To know how a planet attracts a point, one must add up the attraction of each of its pieces. Each piece pulls in a different direction, and adding forces is laborious: they are vectors, with three components each. Eighteenth-century mathematicians discovered a shortcut, and Laplace turned it into a tool.

Adding numbers instead of arrows

For a mass $M$ concentrated at a point, consider the number

$$V = \frac{M}{r},$$

where $r$ is the distance to the mass (we leave out the constant $G$). This number, which later received the name potential, holds all the information about the force: the components of the attraction in each direction are the derivatives of $V$ in that direction. And the great advantage is that, for several masses, the potentials simply add like numbers. First one adds the potential of all the pieces of the planet; then, with a derivative, one gets the force.

The equation

Laplace noticed that, at any point outside the masses, the potential obeys a very simple equation:

$$\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = 0.$$

Today we write $\nabla^2 V = 0$, and the symbol $\nabla^2$ is called the Laplacian. Laplace published the equation in spherical coordinates in 1782, in a work on the shape of the planets, and in the Cartesian form above in 1787, studying the rings of Saturn.

It is worth checking for a single mass at the origin. With $r = \sqrt{x^2+y^2+z^2}$ and $V = \frac{1}{r}$:

$$\frac{\partial V}{\partial x} = -\frac{x}{r^3}, \qquad \frac{\partial^2 V}{\partial x^2} = -\frac{1}{r^3} + \frac{3x^2}{r^5}.$$

Adding the three directions:

$$\nabla^2 V = -\frac{3}{r^3} + \frac{3(x^2+y^2+z^2)}{r^5} = -\frac{3}{r^3} + \frac{3}{r^3} = 0.$$

Since the equation is linear, a sum of solutions is a solution: the potential of any set of masses satisfies it outside them. Functions that obey Laplace’s equation are called harmonic.

What the equation says

There is a simple reading of the equation. Divide the plane into a grid of points. In the discrete version, $\nabla^2 V = 0$ says that the value at each point is the average of the values of its four neighbours:

$$V_{i,j} = \frac{V_{i+1,j} + V_{i-1,j} + V_{i,j+1} + V_{i,j-1}}{4}.$$

A harmonic function has no peaks or valleys in the interior: if a point were larger than all its neighbours, it could not be their average. Maxima and minima are always on the boundary. This also suggests a way to solve the equation: fix the values on the boundary, guess anything in the middle and repeatedly replace each point by the average of its neighbours. The grid relaxes towards the solution.

In the figure below, the top edge of a square plate is at 100 degrees and the other three at 0. Each click performs several rounds of averaging:

One equation for everything

The same equation appears far from gravitation. The temperature of a plate in equilibrium, the electric potential in a vacuum, the flow of a fluid without eddies and the shape of a stretched membrane all obey $\nabla^2 V = 0$. The Laplacian also appears in the heat, wave and Schrödinger equations, and in image processing, where it is used to detect edges.

Inside the masses the equation changes. In 1813 Siméon-Denis Poisson, Laplace’s student, showed that there

$$\nabla^2 V = -4\pi G\rho,$$

where $\rho$ is the density (with the sign convention for the potential used above). This is Poisson’s equation.

Exercise: which are harmonic?

Check which of these functions of the plane satisfy $\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} = 0$: (a) $x^2 - y^2$; (b) $xy$; (c) $x^2 + y^2$.

(a) $2 + (-2) = 0$: harmonic. (b) Both second derivatives are zero: harmonic. (c) $2 + 2 = 4 \neq 0$: not harmonic. The first two are the real and imaginary parts of $(x+iy)^2 = x^2 - y^2 + 2ixy$. It is no coincidence: the real and imaginary parts of any differentiable complex function are harmonic.

Exercise: the centre of the plate

In the figure above, the top edge is at 100 and the others at 0. Why does the temperature at the centre converge to exactly 25?

Consider four problems: in each one a different side is at 100 and the other three at 0. By rotating the plate, the four are the same, and so they give the same value at the centre. Adding the four solutions we get (by linearity) the solution with all four sides at 100, which is the constant 100. So the value at the centre in each problem is $\frac{100}{4} = 25$.

Sources: Pierre-Simon Laplace, “Théorie des attractions des sphéroïdes et de la figure des planètes”, Mémoires de l’Académie royale des sciences (1782, published 1785) · Pierre-Simon Laplace, “Mémoire sur la théorie de l’anneau de Saturne”, Mémoires de l’Académie royale des sciences (1787, published 1789) · Siméon-Denis Poisson, “Remarques sur une équation qui se présente dans la théorie des attractions des sphéroïdes”, Nouveau Bulletin de la Société philomatique 3 (1813) · Oliver D. Kellogg, Foundations of Potential Theory (1929).

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