4 citations · 14 across the 8 of their papers we have counts for
10 papers
The Maxwell Conjecture is False
Philip Arathoon, Gavin Ball, Matthew D. Kvalheim
We exhibit a configuration of five point charges in Euclidean space whose electrostatic potential admits at least 24 critical points all of which are non-degenerate. Maxwell's conj…
Integrable sub-Riemannian geodesic flows on the special orthogonal group
Alejandro Bravo-Doddoli, Philip Arathoon, Anthony M. Bloch
We analyse the geometry of the rubber-rolling distribution on the special orthogonal group and show that almost all the normal geodesics of any right-invariant sub-Riemannian metri…
Relative Equilibria of Mechanical Systems with Rotational Symmetry
Philip Arathoon
We consider the task of classifying relative equilibria for mechanical systems with rotational symmetry. We divide relative equilibria into two natural groups: a generic class whic…
Koopman Embedding and Super-Linearization Counterexamples with Isolated Equilibria
Philip Arathoon, Matthew D. Kvalheim
A frequently repeated claim in the "applied Koopman operator theory'' literature is that a dynamical system with multiple isolated equilibria cannot be linearized in the sense of a…
Linearizability of flows by embeddings
Matthew D. Kvalheim, Philip Arathoon
We consider the problem of determining the class of continuous-time dynamical systems that can be globally linearized in the sense of admitting an embedding into a linear system on…
Unifying the Hyperbolic and Spherical 2-Body Problem with Biquaternions
Philip Arathoon
The 2-body problem on the sphere and hyperbolic space are both real forms of holomorphic Hamiltonian systems defined on the complex sphere. This admits a natural description in ter…
Real Forms of Holomorphic Hamiltonian Systems
Philip Arathoon, Marine Fontaine
By complexifying a Hamiltonian system one obtains dynamics on a holomorphic symplectic manifold. To invert this construction we present a theory of real forms which not only recove…
Singular Reduction of the 2-Body Problem on the 3-Sphere and the 4-Dimensional Spinning Top
Philip Arathoon
We consider the dynamics and symplectic reduction of the 2-body problem on a sphere of arbitrary dimension. It suffices to consider the case for when the sphere is 3-dimensional an…
A Bijection Between the Adjoint and Coadjoint Orbits of a Semidirect Product
Philip Arathoon
We prove that there exists a geometric bijection between the sets of adjoint and coadjoint orbits of a semidirect product, provided a similar bijection holds for particular subgrou…
Hermitian flag manifolds and orbits of the Euclidean group
Philip Arathoon, James Montaldi
We study the adjoint and coadjoint representations of a class of Lie group including the Euclidean group. Despite the fact that these representations are not in general isomorphic,…