Understanding Laplace’s Equation

(This article builds on two earlier ones, Understanding the Heat Equation and Understanding the Wave Equation.)

Having explored the meanings of the Heat Equation and Wave Equation together with you, I now want to tell you about a third differential equation, also highly significant, and closely connected to the other two: Laplace’s Equation. Here it is:

$${\nabla }^{2}F=0$$

In his excellent text Calculus, Gil Strang calls this the “fundamental partial differential equation of equilibrium”.^[1] I found this statement quite mysterious when I first read it; what about ${\nabla }^{2}F=0$ says “equilibrium”? Before we can answer that, let me rewrite the equation using the same notation as the preceding two articles. In the (uninteresting) one-dimensional case, it is:

$$F_{xx}=0$$

Meh. That just says “straight line”. The 2D version is more interesting:

$$F_{xx}+F_{yy}=0$$

And following the same pattern, the 3D version is:

$$F_{xx}+F_{yy}+F_{zz}=0$$

Hopefully it is obvious how to extrapolate to functions with four, five, or more parameters. All of those equations are contained in the short and sweet ${\nabla }^{2}F=0$. (I won’t explain more about the $\nabla$ notation, but you can read more on this page.) But let’s get back to the question: In what way does Laplace’s Equation express the idea of “equilibrium”?

It does so when combined with the Heat Equation or the Wave Equation. So if my previous two articles made sense to you, it should be a very small step to understand Laplace’s Equation now. Let me restate the Heat Equation, with two spacial dimensions:

$$F_t=F_{xx}+F_{yy}$$

And the Wave Equation:

$$F_{tt}=F_{xx}+F_{yy}$$

You may already see what happens when we combine Laplace’s Equation with one of those. For example, say we have some physical system which obeys the Heat Equation. If Laplace’s Equation holds on the distribution of temperatures, then we can use Laplace’s to rewrite the right-hand side of the Heat Equation, giving:

$$F_t=0$$

Which just says: “none of the temperatures are changing”. With the Wave Equation, we would have got:

$$F_{tt}=0$$

Which says: none of the points on our “water surface” (or “guitar string”, or any other medium which can transmit waves) are accelerating up or down. Yep, Gil Strang said it well; Laplace’s is indeed the “fundamental partial differential equation of equilibrium”.

What Do the Solutions Look Like?

What does the space of possible “equilibrium states” which satisfy Laplace’s Equation look like? If we are describing the flow of heat energy through a physical object, then an equilibrium state is one where none of the temperatures are changing. What is the simplest case you can think of where the temperatures at various points in an object are stable and not changing?

That ☝ is not the only solution, by any means. Borrowing ideas from physical intuition again, we can imagine a situation like this:

The illustration represents a plate of some (non-flammable) material, with the bottom-left corner in a fire at 1200°C and the bottom-right corner submerged in ice at 0°C. Let’s ignore the effect of radiation from the fire and convection of air, and also assume that the temperatures of the plate can reach equilibrium before either the fire goes out or the ice melts. You will probably agree that the smooth gradient of temperature which the illustration depicts agrees with your physical intuition.

What if there was a large hole in the middle of the plate? You can imagine that with less area for heat energy to travel from the left corner over to the right corner, the left side would become hotter and the right side would become colder, something like this:

Both of the above illustrations depict solutions to Laplace’s Equation; in fact, I created the illustrations with the help of a computer program which solves Laplace’s Equation. For any similar physical situation which you can imagine (take an object of some shape, fix the temperatures to specific values at some chosen points on the object, allow heat energy to flow until the distribution of temperature stabilizes), there will be some equilibrium state, and that state will be yet another solution to Laplace’s Equation.

In some cases, we can solve Laplace’s Equation analytically to find an equilibrium state; but it is usually easier to solve it numerically, using a computer. In the next post in this series, I will show you how that can be done.

[1] Page 571 in the first edition. Professor Strang has made this textbook freely available for Internet download.⏎