6 Classical mechanics
instanttin time, all of the information about the system is specified by 6Nnumbers
(or 2dNinddimensions). These 6Nnumbers constitute themicroscopic stateof the
system at timet. That these 6Nnumbers are sufficient to characterize the system
follows entirely from the fact that they are all that is needed to seed eqns. (1.2.10),
from which the complete time evolution of the system can be determined.
Suppose, at some instant in time, the positions and momenta of the system are
{r 1 ,...,rN,p 1 ,...,pN}. These 6Nnumbers can be regarded as an ordered 6N-tuple or
a single point in a 6N-dimensional space calledphase space. Although the geometry
of this space can, under certain circumstances, be nontrivial, in itssimplest form, a
phase space is a Cartesian space that can be constructed from 6Nmutually orthogonal
axes. We shall denote a general point in the phase space as
x = (r 1 ,...rN,p 1 ,...,pN) (1.3.3)also known as thephase space vector. (As we will see in Chapter 2, phase spaces play
a central role in classical statistical mechanics.) Solving eqns. (1.2.10) generates a set
of functions
x(t) = (r 1 (t),...,rN(t),p 1 (t),...,pN(t))≡xt, (1.3.4)which describe a parametric path ortrajectoryin the phase space. Therefore, classical
motion can be described by the motion of a point along a trajectory inphase space.
Although phase space trajectories can only be visualized for a one-particle system in
one spatial dimension, it is, nevertheless, instructive to study several such examples.
Consider, first, a free particle with coordinatexand momentump, described by
the one-dimensional analog of eqn. (1.2.7), i.ex(t) =x(0) + (p/m)t, wherepis the
particle’s (constant) momentum. A plot ofpvs.xis simply a straight horizontal line
starting atx(0) and extending in the direction of increasingxifp >0 or decreasingx
ifp <0. This is illustrated in Fig. 1.2. The line is horizontal becausepis constant for
allxvalues visited on the trajectory.
xpx( 0 )p > 0p < 0Fig. 1.2Phase space of a one-dimensional free particle.