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Unformatted text preview: Problem Set 5 Calculus 1 Abdullah Khalid, Hassan Bukhari December 31, 2010 Question 3 1 Consider the square in Figure 1. M = { x 1 ,x 2  ≤ x i ≤ 1 } The position of two vehicles (one on the first road, the other on the second) can be CARS WAGONS Figure 1: Phase space of a pair of vehicles indicated by a point of the square M : it suffices to denote by x i ; the portion of the distance from A to B on road i contained between A and the vehicle on that road. The set of all possible positions of the vehicles corresponds to all possible points of the square M . This square is called the phase space and its points are the phase points. Thus each phase point corresponds to a definite position of the pair of vehicles, and every motion of the vehicles is depicted by the motion of a phase point in the phase space. For example, the initial position of the cars (in city A ) corresponds to the lower left corner of the square ( x 1 = x 2 = 0), and the motion of the cars from A to B is depicted by a curve leading to the opposite corner. In exactly the same way the initial position of the wagons 1 Copied from V. I. Arnold’s amazing book “Ordinary Differential Equations”. The reason why this explanation was copied was that we did not feel like substituting Arnold’s words for our own! 1 corresponds to the lower right corner of the square ( x 1 = 1 ,x 2 = 0), and the motion of the wagons is depicted by a curve leading to the opposite corner of the square. But any two curves in the square joining different pairs of opposite vertices must intersect. Therefore no matter how the wagons move there will come a time at which the pair of wagons occupies a position at which the pair of cars was located at some instant. At that instant the distance between the centers of the wagons will be less than 2 l . Hence they will not be able to pass....
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This note was uploaded on 01/10/2011 for the course EE 100 taught by Professor Razasuleman during the Spring '10 term at University of Engineering & Technology.
 Spring '10
 RazaSuleman

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