Richard Moeckel

Linear stability of relative equilibria with a dominant mass

A criterion for the linear stability of relative equilibria of the Newtoniann-body problem is found in the case whenn−1 of the masses are small. Several stable periodic orbits of the problem are presented as examples.

Measuring the distance between time series

Abstract To evaluate models of dynamical systems, researchers have traditionally used quantitative measures of short term prediction errors. However, for chaotic or stochastic systems, comparison of long term, qualitative behaviors may be more relevant. Let x = (x0,…,xn) be a sequence of real numbers generated by sampling a dynamical system or stochastic process and suppose y = (y0,…,yn) is another sequence, generated by a mathematical model of the process which generated x. In this paper we consider several ways of assigning a distance d(x, y) which measures the difference in long term behavior.

Bifurcation of spatial central configurations from planar ones

Central configurations are important special solutions of the Newtonian N-body problem of celestial mechanics. In this paper a highly symmetrical case is studied. As the masses are varied, spatial central configurations appear through bifurcation from planar ones. In particular, spatial configurations can be found which are arbitrarily close to being planar.

Geodesics on modular surfaces and continued fractions

A connection between the symbolic description of the geodesic flows on certain modular surfaces and the theory of continued fractions is developed. The well-known properties of these dynamical systems then lead to some new results about continued fractions.

Heteroclinic Phenomena in the Isosceles Three-Body Problem

When two of the three particles have equal masses, the three-dimensional three-body problem has a subsystem consisting of motions for which the configuration of the particles is always an isosceles triangle. This subsystem has only two degrees of freedom. Geometrical methods are used to construct an invariant set containing a variety of periodic orbits which exhibit close approaches to triple collision and wild changes of configuration. Furthermore, orbits heteroclinic between these periodic orbits as well as oscillation and capture orbits are found. The whole invariant set is described using symbolic dynamics.

A computer-assisted proof of Saari’s conjecture for the planar three-body problem

The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are the only solutions with constant moment of inertia. This result will be proved here for the planar problem with three nonzero masses with the help of some computational algebra and geometry.

Finiteness of stationary configurations of the four-vortex problem

We show that the number of relative equilibria, equilibria, and rigidly translating configurations in the problem ot tour point vortices is hnite. The proof is based on symbolic and exact integer computations which are carried out by computer. We also provide upper bounds for these classes of stationary configurations.

Linear Stability Analysis of Some Symmetrical Classes of Relative Equilibria

The linear stability of several classes of symmetrical relative equilibria of the Newtonian n-body problem are studied. Most turn out to be unstable; however, a ring of at least seven small equal masses around a sufficiently large central mass is stable.

The inverse problem for collinear central configurations

We consider the problem: given a collinear configuration of n bodies, find the masses which make it central. We prove that for n ≤ 6, each configuration determines a one-parameter family of masses (after normalization of the total mass). The parameter is the center of mass when n is even and the square of the angular velocity of the corresponding circular periodic orbit when n is odd. The result is expected to be true for any n.

Respiratory sinus dysrhythmia persists in transplanted human hearts following autonomic blockade

1. The present study was performed to test whether beat‐to‐beat cardiovascular control in cardiac allograft recipients resides in cholinergic and/or adrenergic nerves that are intrinsic to the heart.