arXiv subjects
Publications and source records attributed to Marshall Hampton.
A new coordinate system on the tangent space to planar configurations is introduced to simplify some calculations on central configurations and relative equilibria in the $N$-body problem with a homogeneous potential, which includes the case of Newtonian gravity. These coordinates are applied to some problems on four-body central configurations to illustrate their utility.
The relative equilibria of planar Newtonian $N$-body problem become coorbital around a central mass in the limit when all but one of the masses becomes zero. We prove a variety of results about the coorbital relative equilibria, with an emphasis on the relation between symmetries of the configurations and symmetries in the masses, or lack thereof. We prove that in the $N=4$, $N=6$, and $N=8$ Newtonian coorbital problems there exist symmetric relative equilibria with asymmetric positive masses. This result can be generalized to other homogeneous potentials, and we conjecture similar results hold for larger even numbers of infinitesimal masses. We prove that some equalities of the masses in the $1+4$ and $1+5$ coorbital problems imply symmetry of a class of convex relative equilibria. We also prove there is at most one convex central configuration of the symmetric $1+5$ problem.
Central configurations and relative equilibria are an important facet of the study of the $N$-body problem, but become very difficult to rigorously analyze for $N>3$. In this paper we focus on a particular but interesting class of configurations of the 5-body problem: the equilateral pentagonal configurations, which have a cycle of five equal edges. We prove a variety of results concerning central configurations with this property, including a computer-assisted proof of the finiteness of such configurations for any positive five masses with a range of rational-exponent homogeneous potentials (including the Newtonian case and point-vortex model), some constraints on their shapes, and we determine some exact solutions for particular N-body potentials.
Central configurations give rise to self-similar solutions to the Newtonian $N$-body problem, and play important roles in understanding its complicated dynamics. Even the simple question of whether or not there are finitely many planar central configurations for $N$ positive masses remains unsolved in most cases. Considering central configurations as critical points of a function $f$, we explicity compute the eigenvalues of the Hessian of $f$ for all $N$ for the point vortex potential for the regular polygon with equal masses. For homogeneous potentials including the Newtonian case we compute bounds on the eigenvalues for the regular polygon with equal masses, and give estimates on where bifurcations occur. These eigenvalue computations imply results on the Morse indices of $f$ for the regular polygon. Explicit formulae for the eigenvalues of the Hessian are also given for all central configurations of the equal mass 4-body problem with a homogeneous potential. Classic results on collinear central configurations are also generalized to the homogeneous potential case. Numerical results, conjectures, and suggestions for future work in the context of a homogeneous potential are given.
Foldable paper constructions known as flexagons have been studied since 1939. In this paper we review the construction of hexagonal flexagons (hexaflexagons) and compute the number of distinct hexaflexagons with n faces.
We examine in detail the relative equilibria in the four-vortex problem where two pairs of vortices have equal strength, that is, Γ_1 = Γ_2 = 1 and Γ_3 = Γ_4 = m where m is a nonzero real parameter. One main result is that for m > 0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for all m but the isosceles trapezoid case exists only when m is positive. In fact, there exist asymmetric convex configurations when m < 0. In contrast to the Newtonian four-body problem with two equal pairs of masses, where the symmetry of all convex central configurations is unproven, the equations in the vortex case are easier to handle, allowing for a complete classification of all solutions. Precise counts on the number and type of solutions (equivalence classes) for different values of m, as well as a description of some of the bifurcations that occur, are provided. Our techniques involve a combination of analysis and modern and computational algebraic geometry.
The main result of this paper is the existence of a new family of central configurations in the Newtonian spatial seven-body problem. This family is unusual in that it is a simplex stacked central configuration, i.e the bodies are arranged as concentric three and two dimensional simplexes.