In order to begin to make a connection between the microscopic and macroscopic worlds, we need to better understand the microscopic world and the laws that govern it. We will begin placing Newton's laws of motion in a formal framework which will be heavily used in our study of classical statistical mechanics.

First, we begin by restricting our discussion to systems for which
the forces are purely *conservative*. Such forces are derivable
from a potential energy function by differentiation:

It is clear that such forces cannot contain dissipative or friction terms. An important property of systems whose forces are conservative is that they conserve the total energy

To see this, simply differentiate the energy with respect to time:

where, the second line, the facts that (Newton's
law) and (conservative force definition)
have been used. This is known as the law of *conservation of energy*.

For conservative systems, there is an elegant formulation of classical
mechanics known as the *Lagrangian* formulation. The *Lagrangian*
function, *L*, for a system is defined to be the difference between the
kinetic and potential energies expressed as a function of positions
and velocities. In order to make the nomenclature more compact, we
shall introduce a shorthand for the complete set of
positions in an *N*-particle system: and
for the velocities: .
Then, the Lagrangian is defined as follows:

In terms of the Lagrangian, the classical equations of motion are
given by the so called *Euler-Lagrange* equation:

The equations that result from application of the Euler-Lagrange
equation to a particular Lagrangian are known as the
*equations of motion*. The solution of the
equations of motion for a given initial condition
is known as a *trajectory* of the system.
The Euler-Lagrange equation results from what is known as
an *action principle*. We shall defer further discussion of
the action principle until we study the Feynman path integral
formulation of quantum statistical mechanics in terms of which
the action principle emerges very naturally. For now, we accept
the Euler-Lagrange equation as a definition.

The Euler-Lagrange formulation is completely equivalent to Newton's second law. In order to see this, note that

Therefore,

which is just Newton's equation of motion.

An important property of the Lagrangian formulation is that it can
be used to obtain the equations of motion of a system in
*any set of coordinates*, not just the standard Cartesian
coordinates, via the Euler-Lagrange equation (see problem set #1).

Wed Jan 8 22:51:23 EST 2003