Simple Harmonic Motion

Consider the motion of a point particle of mass $m$ which is slightly displaced from a stable equilibrium point at $x=0$. Suppose that the particle is moving in the conservative force-field $f(x)$. According to the analysis of Section 3.2, in order for $x=0$ to be a stable equilibrium point we require both

$$f(0) = 0,$$ (72)

and

$$\frac{d f(0)}{dx} < 0.$$ (73)

Now, our particle obeys Newton's second law of motion,

$$m\,\frac{d^2 x}{d t^2} = f(x).$$ (74)

Let us assume that it always stays fairly close to its equilibrium point. In this case, to a good approximation, we can represent $f(x)$ via a truncated Taylor expansion about this point. In other words,

$$f(x) \simeq f(0) + \frac{df(0)}{dx}\,x + {\cal O}(x^2).$$ (75)

However, according to (72) and (73), the above expression can be written

$$f(x) \simeq - m\,\omega_0^{\,2}\,x,$$ (76)

where $df(0)/dx = -m\,\omega_0^{\,2}$. Hence, we conclude that our particle satisfies the following approximate equation of motion,

$$\frac{d^2 x}{dt^2}+ \omega_0^{\,2}\,x\simeq 0,$$ (77)

provided that it does not stray too far from its equilibrium point: i.e., provided $\vert x\vert$ does not become too large.

Equation (77) is called the simple harmonic equation, and governs the motion of all one-dimensional conservative systems which are slightly perturbed from some stable equilibrium state. The solution of Equation (77) is well-known:

$$x(t) = a\,\sin(\omega_0\,t -\phi_0).$$ (78)

The pattern of motion described by above expression, which is called simple harmonic motion, is periodic in time, with repetition period $T_0 = 2\pi/\omega_0$, and oscillates between $x=\pm a$. Here, $a$ is called the amplitude of the motion. The parameter $\phi_0$, known as the phase angle, simply shifts the pattern of motion backward and forward in time. Figure 4 shows some examples of simple harmonic motion. Here, $\phi_0 = 0$, $+\pi/4$, and $-\pi/4$ correspond to the solid, short-dashed, and long-dashed curves, respectively.

Note that the frequency, $\omega_0$--and, hence, the period, $T_0$--of simple harmonic motion is determined by the parameters appearing in the simple harmonic equation, (77). However, the amplitude, $a$, and the phase angle, $\phi_0$, are the two integration constants of this second-order ordinary differential equation, and are, thus, determined by the initial conditions: i.e., by the particle's initial displacement and velocity.

Figure 4: Simple harmonic motion.

Now, from Equations (45) and (76), the potential energy of our particle at position $x$ is approximately

$$U(x) \simeq \frac{1}{2}\,m\,\omega_0^{\,2}\,x^2.$$ (79)

Hence, the total energy is written

$$E = K + U = \frac{1}{2}\,m\left(\frac{dx}{dt}\right)^2+ \frac{1}{2}\,m\,\omega_0^{\,2}\,x^2,$$ (80)

giving

$$E = \frac{1}{2}\,m\,\omega_0^{\,2}\,a^2\,\cos^2(\omega_0\,t-... ...n^2(\omega_0\,t-\phi_0) = \frac{1}{2}\,m\,\omega_0^{\,2}\,a^2,$$ (81)

where use has been made of Equation (78), and the trigonometric identity $\cos^2\theta+\sin^2\theta \equiv 1$. Note that the total energy is constant in time, as is to be expected for a conservative system, and is proportional to the amplitude squared of the motion.