I'm trying to simulate the motion of a particle (a position vector) that is constrained to living on the surface of a unit sphere. Each time-step, $\Delta t$, the particle moves in some direction on the sphere's surface according to some representation of velocity/ acceleration/ etc. Think Euler method in spherical coords. I feel like this should be a straight-forward task, but I'm suffering a major mind block. In fact, I'm not even sure if I'm posing the question correctly.
To state it differently, given two previous positions on a unit sphere, $\vec{r}_{t-1}$ and $\vec{r}_{t}$, integrate forward to find the next position $\vec{r}_{t+1}$, assuming linear dynamics (i.e., the particle is moving in a straight "line" on the surface of the sphere).
To test if the model is implemented correctly, I should be able to select an arbitrary point on the sphere, give it a constant velocity in some arbitrary direction, and it should circle around the sphere and return to its starting position.
What's the optimal way to do this? Should the particle's velocity be in terms of $\theta$ (colatitude) and $\phi$ (azimuth), or something else? Thanks for any pointers!
If you denote by $X(t)$ the position of the particle at time $t$, the set of differential equations that describe the motion is
$$ \ddot X=\frac 1m \left( F - \dfrac{m \dot X^T H \dot X + \ \nabla \Phi^T F}{\|\nabla \Phi \|^2} \nabla \Phi\right), \quad t >0, $$
where $\Phi(x) = x_1^2+x_2^2+x_3^2-1$, $\nabla \Phi$ is its spatial gradient and $H$ its Hessian matrix.
Just pick a initial position on the sphere and an initial velocity tangent to the surface of the sphere. Most ODE solvers you give you nice solutions if you use appropriate time steps (you should rewrite the system as 6 differential first order equations if you want to use existing solvers, e.g. ode45 from matlab or NDSolve from Wolfram).