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Motion in a General OneDimensional Potential
Consider a point particle of mass moving in the direction, say,
under the action of some directed force .
Suppose that is a conservative force: e.g., gravity. In this case,
according to Equation (19), we can write

(45) 
where is the potential energy of the particle at position .
Let the curve take the form shown in Figure 2.
For instance, this curve might represent the gravitational potential energy
of a cyclist freewheeling in a hilly region. Observe that we have set the
potential energy at infinity to zero (which we are generally
free to do, since
potential energy is undefined to an arbitrary additive constant). This is a fairly common convention.
What can we deduce about the motion of the particle in this 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 motionsee Equation (22).
Hence, we can write

(46) 
However, we also know that a kinetic energy can never be negative [since
, and neither
nor can be negative]. Hence, the above
expression tells us that the particle's motion is restricted to the
region (or regions) in which the potential energy curve falls
below the value . This idea is illustrated in Figure 2.
Suppose that the total energy of the system is . It is clear, from
the figure, that the particle 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 particle 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
particle is unbounded: i.e., it can move off to infinity. In conservative 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 .
Figure 2:
A potential energy curve.

The above discussion suggests that the motion of an particle 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 then it appears likely that the
system will settle down in some equilibrium state in which the particle is stationary.
Let us try to identify any prospective equilibrium states in Figure 2.
If the particle remains stationary then it must be subject to zero force (otherwise
it would accelerate). Hence, according to Equation (45), an equilibrium
state is characterized by

(47) 
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 Figure 2 has
three associated equilibrium states located at
, , and .
Let us now make a distinction between stable equilibrium points
and unstable equilibrium points. When the particle is slightly
displaced from a stable equilibrium point then the resultant force acting
on it
must always be such as to return it to this point.
In other words, if is an equilibrium point then we require

(48) 
for stability: i.e., if the particle is displaced to the right, so that ,
then the force must act to the left, so that , and vice versa.
Likewise, if

(49) 
then the equilibrium point is unstable. It follows, from
Equation (45), that stable equilibrium points are
characterized by

(50) 
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,
in Figure 2, and are stable equilibrium points, whereas is an unstable equilibrium point.
Of course, this makes perfect sense if we think of as
a gravitational potential energy curve, so that is
directly proportional to height. In this case, 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 displacement of the mass will cause it
to fall down the hill). Note, finally, that if

(51) 
at any point (or in any region) then we have what is known as a neutral equilibrium
point. We can move the particle slightly away from 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 Figure 3.
Figure 3:
Different types of equilibrium point.

The equation of motion of an particle moving in one dimension
under the action of a conservative force is, in principle, integrable. Since
, the energy
conservation equation (46) can be rearranged to give

(52) 
where the signs correspond to motion to the left and to the right, respectively. However, since
, this expression can be integrated to give

(53) 
where . For sufficiently simple potential functions, , the
above equation can be solved to give as a function of .
For instance, if
, , and the plus sign is chosen, then

(54) 
which can be inverted to give

(55) 
where
and
.
Note that the particle reverses direction each time it reaches one of the socalled
turning points () at which (and, so ).
Next: Velocity Dependent Forces
Up: OneDimensional Motion
Previous: Introduction
Richard Fitzpatrick
20110331