I'm trying to calculate the new position of an object moving over the surface of a 3D sphere after some time Δt, from X1 to X2 always onto the surface of the sphere. The measurements/data I have of such object's displacement and sphere are:
- Translational velocity (cm/seg)
- Rotational velocity (deg/seg)
- Direction of displacement (deg)
- Radius of the sphere (cm) Here is an example of what I mean.
Then the idea is to update for each Δt the position of the object on the sphere. I suspect some linear algebra is needed but pretty lost at the moment. Any help would be greatly appreciated!
This is how the data frame looks like
| time | translational velocity | rotational velocity | direction |
|---|---|---|---|
| 0.01 | 2.36 | 6.45 | 0.78 |
| 0.02 | 1.12 | 1.19 | 0.62 |
| 0.03 | 1.67 | 4.45 | 1.51 |
| 0.04 | 1.25 | 5.39 | 1.67 |
You need to solve Euler-Lagrange equation of motion along the geodesic curve in your metric space: \begin{equation} \frac{d^2x^{\mu}}{dt^2} + \Gamma^{\mu}_{\alpha\beta}\frac{dx^{\alpha}}{dt}\frac{dx^{\beta}}{dt} = 0, \end{equation} where t is the time, $x^{\mu}$ are coordinate components, $\Gamma^{\mu}_{\alpha\beta}$ is the Christoffel symbols of the metric. The Cristoffel symbols $\Gamma^{\mu}_{\alpha\beta}$ are given by the following formula \begin{equation} \Gamma^{\mu}_{\alpha\beta} = \frac{1}{2}g^{\mu\nu}(\frac{\partial g_{\nu\alpha}}{\partial x^{\beta}} + \frac{\partial g_{\nu\beta}}{\partial x^{\alpha}} - \frac{\partial g_{\alpha\beta}}{\partial x^{\nu}}), \end{equation} where $g_{\alpha\beta}$ is the metric tensor. $g_{\alpha\beta}$ and $g^{\beta\gamma}$ are linked by the following equation \begin{equation} g_{\alpha\beta}g^{\beta\gamma} = \delta_{\alpha}^{\,\gamma} = \delta^{\gamma}_{\,\alpha}, \end{equation} where $\delta^{\gamma}_{\,\alpha}$ is the Kronecker symbol: \begin{equation} \delta^{\gamma}_{\,\alpha} = \begin{cases} 1,\text{ if }\gamma=\alpha\\ 0,\text{ otherwise} \end{cases}. \end{equation} So the matrix $g^{\alpha\beta}$ is inverse of the matrix $g_{\alpha\beta}$. The metric tensor is the tensor, which links the length $ds$ of differential minor arc with the coordinate differentials $dx^{\alpha}$: \begin{equation} ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu}. \end{equation} In the case of the sphere \begin{equation} g = \begin{pmatrix} 1 & 0\\ 0 & \sin^2\theta, \end{pmatrix}, \end{equation} so $ds^2$ is given by the following formula \begin{equation} ds^2 = d\theta^2 + \sin^2\theta d\phi, \end{equation} where $\theta$ is a polar angle, $\phi$ is axial angle. $\theta$ and $\phi$ are the components of the vector $x^{\mu}$ in this case. The differential $dx^{mu}$ can be written in the following form $dx^{\mu} = \frac{dx^{\mu}}{dt}dt$, where $dt$ is the time differential and $\frac{dx^{\mu}}{dt}$ is the generalized speed. Let us consider the Lagrange functional of the following form \begin{equation} \mathcal{L}[x^{\mu}] = \int^{s_1}_0 ds = \int^{s_1}_{0}\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}} = \int^{t}_{0}\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{dt}dt\frac{dx^{\nu}}{dt}dt} = \int^{t}_{0}\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{dt}\frac{dx^{\nu}}{dt}}dt. \end{equation} We can use the calculus of variations and find a function $dx^{\mu}(t)$ that minimizes our functional. To do this, you need to write down the Euler-Lagrange equation. In the case of this functional, it has the form \begin{equation} \frac{d^2x^{\mu}}{dt^2} + \Gamma^{\mu}_{\alpha\beta}\frac{dx^{\alpha}}{dt}\frac{dx^{\beta}}{dt} = 0. \end{equation} Since the functional $\mathcal{L}[x^{\mu}] $ has the meaning of length, the solution to the previous equation is a curve of shortest length connecting two points in a given metric space. Such curves are called geodesic curves. The non-zero Christoffel symbols in the case of sphere metric are (see the following link http://einsteinrelativelyeasy.com/index.php/general-relativity/34-christoffel-symbol-exercise-calculation-in-polar-coordinates-part-ii) \begin{equation} \Gamma^{\phi}_{\phi\theta}=\Gamma^{\phi}_{\theta\phi} = \frac{\cos\theta}{\sin\theta}, \\ \Gamma^{\theta}_{\phi\phi} = -\sin\theta\cos\theta, \end{equation} so the system of motion equation has the following form \begin{equation} \begin{cases} \frac{d^2\theta}{dt^2} - \sin\theta\cos\theta\left(\frac{d\phi}{dt}\right)^2 = 0,\\ \frac{d^2\phi}{dt^2} + 2\frac{\cos\theta}{\sin\theta}\frac{d\theta}{dt}\frac{d\phi}{dt} = 0. \end{cases} \end{equation} Let us introduce $\theta$-speed $u_{\theta}$ and $\phi$-speed $u_{\phi}$: \begin{equation} u_{\theta} = \frac{d\theta}{dt},\\ u_{\phi} = \frac{d\phi}{dt}. \end{equation} In this terms the system of motion equations has the following form: \begin{equation} \begin{cases} \frac{du_{\theta}}{dt} - \sin\theta\cos\theta u^2_{\phi} = 0,\\ \frac{du_{\phi}}{dt} + 2\frac{\cos\theta}{\sin\theta}u_{\theta}u_{\phi} = 0. \end{cases} \end{equation}
The useful information about motion along the sphere geodesics can be found here Geodesics of the Unit Sphere using Christoffel symbols.
The simplest way to simulate motion along geodesic curve is to rotate sphere in a such way that the motion will be at a constant $\phi^{\prime}$, for example, $\phi^{\prime} = 0$. In this case only $\theta^{\prime}$ changes with the time. Then calculate $\theta^{\prime}$ as $\theta^{\prime} = \omega t$. Where $\omega$ is a cyclic frequency. Then rotate sphere back into initial basis. There $\theta^{\prime}$ and $\phi^{\prime}$ are coordinates of the point object in the rotated basis.