Suppose I can compute interaction energy of two rigid bodies as a function of their coordinates of centers of masses and Euler rotation angles (total 6 + 6 degrees of freedom). Now I can numerically compute force acting on the center of mass of the body by calculating numerical derivatives e.g. $F_x = (E(x + dx) - E(x - dx)) / (2 * dx)$. But if you do the same for Euler angles this doesn't give you torques. So how do I convert numerical derivatives of energy by Euler angles to the resulting torque on a body?
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The mathematically correct method is to use a Jacobian, which is a matrix of partial derivatives. It works by the chain rule, which for a gradient vector looks like: $$ \begin{pmatrix} \frac{\partial V(\theta,\psi,\phi)}{\partial x} \\ \frac{\partial V(\theta,\psi,\phi)}{\partial y} \\ \frac{\partial V(\theta,\psi,\phi)}{\partial z} \end{pmatrix} = \begin{pmatrix} \frac{\partial \theta}{\partial x} & \frac{\partial \psi}{\partial x} & \frac{\partial \phi}{\partial x} \\ \frac{\partial \theta}{\partial y} & \frac{\partial \psi}{\partial y} & \frac{\partial \phi}{\partial y} \\ \frac{\partial \theta}{\partial z} & \frac{\partial \psi}{\partial z} & \frac{\partial \phi}{\partial z} \end{pmatrix}\begin{pmatrix} \frac{\partial V(\theta,\psi,\phi)}{\partial \theta} \\ \frac{\partial V(\theta,\psi,\phi)}{\partial \psi} \\ \frac{\partial V(\theta,\psi,\phi)}{\partial \phi} \end{pmatrix} $$ The trouble is expressing the derivatives of $\theta$, $\psi$, and $\phi$ in terms of x, y, and z. Rotations about the fixed x, y, and z axes cannot form the Euler angles, so how do you find the derivatives? Seriously, I'm asking, I've been trying to figure it out for two weeks now. |
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OK. I found the answer: $$ \partial V/\partial \theta = N_x \cos \psi - N_y \sin \psi $$ $$ \partial V/\partial \phi = N_x \sin \theta \sin \psi + N_y \sin \theta \cos \psi + N_z \cos \theta$$ $$ \partial V/\partial \psi = N_z $$ Where $\theta, \psi, \phi$ are Euler angles and $N_x, N_y, N_z$ are Torque components. |
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