Moving under the influence of a vector field

2012-04-26T21:18:17Z

I have a continuously varying vector field $v(p)$ on $\mathbb{R}^2$, and a particle at point $p$ in the plane that can move in a direction $u(p)$ as long as $u(p)$ is turned at most $\pi/2$ left of $v(p)$. So at any point $p$, the particle can move in a quarter-circle of directions: from $v(p)$ to $v(p)$ rotated $90^\circ$ counterclockwise.

I would like to identify the points in $\mathbb{R}^2$ reachable from a given start point $p_0$ under this constraint. For example, suppose the vector field is determined by a rotation about a fixed center $c$. Then the reachable points are just those in the disk centered on $c$ with radius $|p_0 - c|$:

I can write down equations, in terms of dot- and cross-product, but they are not revealing to me.

Q. Is there some clean formulation of this problem that suggests a computationally feasible identification of the reachable points?

Thanks for any insights/ideas!

Answer by fuzzytron for Moving under the influence of a vector field

2012-04-27T01:47:08Z

Sounds like a distribution except that instead of having linear subspaces you have cones. There's this paper: Langerock, "Conic Distributions and Accessible Sets," but it sounds an awful lot like your question (and I wonder if that's where you're starting from in the first place!). It also doesn't say anything about the computability of the accessible set, though they do provide some characterization.

Moving under the influence of a vector field - MathOverflow

Moving under the influence of a vector field

Answer by fuzzytron for Moving under the influence of a vector field