I have already posted this question yesterday in Physics Stack Exchange: https://physics.stackexchange.com/q/711270/334688
However, they've suggested me to repost it here where it may be more relevant for the control theory community:
Let's assume to have a smooth curved surface with metric tensor $h$ and a stationary force field $f$ on such surface. Imagine that we now have an overdamped particle moving on it following the equation of motion: \begin{equation} \dot{x}^k = v_0 e^k + \mu f^k \ , \end{equation} with $\mu$ being the particle mobility, $v_0$ its constant self-propulsion speed and $e^k$ the orientation unit vector, i.e. such that $e_ke^k = 1$. I would now like to implement a policy for $e^k$ such that the particle points always towards a fixed target on the surface regardless of the force field. I'd call this the "Straight Policy".
I was thinking to use the fact that in absence of the force field, we know that the Riemannian geodesic equation \begin{equation} \ddot{x}^k+\Gamma_{ij}^k\dot{x}^i\dot{x}^j=0 \Longrightarrow \dot{e}^k+v_0\Gamma_{ij}^ke^ie^j=0 \end{equation} corresponds to such policy because in this case the shortest path is also the "straight" one. However, I can't find a consistent way to make use of this to solve the problem.
My best shot so far has been the following. Differentiating the e.o.m. wrt time we get: \begin{equation} \ddot{x}^k = v_0\dot{e}^k + \mu\dot{x}^i\partial_i f^k \ , \end{equation} then we can use the Riemannian geodesic solution for $\dot{e}^k$ to get: \begin{equation} \ddot{x}^k = -v_0\Gamma_{ij}^k(\dot{x}^i -\mu f^i)(\dot{x}^j -\mu f^j)+ \mu\dot{x}^i\partial_i f^k \ . \end{equation} However, I am pretty sure this is wrong because I have then tried to check that $v_0e^ke_k=(\dot{x}^k -\mu f^k)(\dot{x}_k -\mu f_k)=v_0$ writing a Mathematica code, but turns out this is not the case.
Where am I wrong here? Do you have any suggestion to solve this problem?
In a nutshell: my question amounts to finding the correct expression for the "Straight Policy" (SP) on smooth curved manifolds and in presence of a stationary force field. By SP, once again I mean the one where the active particle keeps pointing "straight" towards a specific target throughout its motion. The corresponding solution in absence of the force would be the Riemannian geodesics, but how can the same protocol be formally implemented when an external force is present?
I hope the question is clear enough and do not hesitate to ask for clarifications in the comments. I can edit the post accordingly. Thanks for your help!