Josep M. Cors, Glen R. Hall and Gareth E. Roberts
To appear in PhysicaD.
Abstract: For a class of potential functions including those used for the planar n-body and n-vortex problems, we investigate co-circular central configurations whose center of mass coincides with the center of the circle containing the bodies. Useful equations are derived that completely describe the problem. Using a topological approach, it is shown that for any choice of positive masses (or circulations), if such a central configuration exists, then it is unique. It quickly follows that if the masses are all equal, then the only solution is the regular n-gon. For the planar n-vortex problem and any choice of the vorticities, we show that the only possible co-circular central configuration with center of vorticity at the center of the circle is the regular n-gon with equal vorticities.
Marshall Hampton, Gareth E. Roberts and Manuele Santoprete
Journal of Nonlinear Science, 24, 39-92, 2014.
Abstract: We examine in detail the relative equilibria in the planar 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.
Gareth E. Roberts
SIAM Journal on Applied Dynamical Systems, 12, no. 2, 1114-1134, 2013.
Abstract: We study the linear and nonlinear stability of relative equilibria in the planar N-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vorticities is also nonlinearly stable. Two symmetric families, the rhombus and the isosceles trapezoid, are analyzed in detail, with stable solutions found in each case.
G. E. Roberts
PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies),
23, no. 9, 785-797, 2013.
Abstract: The notion that undergraduates are capable of making profound and original contribu- tions to mathematical research is rapidly gaining acceptance. Undergraduates bring their enthusiasm, creativity, curiosity and perseverance to bona ﬁde research problems. This article discusses some of the key issues concerning undergraduate mathematical research: selecting good research students, ﬁnding appropriate research questions, mentoring versus collaboration, and presenting and publishing student work. Some useful professional and ﬁnancial resources supporting undergraduate research are also highlighted.
Josep M. Cors and Gareth E. Roberts
Nonlinearity, 25, no. 2, 343-370, 2012.
Abstract: We classify the set of central configurations lying on a common circle in the Newtonian four-body problem. Using mutual distances as coordinates, we show that the set of four-body co-circular central configurations with positive masses is a two-dimensional surface, a graph over two of the exterior side-lengths. Two symmetric families, the kite and isosceles trapezoid, are investigated extensively. We also prove that a co-circular central configuration requires a specific ordering of the masses and find explicit bounds on the mutual distances. In contrast to the general four-body case, we show that if any two masses of a four-body co-circular central configuration are equal, then the configuration has a line of symmetry.
A Maple worksheet (pdf file) containing symbolic calculations about the isosceles trapezoid family
is available here.
L. Bakker, T. Ouyang, S. Simmons, D. Yan and G. E. Roberts
Celestial Mechanics and Dynamical Astronomy, 108, no. 2, 147-164, 2010.
Abstract: We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous binary collision orbits in the symmetric collinear four body problem with masses 1, m, m , 1, and also in a symmetric planar four-body problem with equal masses. In both problems, the assumed symmetries reduce the detemination of linear stability to the numerical computation of a single real number. For the collinear problem, this verifies the earlier numerical results of Sweatman for linear stability with respect to collinear and symmetric perturbations.
J. Kulevich, G. E. Roberts and C. Smith
Qualitative Theory of Dynamical Systems, 8, no. 2, 357-370, 2009.
Note: The final publication is available from Springer.
Abstract: Using BKK theory, we show that the number of equilibria (central configurations) in the planar, circular, restricted four-body problem is finite for any choice of masses. Moreover, the number of such points is bounded above by 196.
Sage worksheets containing all calculations:
General problem and finiteness computations and
Equal Mass case
Sage is a free mathematical software system available here.
G. E. Roberts and L. Melanson
Celestial Mechanics and Dynamical Astronomy, 97, 211-223, 2007.
Note: The final publication is available from Springer.
Abstract: Saari's conjecture adapted to the restricted three-body problem is proven analytically using BKK theory. Specifically, we show that it is not possible for a solution of the planar, circular, restricted three-body problem to travel along a level curve of the amended potential function unless it is fixed at a critical point (one of the five libration points.) Due to the low dimension of the problem, our proof does not rely on the use of a computer.
Ergodic Theory and Dynamical Systems, 27, 1947-1963, 2007.
