which is the sum of kinetic and potential energy, labelledandrespectively. For a one dimensional system, we may write

so

whereNote thatis a function ofonly andis a function ofonly. In generalis a function ofonly andis a function of the coordinates, however they are defined.

The value of the Hamiltonian is the total energy of the system. For a closed system, it is the sum of the kinetic and potential energy in the system and is conserved. The Hamiltonian equations give the time evolution of the system. These are

whereand

The time-derivative of the momentumequals the force acting so the first Hamilton equation means that the force on the particle equals the rate at which it loses potential energy with respect to changes inits position.

The time-derivative ofhere means the velocity: the second Hamilton equation here means that the particle’s velocity equals the derivative of its kinetic energy with respect to its momentum.

We can derive Hamilton's equations by looking at how the total differential of the Lagrangian depends on time, generalized positionsand generalized velocities

(1)
Now useandto give

Substitute these into (1):which we can rewrite asand rearrange to get

The term on the left-hand side is the Hamiltonian so

where the second equality holds because it is equal toAssociating terms from both sides of the equation above yields Hamilton's equations

and