In a system of N classical point particles, for each particle i there will
be a position vector 
and its canonically conjugate momentum
vector 
. Decomposing these vectors into Cartesian coordinates gives 
and 
.
For all N particles, a useful notation for position is 
, and for momentum 
. For extended
particles each particle will have additional internal coordinates and their
canonically conjugate momenta. The complete set of Hamiltonian coordinates of a system is
. For a function, the corresponding notation is
A point particle requires 6 coordinates 
for its complete
description, so an N-particle set of point particles is characterized by 6N
coordinates. Non-point particles, such as most molecules, require some number
m (m > 6) coordinates to be described, so for N extended particles mN
coordinates are needed. Just as the three position coordinates (x, y, z) of
a pointlike atom can be envisioned as a point in a 3-dimensional space, so also
the 6N coordinates needed to describe N pointlike particles can be
imagined as a single point in a 6N-dimensional space. This 6N-dimensional
space is known as phase space. A single point in phase space corresponds
to a list of the values of 6N-coordinates. Since the position of the point
in phase space fixes the 6N variables that comprise a complete microscopic
description of the system, each point in phase space corresponds uniquely to a
single state of the system, and vice versa: every allowed point in phase space
corresponds to a state of the system. A ``point in phase space'' is known
as a phase point, whose customary notation is 
. The value of 
, i. e. the precise state of the
system, was called by Gibbs the configuration in phase. (For extended
particles, the 6N-dimensional phase space is replaced by an mN-dimensional
phase space.) An integral over all 6N coordinates of phase space is a
phase space integral. The subspace spanned by 
is known as
position space; the subspace spanned by 
is known as
momentum space.
For a system of allowed volume V the volume integral has a simplified notation
the rhs being compressed notation for a triple integral over a volume. A further compression expresses an integral over all the particle coordinates, namely
An identical series of replacements transform the integral over particle momenta, namely
The 
is a three-dimensional integral, whose implicit bounds
range from 
to 
in all directions. Also in use is the
notation
This notation will appear gradually in future Lectures.
