# Finding the eigenfrequencies of a vibrating sphere

I'm trying to calculate the eigen-frequencies of a sphere by considering the sphere as a two dimensional lattice of springs. The Lagrangian is $${\label{lag}} L=\sum\limits_{\nu=1}^n \sum\limits_{\mu=1}^{n1}\left(m \dot{\vec{r}}_{\mu,\nu}-\frac{k}{2}((\vec{r}_{\mu, \nu+1}-\vec{r}_{\mu,\nu})^2 + (\vec{r}_{\mu+1,\nu}-\vec{r}_{\mu,\nu})^2\right)$$

In which $\mu$ and $\nu$ are the counting along polar and azimuthal angles.

The generalized momentum is $$\vec{p}_{\mu,\nu}=\frac{\partial L}{\partial \dot{\vec{r}}_{\mu,\nu}}=m \dot{\vec{r}}_{\mu,\nu}$$

Substituting in Euler Lagrange Equations to get the generalized forces and we get

\begin{eqnarray} F_{\mu,\nu}&=& \frac{\partial L}{\partial \vec{r}_{\mu,\nu}}=k(4 \vec{r}_{\mu,\nu} - \vec{r}_{\mu,\nu-1} -\vec{r}_{\mu,\nu+1} - \vec{r}_{\mu-1,\nu} - \vec{r}_{\mu+1,\nu}) \\F_{\mu,\nu}&=&k\left((2 \vec{r}_{\mu,\nu} - \vec{r}_{\mu,\nu-1} -\vec{r}_{\mu,\nu+1}) -(2 \vec{r}_{\mu,\nu}- \vec{r}_{\mu-1,\nu} - \vec{r}_{\mu+1,\nu})\right)\label{eqm} \end{eqnarray} Let the Matrix $R_{\mu,\nu}$ be

$$\begin{bmatrix} \vec{r}_{11} & \vec{r}_{12} & \dots & \dots & \vec{r}_{1n} \\ \vec{r}_{21} & \vec{r}_{22} & \dots & \dots & \vec{r}_{2n} \\ \vdots&\dots & \vec{r}_{\mu,\nu} &\quad&\vdots\\ \vec{r}_{n1, 1} & \vec{r}_{n1, 2} & \dots & \dots & \vec{r}_{n1, n} \end{bmatrix}$$

Then The force can be written as $$=\begin{bmatrix} \vec{r}_{\mu,1}\dots& \vec{r}_{\mu,\nu}& \dots \end{bmatrix} \times \begin{bmatrix} 2 & -1 & 0 & \dots&\dots & -1\\ -1 & 2 & -1 & \dots&\dots&\vdots\\ 0 & -1 & 2 & -1 & \dots& \vdots\\ \vdots& \vdots& \vdots& \ddots &\quad&\vdots\\ \vdots& \vdots & \vdots& \quad& \ddots&\vdots\\ -1& 0& \dots&\dots& -1 &2 \end{bmatrix}\nonumber$$ $$+\begin{bmatrix} 2 & -1 & 0 & \dots&\dots & -1\\ -1 & 2 & -1 & \dots&\dots&\vdots\\ 0 & -1 & 2 & -1 & \dots& \vdots\\ \vdots& \vdots& \vdots& \ddots &\quad&\vdots\\ \vdots& \vdots & \vdots& \quad& \ddots&\vdots\\ -1& 0& \dots&\dots& -1 &2 \end{bmatrix} \times \begin{bmatrix} \vec{r}_{1,\nu}\\ \vdots\\ \vec{r}_{\mu,\nu} \\ \vdots\\ \vdots \end{bmatrix}$$ which is written in the simplified from $$m \ddot{\vec{r}}_{\mu,\nu}=F_{\mu,\nu}=k\left(\sum\limits_{\nu=1}^n R_{\mu,\nu}M_{\nu,\mu} + \sum\limits_{\mu=1}^{n1} \acute{M}_{\nu,\mu} R_{\mu,\nu} \right)$$ where $M_{\nu,\mu}$ and $\acute{M}_{\nu,\mu}$ are matrices given by

\begin{bmatrix} 2 & -1 & 0 & \dots&\dots & -1\\ -1 & 2 & -1 & \dots&\dots&\vdots\\ 0 & -1 & 2 & -1 & \dots& \vdots\\ \vdots& \vdots& \vdots& \ddots &\quad&\vdots\\ \vdots& \vdots & \vdots& \quad& \ddots&\vdots\\ -1& 0& \dots&\dots& -1 &2 \end{bmatrix}

and $M_{\nu,\mu}$ and $\acute{M}_{\nu,\mu}$ have dimensionality $n$ and $n1$ respectively and the rest of the elements in $\acute{M}_{\nu,\mu}$ go to zero. Which maintains the matrix dimensions of $F_{\mu,\nu}$

How can I simplify this to get the eigen-frequencies of the sphere? What are the conserved quantities ? How can I do it better ?

Is it possible to go this via action angle variables ?