胡克定律和泰勒级数（需要微积分）

Hooke's law is a very elegant theorem describing small-amplitude spring vibrations.

Actually, almost everything in life that is in stable equilibrium obeys Hooke's law or something similar to Hooke's law during small vibrations. This includes pendulums, molecules, water levels, and much more.

To understand why, let us consider the force field $F(x)$ acting on an object moving along the $x$ axis very close to an equilibrium position at $x=0$.

This force field's exact expression might be very complicated. However, for sufficiently small displacements from $x=0$, we can use a Taylor series approximation for $F(x)$.

Since $x=0$ is an equilibrium position, $F(0)=0$. Therefore such an approximation for $F(x)$ for small values of $x$ is useless.

The general form of the first order approximation is $F(x) = F'(0)x$. This is a useful approximation and is often accurate in real life situations involving small displacements.

If $F'(0) > 0$, that would mean that if the object made a small positive displacement, the force field would accelerate it further in the positive direction, and if it made a small negative displacement, the force field would further accelerate it in the negative direction. This is not stable equilibrium, but unstable equilibrium.

Usually (but not always) in theoretical physics and in real life, $F'(0)$ is not exactly equal to zero (if it happened to be zero, the first-order approximation would be insufficient, and Hooke's law would not hold).

So, if $F'(0)<0$, as is very often the case, we see that $F(x) = F'(0)x$ is exactly the same thing as Hooke's law.