Simple Harmonic Oscillator Equation

Suppose that a physical system possessing a single degree of freedom—that is, 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 (1.2)],

$$\ddot{s}+ \omega^{\,2}\,s=0,$$ (1.17)

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

$$s(t) = a\,\cos(\omega\,t-\phi),$$ (1.18)

where $a>0$ and $\phi$ are constants. Moreover, this solution 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. The frequency of the oscillation (in hertz) is $f=\omega/2\pi$, and the period is $T=2\pi/\omega$. 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 determined by the initial conditions. [See Equations (1.10)–(1.13).] In fact, $a$ and $\phi$ are the two arbitrary constants of integration of the second-order ordinary differential equation (1.17). Recall, from standard differential equation theory (Riley 1974), 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 to reduce it to zeroth-order, and so obtain the solution. Furthermore, each integration introduces an arbitrary constant. For example, the integral of $\dot{s}=\alpha$, where $\alpha$ is a known constant, is $s=\alpha\,t+\beta$, where $\beta$ is an arbitrary constant.)

Multiplying Equation (1.17) by $\dot{s}$, we obtain

$$\dot{s}\,\ddot{s} + \omega^{\,2}\,\dot{s}\,s=0.$$ (1.19)

However, this can also be written

$$\frac{d}{dt}\!\left(\frac{1}{2}\,\dot{s}^{\,2}\right) +\frac{d}{dt}\!\left( \frac{1}{2}\,\omega^{\,2}\,s^{\,2}\right)=0,$$ (1.20)

or

$$\frac{d{\cal E}}{dt} = 0,$$ (1.21)

where

$${\cal E} = \frac{1}{2}\,\dot{s}^{\,2} + \frac{1}{2}\,\omega^{\,2}\,s^{\,2}.$$ (1.22)

According to Equation (1.21), ${\cal E}$ is a conserved quantity. In other words, it does not vary with time. 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 1.2. The quantity ${\cal E}$ is either zero or positive, because neither of the terms on the right-hand side of Equation (1.22) 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 Equation (1.17) that $s=0$, and from Equation (1.22) 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. Because the equilibrium state is characterized by $s=0$, we deduce that $s$ represents a kind of “displacement” of the system from this state. It is also apparent, from Equation (1.22), that $s$ attains its maximum value when $\dot{s}=0$. In fact,

$$s_{\rm max} = \frac{\sqrt{2\,{\cal E}}}{\omega},$$ (1.23)

where $s_{\rm max}=a$ is the amplitude of the oscillation. Likewise, $\dot{s}$ attains its maximum value,

$$\dot{s}_{\rm max} = \sqrt{2\,{\cal E}},$$ (1.24)

when $s=0$.

The simple harmonic oscillation specified by Equation (1.18) can also be written in the form

$$s(t) = A\,\cos(\omega\,t) + B\,\sin(\omega\,t),$$ (1.25)

where $A= a\,\cos\phi$ and $B=a\,\sin\phi$. Here, we have employed the trigonometric identity $\cos(x-y) \equiv \cos x\,\cos y+\sin x\,\sin y$. (See Appendix B.) Alternatively, Equation (1.18) can be written

$$s(t) = a\,\sin(\omega\,t-\phi'),$$ (1.26)

where $\phi'=\phi-\pi/2$, and use has been made of the trigonometric identity $\cos\theta \equiv \sin(\theta+\pi/2)$. (See Appendix B.) It follows that 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. However, irrespective of its form, a general solution to the simple harmonic oscillator equation must always contain two arbitrary constants. For example, $A$ and $B$ in Equation (1.25), or $a$ and $\phi'$ in Equation (1.26).

The simple harmonic oscillator equation, (1.17), is a linear differential equation, which means that if $s(t)$ is a solution then so is $c\,s(t)$, where $c$ is an arbitrary constant. This can be verified by multiplying the equation by $c$, and then making use of the fact that $c\,d^{\,2}s/dt^{\,2}=d^{\,2}(c\,s)/dt^{\,2}$. Linear differential equations have the very important and useful property that their solutions are superposable. This means that if $s_1(t)$ is a solution to Equation (1.17), so that

$$\ddot{s}_1=-\omega^{\,2}\,s_1,$$ (1.27)

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

$$\ddot{s}_2=-\omega^{\,2}\,s_2,$$ (1.28)

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 $d^{\,2}s_1/dt^{\,2}+d^{\,2} s_2/dt^{\,2}=d^{\,2}(s_1+s_2)/dt^{\,2}$. Furthermore, it can be demonstrated that any linear combination of $s_1$ and $s_2$, such as $\alpha\,s_1+\beta\,s_2$, where $\alpha$ and $\beta$ are arbitrary constants, is also a solution. It is very helpful to know this fact. For instance, the special solution to the simple harmonic oscillator equation, (1.17), with the simple initial conditions $s(0) = 1$ and $\dot{s}(0) = 0$ can easily be shown to be

$$s_1(t) = \cos(\omega\,t).$$ (1.29)

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

$$s_2(t) = \omega^{-1}\,\sin(\omega\,t).$$ (1.30)

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

$$s(t)=s_0\,s_1(t) + \dot{s}_0\,s_2(t),$$ (1.31)

or

$$s(t) = s_0\,\cos(\omega\,t)+ \frac{\dot{s}_0}{\omega}\,\sin(\omega\,t).$$ (1.32)