Inverse matrix mass computation

[ebrattr]

Hi !
I've been thinking this problem a whole and I could not find an answer. I want to solve the following problem: suppose I have $N$ mass particles with absolute coordinates $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N$. Besides, I have the following contraints: for all $i=1,2,\ldots,N-1$, $|\mathbf{x}_{i+1}-\mathbf{x}_i|=L$ where $L$ is a length. In every particle I have a total force over it $\mathbf{F}_i$.

I used spherical coordinates to express every constraints and parametrize $\mathbf{x}_i$. However, when I compute the inverse matrix through these method (https://www.dropbox.com/sh/y25m55jpzrh7kqz/Y8EGX25lOQ), I got troubles. Since, I express a generalized coordiante $q_i$ as $\cos q_i = \dfrac{\mathbf{r}\cdot \mathbf{k}}{L}$ and I get: $\dfrac{\partial q_i}{\partial \mathbf r} = -\dfrac{\mathbf{k}}{L\sin q_i}.$ Therefore, when $q_i = 0,\pi$ the denominator is 0, and -><-.

What can I do ?

Thanks !

 

[Valerie Park]

One possible solution is to use other coordinates that are not affected by singularities. For example, you could try using cylindrical or cartesian coordinates instead of spherical coordinates. This would allow you to avoid the singularities when computing the inverse matrix. Another solution is to choose a different parametrization for your constraint. For example, in the spherical coordinates, you could parametrize the constraint as |x_i - x_{i+1}| = Lcos(theta_i), where theta_i is a parameter. This would allow you to avoid the singularities.