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Simple harmonic motion
Consider the motion of a point particle of mass
, moving in one dimension, that is
slightly displaced from a stable equilibrium point located at
.
Suppose that the particle is moving in the conservative force field
. According to the preceding analysis, in order for
to
correspond to a stable equilibrium point,
we require both
 |
(2.56) |
and
 |
(2.57) |
Our particle obeys Newton's second law of motion,
 |
(2.58) |
Let us assume that the particle always stays fairly close to its equilibrium
point. In this case, to a good approximation, we can represent
via a truncated Taylor expansion about this point. In other words,
 |
(2.59) |
However, according to Equations (2.56) and (2.57), the preceding
expression can be written
 |
(2.60) |
where
.
Hence, we conclude that our particle satisfies the following approximate
equation of motion,
 |
(2.61) |
provided that it does not stray too far from its equilibrium point; in other words,
provided
does not become too large.
Equation (2.61) is called the simple harmonic equation; it
governs the motion of all one-dimensional conservative systems that are slightly
perturbed from some stable equilibrium state. The solution of Equation (2.61)
is well known:
 |
(2.62) |
The pattern of motion described by this expression,
which is called simple harmonic motion,
is periodic in time, with repetition period
, and oscillates between
. Here,
is called the amplitude of the motion. The parameter
,
known as the phase angle,
simply shifts the pattern of motion backward and forward in time.
Figure 2.6 shows some examples of simple harmonic motion.
Here,
,
, and
correspond to the
solid, short-dashed, and long-dashed curves, respectively.
Note that the frequency,
--and, hence, the period,
--of
simple harmonic motion is determined by the parameters appearing in the simple harmonic equation,
Equation (2.61). However, the amplitude,
, and the phase angle,
,
are the two integration constants of this second-order ordinary differential
equation, and are thus determined by the initial conditions; that is, the particle's initial displacement and velocity.
Figure 2.6:
Simple harmonic motion.
 |
From Equations (2.45) and (2.60), the potential
energy of our particle at position
is approximately
 |
(2.63) |
Hence, the total energy is written
 |
(2.64) |
giving
 |
(2.65) |
where use has been made of Equation (2.62), and the trigonometric
identity
. Note that the
total energy is constant in time, as is to be expected for a
conservative system, and is proportional to the amplitude squared
of the motion.
Consider the motion of a point particle of mass
that is
slightly displaced from a unstable equilibrium point at
.
The fact that the equilibrium is unstable implies that
 |
(2.66) |
and
 |
(2.67) |
As long as
remains small, our particle's equation of motion takes the approximate form
 |
(2.68) |
which reduces to
 |
(2.69) |
where
. The most general solution to the preceding equation is
 |
(2.70) |
where
and
are arbitrary constants. Thus, unless the initial conditions are such that
is exactly zero, the particle's displacement
from the unstable equilibrium point grows exponentially in time.
Next: Two-body problem
Up: Newtonian mechanics
Previous: Motion in one-dimensional potential
Richard Fitzpatrick
2016-03-31