What Are the Trajectories of Bugs Chasing Each Other on a Square?

[littleHilbert]

Hello!

I'm thinking about the following problem at the moment:

Four bugs sitting at the corners of the unit square begin to chase one another with constant speed, each maintaining the course in the direction of the one pursued. Describe the trajectories of their motions. What is the law of motion (in cartesian/polar coordinates)?

I heard the problem is fairly known but I think I need some guidance now.

Now I started with polar coordinates and got stuck with what to do with r(t) in: \frac{d}{dt}\vec{r(t)}=\frac{d}{dt}{r(t)}\vec{e_r}+r(t)\frac{d}{dt}\vec{e_r}
I mean the objective is a diff. equation isn't it?... but what should I do with the variable radius? Is the leangth of a side of the square of some importance? What is the implication of the fact that the course is in the direction of the other bug?

 

[berkeman]

Wow, nasty problem. If it were just two bugs, and the chasee stayed along the edge of the square, then the trajectory of the chaser isn't too bad, just simple vector math. But the problem seems nonlinear to me, since the chasee is going to deviate his path as his chasee alters his, etc. I'm not sure how to set this one up in a closed form. I'd probably code it up first to see how the numerical solution played out, and then see if maybe the problem reduces to an infinite series that converges analytically...

 

[littleHilbert]

Well, I suppose it isn't too complicated, I mean it has nothing to do with nonlinearity or something. I suspect that the trajectory is a spiral. My problem is to show that this is so.

Ok, I went on like this: \frac{d}{dt}\vec{r(t)}=\frac{d}{dt}{r(t)}\vec{e_r} +r(t)\frac{d}{dt}\vec{e_r}=\frac{d}{dt}{r(t)}\vec{e_r}+r(t)\frac{d}{dt}\phi\vec{e_{\phi}}
Here we see, that v_r=\frac{d}{dt}r(t),v_{\phi}=r(t)\frac{d}{dt}\phi
Since the angle between the radius-vector and the velocity vector does not change (only the length of the radius-vector changes with time), we get:
v_r=-v\frac{\sqrt{2}}{2},v_{\phi}=v\frac{\sqrt{2}}{2}

I looked up a definition of a spiral, to be exact - the Archimedean spiral: r=a\phi+c;a,c=const. Now, looking at this equation, I can't get which parameter the radius-vector should depend on in the end...is it the angle phi or time?

 

[HallsofIvy]

That's a lovely little math problem but what does it have to do with physics? One problem "chase one another with constant speed" doesn't mean anything. I'm going to assume that each bug chases (moves directly toward) the one on it's right, at constant speed.

I wouldn't put it in polar coordinates. First recognize that the problem is "symmetric". Each bug moves, relative to its starting position, exactly the way all the others do. Set up a coordinate system (until you specify the coordinate system, your equations don't mean anything) with the origin at the starting point of one bug, the others at (1, 0), (0, 1), and (1, 1). If the position, at time t, of one bug is (x,y) then the bug it is chasing is positioned distance y from the x= 1 line, x above the y-axis: its position is (1-y, x). That bug heads directly from (x,y) to (1-y, x) and so the tangent to its path at that instant is \frac{dy}{dx}= \frac{x}{1-y}. That's a fairly easy separable differential equation.