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# Simple Harmonic Oscillator Equation

Suppose that a physical system possesing one degree of freedom--i.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 conditions--see Equations (10)-(13). In fact, and are the two constants of integration of the second-order ordinary differential equation (17). Recall, from standard differential equation theory, that the most general solution of an th-order 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 th-order, and the lowest zeroth-order) 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 zeroth-order, 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 mass-spring system discussed in Section 2.1. Note that is either zero or positive, since neither of the terms on the right-hand 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 2010-10-11