Potential for chasing/pursuit problems There are many interesting kinematics problems, where the velocity vector of one moving body points towards another moving body. For example, consider the well-known problem of a dog chasing a rabbit (which follows a straight line with velocity $v$), where the speed of the dog stays $v$, but $\vec{v}$ always points towards the moving rabbit.
I know that chasing/pursuit problem can be solved, and I am not asking for a full solution.
But from a dynamical point of view, the velocity vector $\vec{v}$ changes with time, so there must be an acceleration, or "force" acting from the moving rabbit on the dog. But I would not know how to find this force. And going even one step further, is it possible to derive a potential in which the dog is moving (for example, by moving in a co-moving reference-frame where the rabbit stays at rest)?
 A: I can't comment on your post because I don't have enough reputation, but the problem is extensively discussed here:
https://arxiv.org/pdf/0711.3293.pdf
On top of that you have to consider if the force field you're looking for is conservative or not. If a force field is conservative, the energy difference between two points is just the difference in energy between the two points.
$\Delta E = E_a - E_b$
and it is curl-less:
$\vec{\nabla}\times\vec{F} = 0$
Then you can write $\vec{F} = -\vec{\nabla}\phi$, where $ \phi = \phi(\vec{x})$ in general (no time dependence assumed).
I imagine if you know the velocity vector at any point you could find the change in momentum in the dog $\dot{\vec{p}} = m\dot{\vec{v}}$ and then you could find $phi$ as
$\phi = -m \int_C \dot{\vec{v}}\cdot d\vec{\ell}$,
which is analogous to the electric potential. This requires that
$\vec{\nabla}\times\dot{\vec{v}} = 0$,
be true, which I believe you can calculate from the arXiv paper I linked. I also believe that because here the time derivative and space derivative operator commute,
$\frac{d}{dt}(\vec{\nabla}\times\vec{v}) = 0$,
which implies that $\vec{\nabla}\times\vec{v}$ is a constant at all times.
A: The typical pursuit problem is entirely kinematic not dynamic. It does not involve forces and inertia at all, and there is no physical interaction between the pursuer and the pursued. Their motion is not caused by forces with known force laws; they do not accelerate as they get closer. The motion and position of the pursuer usually has no influence on the pursued.
The motion is entirely constrained by geometry, like that of rods and pistons, gear wheels and linkages in engineering problems. The participants are massless points, and their motion is defined in terms of their fixed ratio of speed over the ground and the fact that the 1st follows a fixed path while the 2nd always moves towards the 1st. 
If the pursuer and pursued are assigned masses then the forces on them can be deduced very simply from their accelerations - which, if speed is constant, are always tangential/centripetal. The forces are whatever is required for them to follow the constrained motion.  
The kinetic energy of each participant is a constant of the motion, but this is obvious since their speeds are fixed by definition.
