Simple Harmonic Motion: Or How I Learned to Love Quadratic Approximations

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The simple harmonic approximation is perhaps one of the most applied concepts in physics. This is essentially due to the structure of Taylor series expansions around equilibrium points.

Phyisicists often use approximations to reduce systems to something more mathematically modelable. One of the most understood and often applied models is that of simple harmonic motion (SHM). This type of system results from a purely quadratic force term. And, in fact, this is the basic model of an ideal spring.

But why would a seemingly random object provide insight into so many systems at equilibrium? This may be answered by a simple consideration of the Taylor series of a given system.

Springs and Hooke's Law

Every intro physics student (should) know that the force of a spring is directly proportional to it's displacement from equilibrium. Explicitly, for a spring stretched a distance \(\Delta x\) from it's equilibrium point along it's axis, the corresponding force is that below:

\[ F=-k\Delta x \]
where \(k\) is termed the spring constant, which is a characteristic of the spring itself. This relation for the force of a spring is known as Hooke's law and results in simple harmonic motion about the point of equilibrium, \(\Delta x=0\).

As a simple exercise we may also calculate the form of the spring's potential energy by equating it to the work required to stretch the spring a distance \(x\):

\[ W=-\int\vec{F}(\vec{r})\cdot\vec{r}=\int_0^x kx'\ dx'= \frac{1}{2}kx^2 \]
Now that we have the basic relations for a spring in mind, we consider a more general approach to any system in equilibrium.

Systems Around Equilibrium Points

The force of an object subject to a conservative force (and thus one modelable with a scalar potential field) is equal to the derivative (gradient) of this potential with respect to displacement.

Furthermore, at equilibrium position \(x=x_0\) we (should) expect a vanishing force, that is \(F(x_0)=0\) and in which case \(\frac{dV(x)}{dx}=0\). Hence, at points of equilibrium, the first derivative of the potential vanishes, since these are exactly the stationary points of the field.

Now, considering the Taylor expansion of an arbitrary (well-behaved) scalar potential, we get the following:

However, at points of equilibrium we have a vanishing first derivative and thus:
And for ranges where \(x-x_0\) is small, the higher order terms should be neglibible (as long as the deriavtives don't grow faster than the power series). And, hence:
\[V(x)\approx V(x_0)+\frac{d^2V(x)}{dx^2}\vert_{x=x_0}\frac{(x-x_0)^2}{2}\]
where we may further neglect the constant \(V_0\) since it is simply a constant term that will not contribute to dynamics. We also now may consider the form of the resulting potential field if we rename the coefficient of the second order expansion \(k\).
\[V(x)=\frac{1}{2}k(\Delta x)^2\]

Note that this is the expression for the potential energy of a spring! And, indeed taking the negative derivative of this potential gives you Hooke's Law:

\[F(x)=-\frac{d}{d(\Delta x)}\left(\frac{1}{2}k(\Delta x)^2\right)=-k\Delta x\]

This essentially tells us that we can describe many systems in equilibrium as springs!


The above considerations are particularly abstract (though they only rely on basic calculus). While it may be less enlightening, the generality we gain by such abstract considerations are in large part their utility.

Note that this presentation was a little backwards from a certain perspective. In fact, it's not that we can model systems about equilibrium as springs, it's more that the spring is a model system around equilibrium and hence our approximation applies.

Also, the above approximation does have it's shortcomings; SHM generally only works well on sufficiently small neighborhoods of equilibrium points. As such, we can't describe dynamical behaviour for non-equilbrium states. Some systems will even have a vanishing second derivative of potential about certain equilibrium points. This would render the approximation above entirely useless, and require us to model systems about such points with higher order terms.