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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 (2)],

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

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

$\displaystyle s(t) = a\,\cos(\omega\,t-\phi),$ (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 (10)-(13).] In fact, $ a$ and $ \phi$ are the two arbitrary constants of integration of the second-order ordinary differential equation (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}=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

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

However, this can also be written

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

or

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

where

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

According to Equation (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 2.1. The quantity $ {\cal E}$ is either zero or positive, because 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 Equation (17) that $ s=0$ , and from Equation (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 (22), that $ s$ attains it maximum value when $ \dot{s}=0$ . In fact,

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

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

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

when $ s=0$ .

The simple harmonic oscillation specified by Equation (18) can also be written in the form

$\displaystyle s(t) = A \cos(\omega t) + B \sin(\omega t),$ (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$ . (See Appendix B.) Alternatively, Equation (18) can be written

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

where $ \phi'=\phi-\pi/2$ , and use has been made of the trigonometric identity $ \cos\theta \equiv \sin(\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 (25), or $ a$ and $ \phi'$ in Equation (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$ . 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 (17), so that

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

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

$\displaystyle \ddot{s}_2=-\omega^2\,s_2,$ (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 can be 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$ can be shown to be

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

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

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

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

or

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


next up previous
Next: LC Circuits Up: Simple Harmonic Oscillation Previous: Mass on a Spring
Richard Fitzpatrick 2013-04-08