Finding the eigenfrequencies of a vibrating sphere

Question

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 \begin{equation}{\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) \end{equation}

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

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

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{equation} \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} \end{equation}

Then The force can be written as \begin{equation} =\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 \end{equation} \begin{equation} +\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} \end{equation} which is written in the simplified from \begin{equation} 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) \end{equation} 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 ?

Answer 1

I did not check your algebra in detail. In essence you have the problem of small oscillations of a coupled system of mass points with linear forces between them described by a quadratic potential in the differences in coordinates of adjacent mass points. The problem can easily be solved by an orthogonal coordinate transformation so that the quadratic form of the Lagrangian is transformed to its principal axes which yields a system of decoupled oscillators in so-called normal coordinates and their eigenfrequencies. Equivalently, you can diagonalize the symmetric matrix of the system of Lagrange equations by an appropriate orthogonal transformation to find the eigenfrequencies and eigenmodes of your vibrating sphere point particle system.