How do I write the gradient in angular coordinates ($\theta_1$, $\theta_2$, $\theta_3$)?

Question

I have to find $\tau$ by finding the gradient of $U(\theta_1, \theta_2, \theta_3)$, where my coordinates are $(\theta_1, \theta_2, \theta_3)$. I assume the gradient is not the simple Cartesian coordinates formula. However, using the spherical or cylindrical coordinates formula seems wrong since my coordinates are three angles and not an angle and a radius. So how do I express $\nabla U$ in this $\theta_1$, $\theta_2$, $\theta_3$ coordinate system? Is there a Jacobian I need, or some other way?

Edit - To make this question clearer, is there a way to express the $\nabla$ operator in a roll-pitch-yaw $\theta_1, \theta_2, \theta_3$ coordinate space? Analogous to how in Cartesian coordinates $\nabla = \frac{\partial}{\partial x} e_x + \frac{\partial}{\partial y} e_y + \frac{\partial}{\partial z} e_z$. I know that the general way to do something like this is via the Jacobian matrix arising from the metric, but am unsure how to do it in this three angle space.

Answer 1

$\def \b {\mathbf}$

$~\vec \nabla~$ Operator and Jacobian -Matrix

\begin{align*} &\text{Nabla Operator sphere coordinate is }\\\\ &\vec{\nabla}=\vec{e}_r\,\frac{\partial}{\partial r}+\frac{1}{r} \, \vec{e}_\theta\,\frac{\partial}{\partial \theta}+ \frac{1}{r\,\sin(\theta)}\, \vec{e}_\phi\,\frac{\partial}{\partial \phi} \end{align*}

with the sphere position vector $~\vec P~$ \begin{align*} &\vec{P}=\left[ \begin {array}{c} r\cos \left( \phi \right) \sin \left( \theta \right) \\ r\sin \left( \phi \right) \sin \left( \theta \right) \\ r\cos \left( \theta \right) \end {array} \right]\quad, \vec e_r=\frac{\partial\vec P}{\partial r}\quad, \frac{1}{r}\,\vec{e}_\theta=\frac{\frac{\partial\vec P}{\partial \theta}}{|\frac{\partial\vec P}{\partial \theta}|}=\hat{\b{e}}_\theta\quad, \frac{1}{r\,\sin(\theta)}\, \vec{e}_\phi=\hat{\b{e}}_\phi\quad\Rightarrow\\ \end{align*} \begin{align*} &\vec{\nabla}=\underbrace{\begin{bmatrix} \hat{\b{e}}_r & \hat{\b{e}}_\theta & \hat{\b{e}}_\phi \\ \end{bmatrix}}_{\hat{\b{J}}_(\,3\times 3)} \begin{bmatrix} \frac{\partial}{\partial r} \\\\ \frac{\partial}{\partial \theta} \\\\ \frac{\partial}{\partial \phi} \\\\ \end{bmatrix}\quad ,\b J_{i,j}=\frac{\partial P_i}{\partial \b q_j}\quad, \b q=\begin{bmatrix} r & \theta & \phi \\ \end{bmatrix}^T \end{align*} where $~\b J~$ is the Jacobian-Matrix , and "hat" is the matrix column norm

your case

\begin{align*} &\vec{P}=\b R_z(\psi)\,\b R_y(\theta)\,\b R_y(\phi)\, \begin{bmatrix} u_x \\ u_y \\ u_z \\ \end{bmatrix}\quad, \b q=\begin{bmatrix} \psi \\ \theta \\ \phi \\ \end{bmatrix} \end{align*}

---

the torque is:

$$\b\tau=\b r\times\b F= -\b r\times\,\hat{\b{J}}\,\begin{bmatrix} \frac{\partial U}{\partial\psi} \\ \frac{\partial U}{\partial\theta} \\ \frac{\partial U}{\partial\phi} \\ \end{bmatrix}$$

Answer 2

Using roll, pitch, and yaw is overspecified, and does not tell you the distance. For example, telling a bomber jet where to fire, you could specify pitch angle and yaw angle, but roll angle does not add anything. The third thing you need is distance downrange.

Also, force is the (negative) derivative of potential energy of a system with respect to some degree of freedom $q$:

$$F=-\frac {dU}{dq}$$

it only becomes a 3D gradient when the potential energy is a field extending over 3D space (like a gravitational or electric field). In the case of a displaced spring the degree of freedom $q$ might be spring displacement $\Delta x$ or for a pendulum might be angle from vertical $\theta$.

Answer 3

Based on: Is torque always equal to the derivative of potential energy with respect to rotation angle?

Since we have: $$ \hat{n}_{\theta_i}\cdot\vec{\tau} = -\frac{\partial U}{\partial\theta_i}$$

For $i=1,2,3$ , then you may reconstruct the Torque vector via:

$$ \vec{\tau} = -\sum_{i=1}^3 \frac{\partial U}{\partial\theta_i} \hat{n}_{\theta_i} $$

This is not the gradient, at least not in the usual sense of the word, although it has a very similar form. Most importantly: the angles $\theta_i$ may represent rotations from "shifted" axes, in particular this is true for Euler angles. So it's important to know whether the angles refer to an intrinsic or an extrinsic frame, in order to appropriately determine the $\hat{n}_{\theta_i}$'s.

See this for a lot more related information about rotation mechanics of rigid bodies.