Question on Laplace eq. in a ball

[yungman]

For ##\nabla^2 u(r,\theta, \phi)=0##, ##u(r,\theta, \phi)=r^{n}Y_{nm}(\theta,\phi)##.

But I have issue with this, for spherical coordinates:

[tex]\nabla^2u=\frac{\partial^2{u}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial{u}}{\partial {r}}+\frac {1}{r^{2}}\left(\frac{\partial^2{u}}{\partial {\theta}^2}+\cot\theta\frac{\partial{u}}{\partial {\theta}}+\csc\theta\frac{\partial^2{u}}{\partial {\phi}^2}\right)[/tex]

Let ##u=R(r)Y(\theta,\phi)## where ##Y(\theta,\phi)## is the spherical harmonics.

[tex]\Rightarrow\; r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=0[/tex]
and
[tex]\frac{\partial^2{Y}}{\partial {\theta}^2}+\cot\theta\frac{\partial{Y}}{\partial {\theta}}+\csc^2\theta\frac{\partial^2{Y}}{\partial {\phi}^2}+\mu Y=0[/tex]

For Euler equation: ##r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+r\frac{\partial{R}}{\partial {r}}-\mu R=0## where ##\mu=n^2##. and the solution is ##R=r^n##.

Here, because of the condition, only ##\mu=n(n+1)## is used for bounded solution.
[tex] r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-n(n+1) R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+r\frac{\partial{R}}{\partial {r}}-(n+1/2)^2R[/tex]

Which gives

[tex]R=r^{(n+1/2)}[/tex]

What have I done wrong?

 

[jvicens]

Hello,

Thank you for sharing your thoughts on this topic. It seems like you have a good understanding of the mathematics involved, but there are a few things that may have led to your confusion.

Firstly, the equation you have written for the Laplacian in spherical coordinates is correct, but it is not the one that is being used in the forum post. In the post, the Laplacian is written as ##\nabla^2 u(r,\theta,\phi)=0##, which means that the solution is a function of r, θ, and φ that satisfies this equation. This is different from the equation you have written, where the Laplacian is acting on a function that is a product of two separate functions of r and θ, respectively. This difference is important because it means that the solution to the forum post equation will not necessarily be of the form ##R(r)Y(\theta,\phi)##.

Secondly, in your solution for the Euler equation, you have correctly identified ##\mu=n^2##, but you have not taken into account that the solution must also satisfy the boundary conditions of the problem. In this case, the boundary conditions are that the solution must be bounded at the origin and must approach zero as r approaches infinity. This means that the solution cannot simply be ##R=r^n##, as this would not satisfy the boundary conditions. Instead, the solution must be a modified version of this, as you have correctly stated in your last equation.

Overall, it seems like you have a good understanding of the mathematics involved, but you may have made some small mistakes in your calculations and interpretation of the problem. I hope this helps to clarify things for you. Keep up the good work!