417 citations · 1.6k across the 72 of their papers we have counts for
21 papers · 1 filter
Bounded orbits for 3 bodies in
Alain Albouy, Holger R. Dullin
We consider the Newtonian 3-body problem in dimension 4, and fix a value of the angular momentum which is compatible with this dimension. We show that the energy function cannot te…
On the -Regularisation of Simultaneous Binary Collisions in the Planar 4-Body Problem
Nathan Duignan, Holger R. Dullin
The dynamics of the 4-body problem allows for two binary collisions to occur simultaneously. It is known that in the collinear 4-body problem this simultaneous binary collision (SB…
An Evans function for the linearised 2D Euler equations using Hill's determinant
Holger R. Dullin, Robert Marangell
We study the point spectrum of the linearisation of Euler's equation for the ideal fluid on the torus about a shear flow. By separation of variables the problem is reduced to the s…
Relative equilibria of the 3-body problem in
Alain Albouy, Holger R. Dullin
The classical equations of the Newtonian 3-body problem do not only define the familiar 3-dimensional motions. The dimension of the motion may also be 4, and cannot be higher. We p…
On the -Regularisation of Simultaneous Binary Collisions in the Collinear 4-Body Problem
Nathan Duignan, Holger R. Dullin
The singularity at a simultaneous binary collision is explored in the collinear 4-body problem. It is known that any attempt to remove the singularity via block regularisation will…
Symmetry reduction of the 3-body problem in
Holger R. Dullin, Jürgen Scheurle
The 3-body problem in has 24 dimensions and is invariant under translations and rotations. We do the full symplectic symmetry reduction and obtain a reduced Hamilton…
Stability Theory of the 3-Dimensional Euler Equations
Holger R. Dullin, Joachim Worthington
The Euler equations on a three-dimensional periodic domain have a family of shear flow steady states. We show that the linearised system around these steady states decomposes into…
Regularisation for Planar Vector Fields
Nathan Duignan, Holger Dullin
This paper serves as a first foray on regularisation for planar vector fields. Motivated by singularities in celestial mechanics, the block regularisation of a generic class of deg…
Symplectic classification of coupled angular momenta
Jaume Alonso, Holger R. Dullin, Sonja Hohloch
The coupled angular momenta are a family of completely integrable systems that depend on three parameters and have a compact phase space. They correspond to the classical version o…
Stability Results for Idealised Shear Flows on a Rectangular Periodic Domain
Holger Dullin, Joachim Worthington
We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain for , the E…
The Diver with a Rotor
Sudarsh Bharadwaj, Nathan Duignan, Holger R. Dullin +2
We present and analyse a simple model for the twisting somersault. The model is a rigid body with a rotor attached which can be switched on and off. This makes it simple enough to…
Syzygies in the two center problem
Holger R. Dullin, Richard Montgomery
We give a complete symbolic dynamics description of the dynamics of Euler's problem of two fixed centers. By analogy with the 3-body problem we use the collinearities (or syzygies)…
Instability of Equilibria for the 2D Euler Equations on the torus
Joachim Worthington, Holger R. Dullin, Robert Marangell
We consider the hydrodynamics of an incompressible fluid on a 2D periodic domain. There exists a family of stationary solutions with vorticity given by $Ω^*=α\cos (\mathbf{p} \cdot…
The Lie-Poisson Structure of the Symmetry Reduced Regularised n-Body Problem
Suntharan Arunasalam, Holger R. Dullin, Diana M. H. Nguyen
This paper investigates the symmetry reduction of the regularised n-body problem. The three body problem, regularised through quaternions, is examined in detail. We show that for a…
The Lie-Poisson structure of the reduced n-body problem
Holger R. Dullin
The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a…
The degenerate C. Neumann system I: symmetry reduction and convexity
Holger R. Dullin, Heinz Hanßmann
The C. Neumann system describes a particle on the sphere S^n under the influence of a potential that is a quadratic form. We study the case that the quadratic form has l+1 distinct…
The Equilateral Pentagon at Zero Angular Momentum: Maximal Rotation Through Optimal Deformation
William Tong, Holger R. Dullin
A pentagon in the plane with fixed side-lengths has a two-dimensional shape space. Considering the pentagon as a mechanical system with point masses at the corners we answer the qu…
Semi-global symplectic invariants of the spherical pendulum
Holger R. Dullin
We explicitly compute the semi-global symplectic invariants near the focus-focus point of the spherical pendulum. A modified Birkhoff normal form procedure is presented to compute…
On the analytic non-integrability of the Rattleback problem
Holger R. Dullin, Alexei Tsygvintsev
We establish the analytic non-integrability of the nonholonomic ellipsoidal rattleback model for a large class of parameter values. Our approach is based on the study of the monodr…
A new integrable system on the sphere
Holger R. Dullin, Vladimir S. Matveev
We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) two-dimensional sphere. The second integral is cubic in the momenta.
Vanishing Twist near Focus-Focus Points
Holger R. Dullin, Vu Ngoc San
We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-…