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Motion in a general 1dimensional potential
Suppose that the curve in Fig. 43 represents the potential
energy of some mass moving in a 1dimensional conservative forcefield.
For instance, might represent the gravitational potential energy
of a cyclist freewheeling in a hilly region. Note that we have set the
potential energy at infinity to zero. This is a useful, and quite common, convention (recall that
potential energy is undefined to within an arbitrary additive constant).
What can we deduce about the motion of the mass in this potential?
Figure 43:
General 1dimensional potential

Well, we know that the total energy, which is the sum of the kinetic
energy, , and the potential energy, is a constant of the motion.
Hence, we can write

(170) 
Now, we also know that a kinetic energy can never be negative, so the above
expression tells us that the motion of the mass is restricted to the
region (or regions) in which the potential energy curve falls
below the value . This idea is illustrated in Fig. 43.
Suppose that the total energy of the system is . It is clear, from
the figure, that the mass is trapped inside one or other of the two dips
in the potentialthese dips are
generally referred to as potential wells.
Suppose that we now raise the energy to . In this
case, the mass is free to enter or leave each of the potential wells, but
its motion is still bounded to some extent, since it clearly cannot move off to
infinity. Finally, let us raise the energy to . Now the
mass is unbounded: i.e., it can move off to infinity. In systems
in which it makes sense to adopt the convention that the potential
energy at infinity is zero, bounded systems are characterized
by , whereas unbounded systems are characterized by .
The above discussion suggests that the motion of a mass moving in a potential
generally becomes less bounded as the total energy of the system increases.
Conversely, we would expect the motion to become more bounded as decreases.
In fact, if the energy becomes sufficiently small, it appears likely that the
system will settle down in some equilibrium state in which the mass is stationary.
Let us try to identify any prospective equilibrium states in Fig. 43.
If the mass remains stationary then it must be subject to zero force (otherwise
it would accelerate). Hence, according to Eq. (164), an equilibrium
state is characterized by

(171) 
In other words, a equilibrium state corresponds to either a maximum
or a minimum of the potential energy curve . It can
be seen that the curve shown in Fig. 43 has
three associated equilibrium states: these are located at
, , and .
Let us now make a distinction between stable equilibrium points
and unstable equilibrium points. When the system is slightly
perturbed from a stable equilibrium point then the resultant force
should always be such as to attempt to return the system to this point.
In other words, if is an equilibrium point, then we require

(172) 
for stability: i.e., if the system is perturbed to the right, so that ,
then the force must act to the left, so that , and vice versa.
Likewise, if

(173) 
then the equilibrium point is unstable. It follows, from
Eq. (164), that stable equilibrium points are
characterized by

(174) 
In other words, a stable equilibrium point corresponds to a minimum
of the potential energy curve . Likewise, an unstable
equilibrium point corresponds to a maximum of the curve. Hence,
we conclude that and are stable equilibrium points,
in Fig. 43, whereas is an unstable equilibrium point.
Of course, this makes perfect sense if we think of as
a gravitational potential energy curve, in which case is
directly proportional to height. All we are saying is that it is
easy to confine a low energy mass at the bottom of a valley,
but very difficult to balance the same mass on the top of
a hill (since any slight perturbation to the mass will cause it
to fall down the hill). Note, finally, that if

(175) 
at any point (or in any region) then we have what is known as a neutral equilibrium
point. We can move the mass slightly off such a point and it will still
remain in equilibrium (i.e., it will neither attempt to return to
its initial state, nor will it continue to move). A neutral equilibrium point
corresponds to a flat spot in a curve. See Fig. 44.
Figure 44:
Different types of equilibrium

Next: Power
Up: Conservation of energy
Previous: Hooke's law
Richard Fitzpatrick
20060202