Degenerate billiards in celestial mechanics

In an ordinary billiard trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than one, we say that the billiard is degenerate. Degenerate billiards appear as limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems shadowing trajectories of the corresponding degenerate billiards. This research is motivated by the problem of second species solutions of Poincaré.     

Random walk in celestial mechanics

The process of random walk can be demonstrated to occur in the three-body problem. This is shown to be the case for a family of horseshoe orbits. This offers a possible alternate approach to studying special motions of this problem and other problems in celestial mechanics.      

The n-centre problem of celestial mechanics for large energies

We consider the classical three-dimensional motion in a potential which is the sum of n attracting or repelling Coulombic potentials. Assuming a non-collinear configuration of the n centres, we find a universal behaviour for all energies E above a positive threshold. Whereas for n=1 there are no bounded orbits, and for n=2 there is just one closed orbit, for n≥3 the bounded orbits form a Cantor set. We analyze the symbolic dynamics and estimate Hausdorff dimension and topological entropy of this hyperbolic set. Then we set up scattering theory, including symbolic dynamics of the scattering orbits and differential cross section estimates. The theory includes the n–centre problem of celestial mechanics, and prepares for a geometric understanding of a class of restricted n-body problems. To allow for applications in semiclassical molecular scattering, we include an additional smooth (electronic) potential which is arbitrary except its Coulombic decay at infinity. Up to a (optimal) relative error of order 1/E, all estimates are independent of that potential but only depend on the relative positions and strengths of the centres. Finally we show that different, non-universal, phenomena occur for collinear configurations. Received October 16, 2000 / final version received June 18, 2001?Published online August 15, 2001     

Integrable problems of celestial mechanics in spaces of constant curvature

The technique of topological analysis of integrable problems developed by A. T. Fomenko is applied for studying certain problems of celestial mechanics.     

Macrocosmos/microcosmos: celestial mechanics and old quantum theory

The Bohr atom was a solar system in miniature. Despite many deep foundational questions related to the origin of quantized motion, rapid progress was made in its mathematical development and its apparently successful application to spectral line series. In United States, where celestial mechanics flourished throughout the 19th and well into the 20th century, mathematicians and physicists were well prepared for just this sort of problem and made it their own far faster than many areas of the new physics. This paper examines the link between classical problems of perturbation theory, three-body and N-body orbital trajectories, the Hamilton–Jacobi equation, and the old quantum theory. I discuss why it was comparatively easy for American applied mathematicians, astronomers, and mathematical physicists to make significant contributions quickly to quantum theory and why further progress toward quantum mechanics by the same cohort was, in contrast, so slow.     

Euler's problem and its applications in celestial mechanics and space dynamics

The numerous generalizations of the classical problem of two fixed centres are analysed, starting from the formulation of the problem and its solution by Euler in 1760 to the present day. The role played by numerous researchers in analysing this problem is noted. The publications cited indicate conclusively that the main results of generalizations of the problem and analytical and qualitative investigations had already been obtained in the nineteenth century and at the beginning of the twentieth century. Present-day researchers can only lay claim to a few occasionally productive and at the same time effective applications of individual generalizations (the Gredeaks problem, for example).     

On subharmonic functions dominated by certain functions

Given two kinds of functionsf(X) andh(y) defined on them-dimensional Euclidean spaceR ^m (m≧1) and the set of positive real numbers respectively, we give an estimation of growth of subharmonic functionsu(P) defined onR ^m+n (n≧1) such that $$u(P) \leqq f\left( X \right)h\left( {\left\| Y \right\|} \right)$$ for anyP=(X, Y),X ∈R ^m, Y ∈R ⁿ, where ‖Y ‖ denotes the usual norm ofY. Using an obtained result, we give a sharpened form of an ordinary Phragmén-Lindelöf theorem with respect to the generalized cylinderD ×R ⁿ, with a bounded domainD inR ^m.     

Avoiding collisions under topological constraints in variational problems coming from celestial mechanics

In a singular potential setting, we generalize a method which allows to show that minimizers under topological constraints of the action functional (or of the Maupertuis one) are collision-free. This methods applies to 3-dimensional problems of celestial mechanics exhibiting a particular cylindrical symmetry, as well as to planar problems of N-centre type, where it gives optimal results.     

Pseudo-rank functions on certain lattices

Pseudo-rank functions on bounded lattices are introduced and their properties are studied. It is shown that if the set of all pseudo-rank functions on a Boolean lattice is nonempty then it is a Choquet simplex.