Gareth E. Roberts
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Gareth E. Roberts has a total of 22 co-authors. Here are the top co-authors by number of shared publications:
- Josep Maria Cors
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Manresa, Catalonia
6 shared publications - Montserrat Corbera
Facultat de Ciències, Tecnologia i Enginyeries, Universitat de Vic - Universitat Central de Catalunya (UVic-UCC), Vic, Spain
4 shared publications Clara Bodelón
Department of Applied Mathematics , University of Colorado , Boulder, Co , 80309, Usa
2 shared publications- Duokui Yan
Nankai University
2 shared publications John H. Hubbard
Department of Mathematics , Cornell University , Ithaca, Ny , 14853 , Usa
2 shared publications- Lennard F. Bakker
Brigham Young University
2 shared publications
Latest Publications
- Open Access
- 30 May 2025
- Published by IOP Publishing in Nonlinearity
- Vol. 38 (7) , 075001
On kite central configurations*
We study kite central configurations (c.c.’s) in the Newtonian four-body problem. We present a new proof that there exists a unique convex kite c.c. for a given choice of positive masses and a particular ordering of the bodies. Our proof uses tools from differential topology (e.g. the Poincaré–Hopf index theorem) and computational algebraic geometry (e.g. Gröbner bases). We also discuss concave kite c.c.’s, including degenerate examples and bifurcations. Finally, we numerically explore the linear stability of the corresponding kite relative equilibria, finding that the heaviest body must be at least 25 times larger than the combined masses of the other three bodies in order to be linearly stable.
- 30 May 2025
- Published by IOP Publishing in Nonlinearity
- Vol. 38 (7) , 075001
Open AccessOn kite central configurations*
We study kite central configurations (c.c.’s) in the Newtonian four-body problem. We present a new proof that there exists a unique convex kite c.c. for a given choice of positive masses and a particular ordering of the bodies. Our proof uses tools from differential topology (e.g. the Poincaré–Hopf index theorem) and computational algebraic geometry (e.g. Gröbner bases). We also discuss concave kite c.c.’s, including degenerate examples and bifurcations. Finally, we numerically explore the linear stability of the corresponding kite relative equilibria, finding that the heaviest body must be at least 25 times larger than the combined masses of the other three bodies in order to be linearly stable.
- Open Access
On Kite Central Configurations
We study kite central configurations in the Newtonian four-body problem. We present a new proof that there exists a unique convex kite central configuration for a given choice of positive masses and a particular ordering of the bodies. Our proof uses tools from differential topology (e.g., the Poincaré-Hopf Index Theorem) and computational algebraic geometry (e.g., Gröbner bases). We also discuss concave kite central configurations, including degenerate examples and bifurcations. Finally, we numerically explore the linear stability of the corresponding kite relative equilibria, finding that the heaviest body must be at least 25 times larger than the combined masses of the other three bodies in order to be linearly stable.
Open AccessOn Kite Central Configurations
We study kite central configurations in the Newtonian four-body problem. We present a new proof that there exists a unique convex kite central configuration for a given choice of positive masses and a particular ordering of the bodies. Our proof uses tools from differential topology (e.g., the Poincaré-Hopf Index Theorem) and computational algebraic geometry (e.g., Gröbner bases). We also discuss concave kite central configurations, including degenerate examples and bifurcations. Finally, we numerically explore the linear stability of the corresponding kite relative equilibria, finding that the heaviest body must be at least 25 times larger than the combined masses of the other three bodies in order to be linearly stable.
