MAA Program
向量場的势滿足球對稱性和拉普拉斯方程，蘊含向量場滿足牛頓平方反比定律。
And if in the ordinary three-dimensional continuum, time aside, one adopts a scalar-field, and looks for the simplest laws which such a field can satisfy, namely, spherical symmetry and Laplace's equation, then one arrives at Newton's inverse square law.
The argument is simple indeed. Let the scalar-field be $\phi(x, y, z)$. If it is spherically symmetric, then $\phi=\phi(r)$, where $r=\sqrt{x^{2}+y^{2}+z^{2}}$. The analogue of relativity's vanishing curvature condition is Laplace's equation:$$\phi_{x x}+\phi_{y y}+\phi_{z z}=0$$Computing gives
$$
\phi_{x x}=\frac{\phi^{\prime}}{r}+\frac{x^{2} \phi^{\prime \prime}}{r^{2}}-\frac{x^{2} \phi^{\prime}}{r^{3}}
$$
and similarly for $\phi_{y y}$ and $\phi_{z z}$.
Thus Laplace's equation becomes$$\phi^{\prime \prime}+{2 \phi^{\prime} \over r}=0\tag1$$which, upon integration, yields $\phi=-1 / r$. This potential has force-field,
$$
-\left(\phi_{x}, \phi_{y}, \phi_{z}\right)=-\left(\frac{x}{r^{3}}, \frac{y}{r^{3}}, \frac{z}{r^{3}}\right)
$$
whose magnitude is $1 / r^{2}$.
式(1)又见:
Laplacian of spherically symmetric function | 
