Proving the Dogs Meet at a Single Point: A Mathematical Analysis

In summary, the conversation discusses the problem of N dogs chasing each other's tails on a plane with a speed of 1, and aims to prove that they will eventually meet at a single point. The conversation includes an attempt at a solution using a distance function and the chain rule, as well as a discussion about the sum of angles. The expert summarizer notes a small mistake in the second line that leads to a positive derivative, but points out that with the correction, the derivative is always negative.
  • #1
8daysAweek
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Homework Statement


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?

Thank you.



3. Relevant equations
attachment.php?attachmentid=31811&stc=1&d=1296560206.jpg

[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.
 

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  • #2
I do not understand where the second equality in your second lines comes from.
You regard your u's as vectors, but treat your p as 1 dimensional point.
It might be a good idea to write the distances explicitly in terms of the x and y coordinates of the positions of the dog. You can then also express the u in terms of the positions.
 
  • #3
p is a 2D point. I forgot to mention that d is the distance function between two points.
I got this result using the chain rule:
[tex]
\[\frac{{\partial d\left( {{p_i},{p_j}} \right)}}{{\partial {p_i}}} = \frac{{\partial \sqrt {{{\left( {{p_i} - {p_j}} \right)}^2}} }}{{\partial {p_i}}} = \frac{{\frac{1}{2}}}{{\sqrt {{{\left( {{p_i} - {p_j}} \right)}^2}} }} \cdot 2\left( {\left( {{p_i} - {p_j}} \right)} \right) \cdot 1 = \frac{{{p_i} - {p_j}}}{{d\left( {{p_i} - {p_j}} \right)}} = {u_{i \to j}}\]
[/tex]
 
  • #4
Ok, then your definitions are alright.

I don't have much time right now.
Did you remember that the sum of your angles has to be 2Pi (maybe a multiple of that). Maybe this will help you.
 
  • #5
Actually as your result goes, the derivative is always positive.

But you made a small mistake in your second line, that also shows in your last post.

As you defined it, [tex]\vec u_{k\to k+1}[/tex] is the vector going from [tex]p_k[/tex] to [tex]p_{k+1}[/tex], or in vector notation [tex]\vec u_{k\to k+1}=\vec p_{k+1}-\vec p_k[/tex].
Therefore
[tex]

\[\frac{\partial d\left( p_i,p_j \right)}{\partial p_i} =- u_{i \to j}\]

[/tex]
This will give you a minus in front of the N in your final result and then the derivative is always negative.
 

1. How does mathematical analysis prove that dogs meet at a single point?

Mathematical analysis uses mathematical principles and techniques to study and understand the behavior of various phenomena, including the movements of dogs. By analyzing the positions and movements of two or more dogs, mathematical analysis can determine whether or not they will eventually meet at a single point.

2. What types of data are needed for this mathematical analysis?

To prove that dogs meet at a single point, mathematical analysis requires data on the positions and movements of the dogs, such as their coordinates and velocities. This data can be collected through observations or experiments, or can be simulated using mathematical models.

3. Can this analysis be applied to any number of dogs?

Yes, this analysis can be applied to any number of dogs. The mathematical principles and techniques used in this analysis can be extended to any number of objects, making it applicable to multiple dogs.

4. What assumptions are made in this mathematical analysis?

This analysis makes several assumptions, including that the dogs are point objects with no size or shape, and that they follow a linear path without any external forces acting upon them. It also assumes that the dogs have consistent speeds and do not change direction suddenly.

5. How can this analysis be useful in real-life situations?

This mathematical analysis can have practical applications in various fields, such as biology and physics. For example, it can be used to study the movement patterns of animals, or to analyze the trajectories of objects in motion. It can also be used to make predictions and inform decision-making in situations involving multiple moving objects.

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