8 Classical mechanics
conditions give rise to different values ofC, which changes the size of the ellipse.
Changing the mass and frequency changes the shape of the ellipse.Phase space plots
determine the values of position and momentum the system will visit along a trajectory
for a given set of initial conditions. These values constitute theaccessible phase space.
For a free particle, the accessible phase space is unbounded sincexlies in the interval
x∈[x(0),∞) forp >0 orx∈(−∞,x(0)] forp <0. The harmonic oscillator, by
contrast, provides an example of a phase space that is bounded.
p -p(a)xpxpx(b) (c) (d) pFig. 1.4Phase space of a one-dimensional particle subject to the “hill” potential: (a) Two
particles approach the hill, one from the left, one from the right. (b) Phase space plot if the
particles have insufficient energy to roll over the hill. (c) Same if the energy is just sufficient
for a particle to reach the top of the hill and come to rest there. (d) Same if the energy is
greater than that needed to roll over the hill.
Consider, finally, the example of a particle of massmrolling over a hill under the
influence of gravity, as illustrated in Fig. 1.4(a). (This example is a one-dimensional
idealization of a situation that should be familiar to anyone who has ever played
miniature golf and also serves as a paradigm for chemical reactions.)We will assume
that the top of the hill corresponds to a positionx= 0. The force law for this problem
is non-linear, so that a simple, closed-form, analytical solution to Newton’s second
law is not readily available. However, an analytical solution is not needed in order to
visualize the motion using a phase space picture. Several kinds of motion are possible
depending on the initial conditions. First, if the particle is not rolled quickly enough,
it cannot roll completely over the hill. Rather, it will climb part way up the hill and
then roll back down the same side. This type of motion is depicted in the phase space
plot of Fig. 1.4(b). Note that the plot only shows the motion in a regionclose to the
hill. A full phase space plot would extend tox=±∞. On the other hand, if the initial
speed is high enough, the particle can reach the top of the hill and roll down the other
side as depicted in Fig. 1.4(d). The crossover between these two scenarios occurs for