The Lagrangian formulation of mechanics will be useful later when we
study the Feynman path integral. For our purposes now, the Lagrangian
formulation is an important springboard from which to develop
another useful formulation of classical mechanics known
as the *Hamiltonian* formulation. The Hamiltonian of a system
is defined to be the sum of the kinetic and potential energies
expressed as a function of positions and their *conjugate momenta*.
What are conjugate momenta?

Recall from elementary physics that momentum of a particle, , is defined in terms of its velocity by

In fact, the more general definition of conjugate momentum, valid for any set of coordinates, is given in terms of the Lagrangian:

Note that these two definitions are equivalent for Cartesian variables.

In terms of Cartesian momenta, the kinetic energy is given by

Then, the Hamiltonian, which is defined to be the sum, *K*+*U*, expressed
as a function of positions and momenta, will be given by

where . In terms of the Hamiltonian, the
equations of motion of a system are given by *Hamilton's equations*:

The solution of Hamilton's equations of motion will yield a trajectory in terms of positions and momenta as functions of time. Again, Hamilton's equations can be easily shown to be equivalent to Newton's equations, and, like the Lagrangian formulation, Hamilton's equations can be used to determine the equations of motion of a system in any set of coordinates.

The Hamiltonian and Lagrangian formulations possess an interesting
connection.
The Hamiltonian can be directly obtained from the Lagrangian by
a transformation known as a *Legendre transform*. We will
say more about Legendre transforms in a later lecture. For now,
note that the connection is given by

which, when the fact that is used, becomes

Because a system described by conservative forces conserves the total energy, it follows that Hamilton's equations of motion conserve the total Hamiltonian. Hamilton's equations of motion conserve the Hamiltonian

**Proof**:

QED. This, then, provides another expression of the law of conservation of energy.

Wed Jan 8 22:51:23 EST 2003