There are N dogs on the plane. Each dog chases the tail of the next dog with a speed of 1 (and the last dog is chasing the first). I want to prove that the dogs will eventually meet at a single point.
2. The attempt at a solution
I defined a function that is sum of distances between each dog and the next one. If I show that the derivative of this function over time is negative, then this distances will eventually become zero.
I have attached a scan of my attempt. As you can see, the result is not always negative. I know there are better way to solve this problem, I just want to understand what is wrong in my approach. Maybe the derivative at line 2 is wrong?
3. Relevant equations
[tex]p_k[/tex] is the location of the kth dog on 2D plane.
[tex]u_k[/tex] is the unit vector that points to the next dog, it is also the velocity of the kth dog.