Abstract: We show that the well-known figure-eight orbit of the three-body problem is linearly stable. Building on the strong amount of symmetry present, the monodromy matrix for the figure-eight is factored so that its stability can be determined from the first twelfth of the orbit. Using a clever change of coordinates, the problem is then reduced to a 2 X 2 matrix whose entries depend on solutions of the associated linear differential system. These entries are estimated rigorously using only a few steps of a Runge-Kutta-Fehlberg algorithm. From this we conclude that the characteristic multipliers are distinct and lie on the unit circle. The methods and results presented are applicable to a wide range of Hamiltonian systems containing symmetric periodic solutions.
Transactions of the American Mathematical Society, 358, No. 1, 251-265, 2006.
Abstract: For the Newtonian N-body problem, Saari's conjecture states that the only solutions with a constant moment of inertia are relative equilibria, solutions rigidly rotating about their center of mass. We consider the same conjecture applied to Hamiltonian systems with power-law potential functions. A family of counterexamples is given in the five-body problem (including the Newtonian case) where one of the masses is taken to be negative. The conjecture is also shown to be false in the case of the inverse square potential and two kinds of counterexamples are presented. One type includes solutions with collisions, derived analytically, while the other consists of periodic solutions shown to exist using standard variational methods.
Abstract: We compare the iterative root-finding methods of Newton and Halley applied to cubic polynomials in the complex plane. Of specific interest are those ``bad'' polynomials for which a given numerical method contains an attracting cycle distinct from the roots. This implies the existence of an open set of initial guesses whose iterates do not converge to one of the roots (ie. the numerical method fails). Searching for a set of bad parameter values leads to Mandelbrot-like sets and interesting figures in the parameter plane. We provide some analytic and geometric arguments to explain the contrasting parameter plane pictures. In particular, we show that there exists a sequence of parameter values c_{n} for which the corresponding numerical method has a superattracting n cycle. The c_{n} lie at the centers of a converging sequence of Mandelbrot-like sets.
Journal of Differential Equations 182, 191-218, 2002.
Abstract:
This paper concerns the linear stability of the well-known periodic orbits of Lagrange
in the three-body problem. Given any three masses, there exists a family of periodic solutions
for which each body is at the vertex of an equilateral triangle and traveling along
an elliptic Kepler orbit.
Reductions are performed to derive equations which determine the linear stability
of the periodic solutions. These equations depend on
two parameters:
the eccentricity e of the orbit, and the mass parameter
b = 27(m_{1} m_{2} + m_{1} m_{3} + m_{2} m_{3})/
(m_{1} + m_{2} + m_{3})^{2}.
A combination of numerical
and analytic methods are used to find the regions of stability in the be-plane.
In particular, using perturbation techniques it is rigorously proven that there are mass values
where the truly elliptic orbits are linearly stable even though the circular orbits are not.
Nonlinearity 12 757-769, 1999.
Abstract: An inequality is derived which must be satisfied for a
relative equilibrium of the N-body problem to be spectrally stable.
This inequality is studied in the equal mass case
and using simple geometric
estimates, it is shown that any relative equilibrium of N equal masses is not
spectrally stable for N > 24,305.
Physica D 127 141-145, 1999.
Abstract: It is generally believed that the set of relative
equilibria equivalence classes in the Newtonian N-body problem,
for a given set of positive masses, is finite.
However, the result
has only been proven for N=3 and remains a difficult, open
question for N > 4.
We demonstrate that the condition
for the masses being positive is a necessary one by finding a
continuum of relative equilibria in the five-body problem which (unfortunately)
includes one negative mass. This family persists in similar potential functions,
including the logarithmic potential used to describe
the motion of point vortices in a plane of fluid.
Hamiltonian
Systems and Celestial Mechanics (HAMSYS-98), Proceedings
of the III International Symposium, Patzcuaro, Mexico.
World Scientific Monograph Series in Mathematics 6 303-330, 2000.
Abstract: We study the linear stability of the
relative equilibria consisting of n bodies at the vertices
of a regular n-gon with an additional body of mass m at the center.
This configuration is shown to be linearly unstable when
n < 7. For n > 6, a value c_n is found such that
the configuration is linearly stable if and only if m > c_n.
This value is shown to increase proportionately to n^3.