From these equations it follows that X = A2 = 1/2 and V2 = -V1. What this means is that all we can do is choose in the center-of-mass frame a new direction vector for one of the two colliding particles. Everything else is then determined. The other particle goes in the opposite direction, and the total kinetic energy in the center-ofmass frame is divided evenly between the two particles. Thus, the only element of randomness is in how the new direction vector is chosen. If all directions are assumed to be equiprobable, then it can be shown that no matter what the initial distribution of velocity is, the system tends under iteration to a limiting distribution that is the standard normal distribution in three-dimensional Euclidean space R3. We have thus rederived the Maxwell-Boltzmann distribution of velocities. Here again we can go further and consider more complicated redistribution laws. Suppose one allows ternary collisions instead of binary collisions. Then there are more degrees of freedom, and the problem again becomes interesting mathematically. The results of our analysis show that the situation is much like the redistribution of energy in that the limiting distribution of velocity depends on the law of redistribution of velocity. Problem 2. Geometry, Invariant Measures, and Dynamical Systems The intimate relationship among geometry, measures, and dynamical systems that was elucidated in the last century continues to deepen and hold our attention today. Poincaré made several monumental contributions to this development in his treatise Les Méthodes Nouvelles de la Mécanique Céleste. One major issue he considered concerned the stability of motion in a gravitational field such as that of our solar system. Would small perturbations from any given set of initial orbits lead to a collision of the planets? A tremendous amount of work had been done on this dynamical system, but the governing system of differential equations remained unsolved. Faced with this situation, Poincaré made a wonderful flanking maneuver by introducing "qualitative" methods that involved measures. For the setting consider the motion of N bodies and the corresponding phase space S, whose 6N coordinates code the position and momentum of each of the N bodies. The phase space is a subset of Euclidean 6N-space and each point of S corresponds to a state of the system. Consider T, the time-one map of S. That is, if s is the initial state of the system, then T(s) is the state of the system one time unit later. Now, various notions of stability can be given in terms of the properties of T. One of these is recurrence, or, as Poincaré said, "stabilité à la Poisson." A state s is said to be recurrent provided that if the system is ever in s, then it will return arbitrarily close to s infinitely often. Formally, s is recurrent provided that for every open region U about s there are infinitely many positive integers n such that T"(s) is in U. Poisson had earlier attempted to show this kind of stability for the restricted three-body problem. Poincaré used the fundamental tenet of measure theory, countable additivity, to prove that the set of all points s in the phase space for which recurrence does not occur is of measure zero. Recurrence Theorem: Let B = (s € S❘s is not recurrent). Then B has measure zero. 79 currence Theorem") is a shining jewel that made clear to the mathematical world the importance of countable additivity in the development of measure. But what measure did Poincaré have in mind here? After all, there is an entire grab bag of measures on the subsets of S. In the case of the N-body problem, since the system is a Hamiltonian system, the geometry of the phase space clearly indicates the correct measure. Let us see why. Liouville had proved the seminal result that if the map T that describes the time evolution of the system is a Hamiltonian, then T is volume-preserving in the phase space. That is, if U is an open set or region, then v(U)= v(T(U)), where v(E) is the volume of E. Poincaré carried out his analysis on a "surface of constant energy." Since the N-body problem is a conservative system, the function T leaves the total energy invariant and therefore maps each such surface into itself. Moreover, since T is a Hamiltonian, it is volume-preserving on this surface. Consequently, the geometric structure of the surface determines the appropriate measure to use. Since the surface is a manifold, by definition there is a positive integer m such that each point of S lies in a region that is geometrically the same as a piece of Euclidean m-dimensional space. So, the measure to use on the manifold S is the one we naturally associate with Euclidean m-dimensional space, namely, m-dimensional volume. Geometry and Dynamical Systems To summarize, the N-body problem is a classical dynamical system in which the time-one map T is a continuous one-to-one map of the phase space X onto itself. The inverse map, T-1, is also continuous. Thus, T is a homeomorphism. There is a natural measure on the phase space X that is invariant under T. From one point of view, this measure is the volume element corresponding to the dimension of the phase space. From another viewpoint the natural invariant measure expresses the fact that the system is a Hamiltonian system. In the phase space X a surface S of constant energy forms an invariant set, and again there is an invariant measure on S corresponding to our ordinary notion of volume. The set B of all points that are not recurrent is also an invariant set with respect to T. However, it is not at all clear that we can define some natural invariant measure on B that is both nonzero and invariant under T. Many dynamical systems being studied today "live" on invariant sets that, like B, are not manifolds. Instead they are "pathological" sets, sets that at one time were thought to be the private domain of the purest and most abstract mathematicians. The examples range from Cantor sets to nowhere-differentiable curves to indecomposable continua. Many of these pathological invariant sets are "strange attractors" of dynamical systems; the system is "attracted" in the sense that it will eventually end up on the set from any starting point. (The discovery of one of the first strange attractors is described in the section Cubic Maps and Chaos of the article "Iteration of Maps, Strange Attractors, and Number Theory-An