Simple Harmonic Oscillator Equation

Suppose that a physical system possesing one degree of freedom--i.e., a system whose instantaneous state at time $t$ is fully described by a single dependent variable, $s(t)$--obeys the following time evolution equation [cf., Equation (2)]:

$$\ddot{s}+ \omega^2\,s=0,$$ (17)

where $\omega>0$ is a constant. As we have seen, this differential equation is called the simple harmonic oscillator equation, and has the following solution

$$s(t) = a\,\cos(\omega\,t-\phi),$$ (18)

where $a>0$ and $\phi$ are constants. Moreover, the above equation describes a type of oscillation characterized by a constant amplitude, $a$, and a constant angular frequency, $\omega$. The phase angle, $\phi$, determines the times at which the oscillation attains its maximum value. Finally, the frequency of the oscillation (in Hertz) is $f=\omega/2\pi$, and the period is $T=2\pi/\omega$. Note that the frequency and period of the oscillation are both determined by the constant $\omega$, which appears in the simple harmonic oscillator equation, whereas the amplitude, $a$, and phase angle, $\phi$, are both determined by the initial conditions--see Equations (10)-(13). In fact, $a$ and $\phi$ 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 $n$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 $n$th-order, and the lowest zeroth-order) involves $n$ arbitrary constants of integration. (Essentially, this is because we have to integrate the equation $n$ 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 $\dot{s}=a$, where $a$ is a known constant, is $s=a\,t+b$, where $b$ is an arbitrary constant.)

Multiplying Equation (17) by $\dot{s}$, we obtain

$$\dot{s}\,\ddot{s} + \omega^2\,\dot{s}\,s=0.$$ (19)

However, this can also be written in the form

$$\frac{d}{dt}\!\left(\frac{1}{2}\,\dot{s}^{\,2}\right) +\frac{d}{dt}\!\left( \frac{1}{2}\,\omega^2\,s^2\right)=0,$$ (20)

or

$$\frac{d{\cal E}}{dt} = 0,$$ (21)

where

$${\cal E} = \frac{1}{2}\,\dot{s}^{\,2} + \frac{1}{2}\,\omega^2\,s^2.$$ (22)

Clearly, ${\cal E}$ 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, ${\cal E}$ would be the energy divided by the mass in the mass-spring system discussed in Section 2.1. Note that ${\cal E}$ 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 $s= {\rm constant}$, so that $\dot{s}=\ddot{s}=0$. It follows from (17) that $s=0$, and from (22) that ${\cal E}=0$. We conclude that the system can only remain permanently at rest when ${\cal E}=0$. Conversely, the system can never permanently come to rest when ${\cal E}>0$, and must, therefore, keep moving for ever. Furthermore, since the equilibrium state is characterized by $s=0$, it follows that $s$ represents a kind of ``displacement'' of the system from this state. It is also apparent, from (22), that $s$ attains it maximum value when $\dot{s}=0$. In fact,

$$s_{\rm max} = \frac{\sqrt{2\,{\cal E}}}{\omega}.$$ (23)

This, of course, is the amplitude of the oscillation: i.e., $s_{\rm max}=a$. Likewise, $\dot{s}$ attains its maximum value when $x=0$, and

$$\dot{s}_{\rm max} = \sqrt{2\,{\cal E}}.$$ (24)

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

$$s(t) = A\,\cos(\omega\,t) + B\,\sin(\omega\,t),$$ (25)

where $A= a\,\cos\phi$ and $B=a\,\sin\phi$. Here, we have employed the trigonometric identity $\cos(x-y) \equiv \cos x\,\cos y+\sin x\,\sin y$. Alternatively, (18) can be written

$$s(t) = a\,\sin(\omega\,t-\phi'),$$ (26)

where $\phi'=\phi-\pi/2$, and use has been made of the trigonometric identity $\cos\theta \equiv \sin(\pi/2+\theta)$. 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 $\omega\,t+c$, where $c$ 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., $A$ and $B$ in (25) or $a$ and $\phi'$ in (26).

The simple harmonic oscillator equation, (17), is a linear differential equation, which means that if $s(t)$ is a solution then so is $a\,s(t)$, where $a$ is an arbitrary constant. This can be verified by multiplying the equation by $a$, and then making use of the fact that $a\,d^2s/dt^2=d^2(a\,s)/dt^2$. Now, linear differential equations have a very important and useful property: i.e., their solutions are superposable. This means that if $s_1(t)$ is a solution to Equation (17), so that

$$\ddot{s}_1=-\omega^2\,s_1,$$ (27)

and $s_2(t)$ is a different solution, so that

$$\ddot{s}_2=-\omega^2\,s_2,$$ (28)

then $s_1(t)+s_2(t)$ is also a solution. This can be verified by adding the previous two equations, and making use of the fact that $d^2s_1/dt^2+d^2 s_2/dt^2=d^2(s_1+s_2)/dt^2$. Furthermore, it is easily demonstrated that any linear combination of $s_1$ and $s_2$, such as $a\,s_1+b\,s_2$, where $a$ and $b$ 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 $s(0) = 1$ and $\dot{s}(0) = 0$ is easily shown to be

$$s_1(t) = \cos(\omega\,t).$$ (29)

Likewise, the special solution with the simple initial conditions $s(0)=0$ and $\dot{s}(0)=1$ is clearly

$$s_2(t) = \omega^{-1}\,\sin(\omega\,t).$$ (30)

Thus, since the solutions to the simple harmonic oscillator equation are superposable, the solution with the general initial conditions $s(0)=s_0$ and $\dot{s}(0)=\dot{s}_0$ is

$$s(t)=s_0\,s_1(t) + \dot{s}_0\,s_2(t),$$ (31)

or

$$s(t) = s_0\,\cos(\omega\,t)+ \frac{\dot{s}_0}{\omega}\,\sin(\omega\,t).$$ (32)