Harry Pollard

Singularities of the n-body problem. I

Little is known about the nature of the singularities of the n-body problem. While it is plausible to suppose that they are due to collisions, this has never been established, except when n = 2 or n = 3. In the general case the best that can be said at present is the fact, due to PAINLEV~ [5], that a singularity occurs at the time to if and only if the minimum of the mutual distances between pairs of particles approaches zero as the time t approaches to. In the present paper we shall investigate the problem of singularities due to collisions. We define a singularity at time to to be due to collisions i f as t ~ to each particle approaches a definite position in the inertial coordinate f rame. This means, in view of PAINLEV~S theorem, that at least two particles approach the same point. In 1908 VON ZEIPEL [4] published a statement to the effect that if the system remains bounded as t--+ to, then a singularity at time t o is due to collisions. His proof is erroneous, and the assertion still stands as a conjecture. The purpose of the present paper is to obtain necessary and sufficient conditions for a singularity due to collisions. It will be supposed that the origin of coordinates is fixed at the center of mass, and that the singularity occurs as t ~ 0 +. The following notation will be used. The symbols m k, re, vk denote respectively the mass, position and velocity of the kth particle. We define further

Escape from a gravitational system of positive energy

It is proved in this article that under the condition defined in Equation (6), whent→∞, at least two particles must escape from a Newtonian gravitational system of positive energy.

A variational problem related to an optimal filter problem with self-correlated noise

in the class Y. The filter problem that generates the variational problem will now be described. A precise mathematical formulation of the filter problem and its relation to the variational problem will be given in succeeding sections. Let (D denote the class of functionsf of class C(2) on (-oo, oo) such that If(t)I < 1 for all t. A functionf in (D represents an incoming signal. This signal is accompanied by a noise g. The problem is to filter out the noise for the whole class (D in an optimal fashion using the following criterion of optimality. If the function K represents the filter, then at time t the deviation between the actual signal f(t) and the filtered signal is given by

Addenda to “A variational problem related to an optimal filter problem with self-correlated noise”

The explicit solution is given of a nonclassical variational problem that is related to an optimal filter problem.

Disintegration and Escape

For the purpose of this paper a gravitational system is a set of a finite number n of mass-particles governed by Newton’s Law, and moving through all future time without singularity.

Gaussian representations related to heat conduction

A question which arises in statistics is whether the representation (1. l) o f f ( x ) is valid for more than one value of a, and if so, what is the maximum value for a. We have at once from (1.1) that ONf(x)N(4na) -~. Since a frequency function cannot vanish identically, we see at once that there must be a finite least upper bound for acceptable values of a. Consider the ex am p le f (x )= (n ) -~e -x~. It has the representation (1.1) for 0 < a < 88 with

THE CURRENT STATE OF THE N-BODY PROBLEM

This chapter describes the current state of the N -body problem. It highlights the physical conditions corresponding to the occurrence of a singularity in the analytic solution of the problem. The major classical result concerning problem asserts that a singularity occurs at time t 1 if and only if r → 0 as t → t , where r is the minimum distance between pairs of particles. This does not assert the occurrence of a collision. The chapter presents an assumption that T is the kinetic energy, h = T − U the total energy, I is one half the moment of inertia related to the preceding quantities by the Lagrange–Jacobi identity I = 2T − U . A standard theorem states that if I and T remain bound for all time, then the time averages and U exist. A particle must escape from a system of positive energy. The chapter presents a proof that if U is square-integrable, then a particle must escape.