Ulamian Potpourri.") Properties of Invariant Sets. Let us now indicate some of the problems and techniques used in studying such sets in the context of dynamical systems. We will consider discrete dynamical systems, that is, systems in which the time evolution is described by discrete steps. We consider a function T that maps a space X into itself and the iterates of T, that is, T', T2, T3,..., where Tn+1(x) = T(T" (x)). We are interested in an invariant set—a subset M of X such that T(M) CM. The simplest invariant set consists of a fixed point x such that T(x) = x; a more complicated invariant set is a periodic orbit, a set consisting of the points x,T(x),...,T"-1(x), and T”(x) = x. Invariant sets are further classified according to how points near the invariant set behave under T. An invariant set M is called an attractor if there is a region U surrounding M such that if x Є U, then T" (x) gets closer and closer to M as n increases. On the other hand, M is called a repeller if there is a region U surrounding M such that if x Є (UM), then T" (x) is not in M for n sufficiently large. For example, if X is the real number line, then O is an attracting fixed point for T(x) = x/2 and a repelling fixed point for T-1(x) = 2x. The intrinsic properties of an invariant set are also of interest. For example, one might want to know whether there is a point x of M such that the orbit of x, that is, x,T(x), T2(x),..., is dense in M. If T is an irrational rotation of the plane, then the unit circle is invariant and the orbit of every point on the circle is dense in the circle. Another possibility is that T is topologically mixing on M; that is, for every region U of M there is some n such that M CT" (U). One central problem we will look at in some depth is the construction of “natural” or useful invariant measures for the sets M. In particular we want a measure μ such that μ(X – M ) = 0 and μ (T−1(B)) = μ(B) for each measurable subset B of M. That is, the measure is zero for points outside the invariant set M and is invariant with respect to the inverse of T. Cantor's Set as an Invariant Set. Let us consider a simple example of a map whose invariant set is Cantor's middle-third set. Let X be the real number line and let T(x) = (3/2)(1 − 2x 1). Then T is a two-to-one map of X into itself, the "triangle function" whose graph is shown in Fig. 7. This transformation can also be written in the following form: T(x)= { 3x if x 3(1-x) if x 1/2 1/2. = Now consider what happens to x under the iterates of T. If x < 0, then T" (x) = 3"x and T" (x)--∞. If 1 < x, then T(x) < 0 and higher iterates are given by 3"(1 − x). Again, T" (x) -∞. Thus, the iterates of all points outside the interval [0,1] are repelled. On the other hand, x = 0 is a fixed point, and, since T(1/4) = 3/4 and T(3/4) 1/4, the set {1/4, 3/4} forms a periodic orbit of order 2. It turns out that there is a natural invariant set under the iterates of T that lies in the interval [0,1]. To find it we consider successive iterations of T and keep track of the parts of the interval [0,1] that are mapped outside the interval by each interation. The first few iterations of T are illustrated in Fig. 8 and are described below. If x is in the open interval (1/3,2/3), T(x) > 1, and thus T maps this open interval out of the interval [0,1]. The two intervals J1 = [0, 1/3] and J2 = [2/3, 1] are each mapped onto [0,1]. Thus, J UJ2 consists of all points remaining in the interval [0,1] after one iteration. What points of J, remain in [0,1] after the second iteration? The middle third of J1, namely (1/9,2/9) is mapped out of the interval [0,1] by the second iteration of T, and the two subintervals J11 [0, 1/9] and J12 = [2/9, 1/3] make up the points of J, that remain in [0, 1] after two iterations of T. Similarly, the middle third of J2, (7/9, 8/9), is mapped out of [0,1] by T2, and the two subintervals of J2, J21 = [2/3,7/9] and J22 [8/9, 1], make up the points of J2 that remain in [0,1] under T2. Continuing this analysis, we find that the points of [0,1] that remain in [0,1] after n iterations of T consist of 2" intervals. Moreover, they are precisely the same 2" intervals that appear in the construction of Cantor's famous middle-third set. Thus, Cantor's middle-third set, call it M, is invariant under T, and if x M, then for some k, Tk (x) is not in [0,1]. Thus, if x M,T" (x) -∞. The Cantor set is a repellent invariant set of T, and this map is also topologically mixing on M. = = -> Hausdorff Measure and Dimension. If we think of T as an analog of a dynamical system whose motion in phase space is restricted to a Cantor set we might like to find a natural measure on this set. Our problem is: Which one of the many possible invariant measures is useful? One clue for determining the appropriate measure for the N-body problem was the fact that the phase space is a manifold and we therefore know the dimension of the space. We could then use the corresponding volume in the Euclidean space of that dimension to guide us to the correct measure. But what do we do with the Cantor set of our example? What is its dimension? In the early part of this century Felix Hausdorff developed an approach for determining the dimension of a general metric space (a space with a notion of a metric, or distance, between points) in terms of measures associated with the metric. It is perhaps surprising at first that the dimension of a space may not be an integer. Such spaces have been christened fractals by Mandelbrot, and he has provided many examples of their occurrence in physical phenomena. The idea behind Hausdorff's generalization of dimension is very simple Fig. 8. An invariant set for the transformation defined in Fig. 7 consists of all the closed subintervals of [0, 1] that are not mapped outside [0, 1] after an infinity of iterations of T. The idea is illustrated here with the first three iterations of T. The intervals J1 = [0,1/3] and J2 = [2/3, 1] are not mapped outisde [0, 1] by the first iteration; the intervals J11 = [0, 1/9], J12 = [2/9, 1/3], J21 = [2/3,7/9], and J22 = [8/9,1] are not mapped outside [0, 1] by the second iteration; and the intervals J111 = [0,1/27], J112 [2/27, 1/9], J121 = [2/9,7/27], ..., and J222 = [26/27, 1] are not mapped outside [0, 1] by the third iteration. If this process is continued indefinitely, an invariant set for T is found to be Cantor's middle-third set. 312 |