Unit 1 - Poisson's and Laplace's Equations

One thing that makes understanding these equations important is that they reappear, with variations, in other contexts. For instances, in quantum theory, Schrodinger's equation is a Poisson's equation with either an imaginary constant times the time derivative of, or a multiple of, the unknown function replacing the density; the diffusion equation is Poisson's equation with a real constant times the time derivative of the unknown.

An important property that both these equations have is that they are linear. Linear equations are consistently easier to solve than nonlinear ones; much of what we can explicitly do about nonlinear equations is based on what we have learned from linear approximations. (Unfortunately, a lot of the mistakes we make in trying to understand nonlinear problems are also based on what we learned from linear problems.)

Laplace's equation is
del² f = 0.
Linearity asserts, for a homogeneous (that's a fancy word to say the right-hand side is zero) equation like this, that if a₁ and a₂ are constants, and f₁ and f₂ are solutions, then a₁ f₁ + a₂ f₂ is also a solution. In other words, once you have some solutions, you can get lots more by adding. So you try to find a large family of easily-managed solutions, figuring that any particular solution you may need can be constructed from the family.

Poisson's equation is the more general one
del² f = g.
Linearity, for an inhomogeneous equation like this, means that if f₁ is a solution for right-hand-side equal to g₁, and f₂ is a solution for right-hand-side equal to g₂, then f₁ + f₂ is a solution for right-hand-side equal to g₁ + g₂. Of course, if g₂ = 0, f₂ can be any solution of Laplace's equation. That is, if f_P is any solution of Poisson's equation, then so is f_P plus any solution of Laplace's equation.

Linearity of the equation basically says that you don't have to work very hard to cope with even a really messy right-hand-side, or a really messy boundary condition, as long as you can express that messy part as a sum of simple parts for each of which you can solve the problem. Basically, linearity is the mathematical statement of the superposition property that we use all over the place in physics.

Another property that these equations have, is uniqueness theorems. Of course, those theorems aren't about the solutions to the differential equation by itself. Any differential equation has a family of solutions; physical problems won't correspond to just a differential equation, but to a differential equation plus additional conditions. In the case of ordinary differential equations, with one independent variable, such as arise in mechanics of point particles and rigid bodies (where the independent variable is time), the additional conditions are usually initial conditions. Thus, if you are given the forces, you also need initial position and initial velocity in order to determine later values. For partial differential equation problems, where typically at least all three space coordinates are independent variables, the additional information is normally some condition that applies at the 'edge', or boundary, of the region you are concerned with (for physics on a surface, 'edge' will be the edge of the surface; for physics in a three-dimensional region, 'edge' will be the surface enclosing the region). Hence, we refer to 'boundary conditions'.

The shape function for a stiff but flexible wire will obey some sort of differential equation; will that equation obey a uniqueness theorem? No: a two-foot wire, attached at its ends to anchors one foot apart, has several possible positions (if the ends are at equal height, two positions are an arch shape and a U shape). There's usually no such multiple-solution situation in electrostatics, and the uniqueness theorems spell that out.

Griffiths spends an entire section on the method of images. In one sense this is a lot, because this method actually works predictably for only two problems plus superpositions: infinite flat conductors and perfect sphere conductors; there's also a solution for infinite circular-cylinder conductors if the charges are lines parallel to the cylinder. But there are two reasons for so much discussion, one conceptual, one practical.

The conceptual reason for considering images is as an example of the power of uniqueness. Here we have a problem, that's supposed to be electrostatics, but we have no way to tell where the charges are. But we turn our back on that problem, consider a different problem, and then look at a piece of its solution. We take that function form, which as far as the original problem is concerned came out of nowhere; we try it on our original problem, and it works. Therefore it is the one and only solution. (In partial differential equations, in some cases you can get a candidate solution by a systematic approach; in other cases by educated trial and error; in more cases it will take numerical methods, or divine revelation, or peeking at the answer. But if you plug it in and it works, it's an answer no matter how you got it; if there's an applicable uniqueness theorem, then not only is it an answer but it's the only answer.)

The practical reason for covering the method of images is that there are lots of conductors in the real world, and when you are near enough to any one of them it will be either pretty flat, in which case an infinite plane is a rough approximation, or definitely curved, in which case a sphere is a rough approximation (or sometimes the cylinder case is near enough). In either case thinking of (approximate) image charges will give you a start on what to expect from the real conductor.

Griffiths spends another section on separation of variables. This is the only (fairly) straightforward method of constructing general explicit (as opposed to numerical) solutions to common linear partial differential equations. It does require symmetry in the boundary shape of the regions involved (but 'all space' is a pretty common case of region involved, and 'all space' has plenty of boundary symmetry). However, because of superposition, this method does not require symmetry of the boundary values.

Since superposition is generally involved in matching boundary conditions, applying this method usually involves techniques for writing a given function as a sum of members of a specified family. (The specified family will be those functions for which the boundary condition, or the right-hand-side in Poisson's equation, leads to a specific solution; then the solution for having the given function as boundary condition or RHS will be the corresponding sum of the specific solutions.) To illustrate in a simpler, more familiar context, consider motion with zero net force. One solution is:
For x_initial = x_o and v_initial = 0, then x(t) = x_o.
Another solution is:
For x_initial = 0 and v_initial = v_o, then x(t) = v_o t.
Hence by superposition, for x_initial = x_o = x_o + 0, and v_initial = v_o = 0 + v_o, then x(t) = x_o + v_o t.
Now the concept is the same, but instead of having just two initial conditions, which are always just two constants, leading to superposition of two terms, we have boundary conditions which are functions, infinite sets of values, for which we use basis functions such as sin x, sin 2x, etc. (These are comparable to basis vectors such as i, j, k, of which there are only three; for functions we need an infinite set.) Since a complicated function will typically require summing an infinite number of the basis functions (unlike ordinary vectors where three components will always be sufficient), one ends up usually making an approximation by stopping the summing after a reasonable number of contributions.

Finally, Griffiths presents a section on multipole expansion. This is an approximation scheme, similar to perturbation expansion schemes (which you may or may not have heard of). That means, basically, that it involves replacing your original problem with a simpler one whose answer will be close to the answer of your original problem. Further, you can estimate how far off the answer might be, and if that estimate isn't satisfactory you can extend the method: make the replacement a step less simple but the answer a step closer to the answer of the original problem. And you can extend that improvement as many times as you need (or can afford). In class situations, the level of approximation will usually be specified when the problem is given. In other situations, there will usually be some implicit or explicit notion to say how good will be good enough; for a few problems, successive generations of researchers (or their graduate students) have labored to make 'good enough' a step better.

The method of images Solution by separation of variables The multipole expansion

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Modern Physics 1.