May 8th: Invariance of Laplacians, Spherically Symmetric Harmonics in R^2 and R^3

Goal 1: Verify that the Laplacian is invariant under rotation and translation (we will do this for $\mathbb{R}^2$ ).

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be arbitrary. We know show that:

$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 f}{\partial u^2} + \frac{\partial^2 f}{\partial v^2}$ , where

$u(x, y) = xcos(\theta) - ysin(\theta)$ and

$v(x, y) = xsin(\theta) + xcos(\theta)$ .

We use the chain rule. It follows that:

$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}$

$= \frac{\partial}{\partial x}(\frac{\partial f}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial x}) + \frac{\partial}{\partial y}(\frac{\partial f}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial f}{\partial v}\frac{\partial v}{\partial y})$

$= \frac{\partial}{\partial x}(\frac{\partial f}{\partial u}cos(\theta) + \frac{\partial f}{\partial v}sin(\theta)) + \frac{\partial}{\partial y}(-\frac{\partial f}{\partial u}sin(\theta) + \frac{\partial f}{\partial v}cos(\theta))$

$= \frac{\partial (\frac{\partial f}{\partial u}cos(\theta) + \frac{\partial f}{\partial v}sin(\theta))}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial(\frac{\partial f}{\partial u}cos(\theta) + \frac{\partial f}{\partial v}sin(\theta))}{\partial v}\frac{\partial v}{\partial x} + \frac{\partial(-\frac{\partial f}{\partial u}sin(\theta) + \frac{\partial f}{\partial v}cos(\theta))}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial((-\frac{\partial f}{\partial u}sin(\theta) + \frac{\partial f}{\partial v}cos(\theta))}{\partial v}\frac{\partial v}{\partial y}$

$= \frac{\partial^2 f}{\partial u^2}(cos^2(\theta) + sin^2(\theta)) + \frac{\partial^2 f}{\partial v \partial u}cos(\theta)sin(\theta) + \frac{\partial^2 f}{\partial u \partial v}cos(\theta)sin(\theta) + \frac{\partial^2 f}{\partial v^2}(cos^2(\theta) + sin^2(\theta)) - (\frac{\partial^2 f}{\partial v \partial u}cos(\theta)sin(\theta) + \frac{\partial^2 f}{\partial u \partial v}cos(\theta)sin(\theta))$

$= \frac{\partial^2 f}{\partial u^2} + \frac{\partial^2 f}{\partial v^2}$ , which is what we want.

We can use similar method to show that the Laplacian is invariant under translation (I’m too lazy to write the rest out).

Goal 2: What do the spherically symmetric harmonic functions look like in $\mathbb{R}^2, \mathbb{R}^3$ ?

We know that in $\mathbb{R}^2$ , $f(r) = log(r)$ is a spherically symmetric harmonic function and in $\mathbb{R}^3$ , $f(r) = \frac{1}{r}$ is a spherically symmetric harmonic functions. Are there others?

We will solve the Laplacian equation in polar coordinates to figure out.

In $\mathbb{R}^2$ , a spherically symmetric harmonic function would satisfy:

$f_{\theta} = 0$ and

$f_{rr} + \frac{1}{r}f_{r} + \frac{1}{r^2}f_{\theta\theta} = 0$ .

This reduces to:

$f_{rr} + \frac{1}{r}f_{r} = 0$ (a second-order ODE).

Solving this gives:

$f(r) = C_1log(r) + C_2$ , where $C_1, C_2 \in \mathbb{R}$ . This is what spherically symmetric harmonics in $\mathbb{R}^2$ look like.

What about in $\mathbb{R}^3$ ?

They would need to satisfy:

$f_{\phi} = f_{\theta} = 0$ and

$\frac{1}{r^2}\frac{\partial }{\partial r}(r^2\frac{\partial f}{\partial r}) + \frac{1}{r^2sin^2(\phi)}\frac{\partial^2 f}{\partial \theta^2} + \frac{1}{r^2 sin(\phi)}\frac{\partial}{\partial \phi}(sin(\phi)\frac{\partial}{\partial \phi}) = 0$ .