- 15 July 2019
- Published by Springer Nature in Celestial Mechanics and Dynamical Astronomy
- Vol. 131 (7) , 1-27
Classifying four-body convex central configurations
- 15 July 2019
- Published by Springer Nature in Celestial Mechanics and Dynamical Astronomy
- Vol. 131 (7) , 1-27
Classifying four-body convex central configurations
- Open Access
Classifying Four-Body Convex Central Configurations
We classify the full set of convex central configurations in the Newtonian four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include kite, trapezoidal, co-circular, equidiagonal, orthodiagonal, and bisecting-diagonal configurations. Good coordinates for describing the set are established. We use them to prove that the set of four-body convex central configurations with positive masses is three-dimensional, a graph over a domain $D$ that is the union of elementary regions in $\mathbb{R}^{+^3}$.Open AccessClassifying Four-Body Convex Central Configurations
We classify the full set of convex central configurations in the Newtonian four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include kite, trapezoidal, co-circular, equidiagonal, orthodiagonal, and bisecting-diagonal configurations. Good coordinates for describing the set are established. We use them to prove that the set of four-body convex central configurations with positive masses is three-dimensional, a graph over a domain $D$ that is the union of elementary regions in $\mathbb{R}^{+^3}$. - 27 October 2017
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 228 (1) , 209-236
Morse Theory and Relative Equilibria in the Planar n-Vortex Problem
- 27 October 2017
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 228 (1) , 209-236
Morse Theory and Relative Equilibria in the Planar n-Vortex Problem
- Open Access
Morse Theory and Relative Equilibria in the Planar $n$-Vortex Problem
Morse theoretical ideas are applied to the study of relative equilibria in the planar $n$-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the Hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution. The Morse inequalities are then used to prove the instability of some families of relative equilibria in the four-vortex problem with two pairs of equal vorticities. We also show that, for positive circulations, relative equilibria cannot accumulate on the collision set.
Open AccessMorse Theory and Relative Equilibria in the Planar $n$-Vortex Problem
Morse theoretical ideas are applied to the study of relative equilibria in the planar $n$-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the Hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution. The Morse inequalities are then used to prove the instability of some families of relative equilibria in the four-vortex problem with two pairs of equal vorticities. We also show that, for positive circulations, relative equilibria cannot accumulate on the collision set.
- Open Access
Existence and Stability of Four-Vortex Collinear Relative Equilibria with Three Equal Vorticities
We study collinear relative equilibria of the planar four-vortex problem where three of the four vortex strengths are identical. The $S_3$ invariance obtained from the equality of vorticities is used to reduce the defining equations and organize the solutions into two distinct groups based on the ordering of the vortices along the line. The number and type of solutions are given, along with a discussion of the bifurcations that occur. The linear stability of all solutions is investigated rigorously and stable solutions are found to exist for cases where the vorticities have mixed signs. We employ a combination of analysis and computational algebraic geometry to prove our results.Open AccessExistence and Stability of Four-Vortex Collinear Relative Equilibria with Three Equal Vorticities
We study collinear relative equilibria of the planar four-vortex problem where three of the four vortex strengths are identical. The $S_3$ invariance obtained from the equality of vorticities is used to reduce the defining equations and organize the solutions into two distinct groups based on the ordering of the vortices along the line. The number and type of solutions are given, along with a discussion of the bifurcations that occur. The linear stability of all solutions is investigated rigorously and stable solutions are found to exist for cases where the vorticities have mixed signs. We employ a combination of analysis and computational algebraic geometry to prove our results. - Open Access
- 17 April 2017
- Published by Springer Nature in Qualitative Theory of Dynamical Systems
- Vol. 17 (2) , 367-374
A Four-Body Convex Central Configuration with Perpendicular Diagonals is Necessarily a Kite
- 17 April 2017
- Published by Springer Nature in Qualitative Theory of Dynamical Systems
- Vol. 17 (2) , 367-374
Open AccessA Four-Body Convex Central Configuration with Perpendicular Diagonals is Necessarily a Kite
- Open Access
A Four-Body Convex Central Configuration with Perpendicular Diagonals Is Necessarily a Kite
We prove that any four-body convex central configuration with perpendicular diagonals must be a kite configuration. The result extends to general power-law potential functions, including the planar four-vortex problem.
Open AccessA Four-Body Convex Central Configuration with Perpendicular Diagonals Is Necessarily a Kite
We prove that any four-body convex central configuration with perpendicular diagonals must be a kite configuration. The result extends to general power-law potential functions, including the planar four-vortex problem.
- Open Access
- 1 July 2014
- Published by Elsevier in Physica D: Nonlinear Phenomena
- Vol. 280-281, 44-47
Uniqueness results for co-circular central configurations for power-law potentials
- 1 July 2014
- Published by Elsevier in Physica D: Nonlinear Phenomena
- Vol. 280-281, 44-47
Open AccessUniqueness results for co-circular central configurations for power-law potentials