Next: Circuit
Up: Simple Harmonic Oscillation
Previous: Mass on a Spring
Suppose that a physical system possesing one degree of freedomi.e., a
system whose instantaneous state at time is fully described by a single dependent variable, obeys the following time evolution equation [cf., Equation (2)]:

(17) 
where is a constant. As we have seen, this differential
equation is called the simple harmonic oscillator equation, and has the
following solution

(18) 
where and are constants. Moreover, the above equation describes a type of oscillation
characterized by a constant amplitude, , and a constant angular frequency, .
The phase angle, , determines the times at which the oscillation attains its
maximum value. Finally, the frequency of the oscillation (in Hertz) is ,
and the period is . Note that the frequency and period of the
oscillation are both determined by the constant , which appears in the simple harmonic oscillator equation, whereas the amplitude, , and phase angle, , are both determined by the initial conditionssee Equations (10)(13). In fact, and are the two constants of integration of the
secondorder ordinary differential equation (17). Recall, from standard differential equation theory, that the
most general solution of an thorder ordinary differential equation (i.e.,
an equation involving a single independent variable, and a single dependent variable, in which the highest derivative of the dependent with respect to the
independent variable is
thorder, and the lowest zerothorder) involves arbitrary constants of integration. (Essentially, this is because
we have to integrate the equation times with respect to the independent variable in order to reduce it to zerothorder, and so
obtain the solution. Furthermore, each integration introduces an arbitrary constant: e.g.,
the integral of , where is a known constant, is , where is an arbitrary constant.)
Multiplying Equation (17) by , we obtain

(19) 
However, this can also be written in the form

(20) 
or

(21) 
where

(22) 
Clearly, is a conserved quantity: i.e., it does not vary with time. In fact, this quantity is
generally proportional to the overall energy of the system. For instance, would be the energy divided by the mass in
the massspring system discussed in Section 2.1. Note that is either
zero or positive, since neither of the terms on the righthand side of Equation (22) can be negative. Let us search for an equilibrium state. Such a state is
characterized by
, so that
. It follows
from (17) that , and from (22) that . We conclude that the system can only remain permanently at
rest when .
Conversely, the system can
never permanently come to rest when , and must, therefore, keep moving for ever. Furthermore, since the equilibrium state is characterized by , it follows that
represents a kind of ``displacement'' of the system from this state.
It is also apparent, from (22), that attains it maximum value when .
In fact,

(23) 
This, of course, is the amplitude of the oscillation: i.e., .
Likewise, attains its maximum value when , and

(24) 
Note that the simple harmonic oscillation (18) can also
be written in the form

(25) 
where
and . Here, we have employed the trigonometric identity
.
Alternatively, (18) can be written

(26) 
where
, and use has been made of the trigonometric
identity
. Clearly, there are many different
ways of representing a simple harmonic oscillation, but they all involve
linear combinations of sine and cosine functions whose arguments
take the form , where is some constant. Note, however, that,
whatever form it takes,
a general solution to the simple harmonic oscillator equation must always contain two arbitrary constants: i.e., and in (25) or
and in (26).
The simple harmonic oscillator equation, (17), is a linear differential equation,
which means that
if is a solution then so is , where is
an arbitrary constant. This can be verified by multiplying the equation by ,
and then making use of the fact that
. Now, linear
differential equations have a very important and useful property: i.e., their
solutions are superposable. This means that if is a
solution to Equation (17), so
that

(27) 
and
is a different solution, so that

(28) 
then is also a solution. This can be verified by adding the previous
two equations, and making use of the fact that
. Furthermore, it is easily demonstrated that any linear combination of and ,
such as , where and are constants, is also a solution.
It is very helpful to know this fact.
For instance, the special solution to the simple harmonic oscillator equation (17) with the simple initial
conditions and
is easily shown to be

(29) 
Likewise, the special solution with the simple initial conditions and is clearly

(30) 
Thus, since the solutions to the simple harmonic oscillator equation are superposable, the
solution with the general initial conditions and
is

(31) 
or

(32) 
Next: Circuit
Up: Simple Harmonic Oscillation
Previous: Mass on a Spring
Richard Fitzpatrick
20101011