arXiv subjects

Holger Dullin

Publications and source records attributed to Holger Dullin.

8 recordsLinked to original sources

arXiv2025

The N-Body Problem on Coadjoint Orbits

We show (Theorem 3) that the symplectic reduction of the spatial $n$-body problem at non-zero angular momentum is a singular symplectic space consisting of two symplectic strata, one for spatial motions and the other for planar motions. Each stratum is realized as coadjoint orbit in the dual of the Lie algebra of the linear symplectic group $Sp(2n-2)$. The planar stratum arises as the frontier upon taking the closure of the spatial stratum. We reduce by going to center-of-mass coordinates to reduce by translations and boosts and then performing symplectic reduction with respect to the orthogonal group $O(3)$. The theorem is a special case of a general theorem (Theorem 2) which holds for the $n$-body problem in any dimension $d$. This theorem follows largely from a ``Poisson reduction'' theorem, Theorem 1. We achieve our reduction theorems by combining the Howe dual pair perspective of reduction espoused by Lerman-Montgomery-Sjamaar with a normal form arising from a symplectic singular value decomposition due to Xu. We begin the paper by showing how Poisson reduction by the Galilean group rewrites Newton's equations for the $n$-body problem as a Lax pair. In section 6.4 we show that this Lax pair representation of the $n$-body equations is equivalent to the Albouy-Chenciner representation in terms of symmetric matrices.

math-ph↗

arXiv2024

Quantum Integrable Systems arising from Separation of Variables on S3

We study the family of quantum integrable systems that arise from separating the Schrödinger equation in all 6 separable orthogonal coordinates on the 3 sphere: ellipsoidal, prolate, oblate, Lamé, spherical and cylindrical. On the one hand each separating coordinate system gives rise to a quantum integrable system on S2 x S2, on the other hand it also leads to families of harmonic polynomials in R4. We show that separation in ellipsoidal coordinates yields a generalised Lamé equation - a Fuchsian ODE with 5 regular singular points. We seek polynomial solutions so that the eigenfunctions are analytic at all finite singularities. We classify eigenfunctions by their discrete symmetry and compute the joint spectrum for each symmetry class. The latter 5 separable coordinate systems are all degenerations of the ellipsoidal coordinates. We perform similar analyses on these systems and show how the ODEs degenerate in a fashion akin to their respective coordinates. For the prolate system we show that there exists a defect in the joint spectrum which prohibits a global assignment of quantum numbers: the system has quantum monodromy. This is a companion paper to our previous work where the respective classical systems were studied.

math-ph↗

arXiv2023

Probabilistic description of dissipative chaotic scattering

We investigate the extent to which the probabilistic properties of a chaotic scattering system with dissipation can be understood from the properties of the dissipation-free system. For large energies $E$, a fully chaotic scattering leads to an exponential decay of the survival probability $P(t) \sim e^{-κt}$ with an escape rate $κ$ that decreases with $E$. Dissipation $γ>0$ leads to the appearance of different finite-time regimes in $P(t)$. We show how these different regimes can be understood for small $γ\ll 1$ and $t\gg 1/κ_0$ from the effective escape rate $κ_γ(t)=κ_0(E(t))$ (including the non-hyperbolic regime) until the energy reaches a critical value $E_c$ at which no escape is possible. More generally, we argue that for small dissipation $γ$ and long times $t$ the surviving trajectories in the dissipative system are distributed according to the conditionally invariant measure of the conservative system at the corresponding energy $E(t)<E(0)$. Quantitative predictions of our general theory are compared with numerical simulations in the Henon-Heiles model.

cond-mat.stat-mech↗

arXiv2019

Instability of unidirectional flows for the 2D $α$-Euler equations

We study stability of unidirectional flows for the linearized 2D $α$-Euler equations on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a vector $\mathbf p \in \mathbb Z^{2}$. We linearize the $α$-Euler equation and write the linearized operator $L_{B} $ in $\ell^{2}(\mathbb Z^{2})$ as a direct sum of one-dimensional difference operators $L_{B,\mathbf q}$ in $\ell^{2}(\mathbb Z)$ parametrized by some vectors $\mathbf q\in\mathbb Z^2$ such that the set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ covers the entire grid $\mathbb Z^{2}$. The set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ can have zero, one, or two points inside the disk of radius $\|\mathbf p\|$. We consider the case where the set $\{\mathbf q +n \mathbf p:n \in \mathbb Z\}$ has exactly one point in the open disc of radius $\mathbf p$. We show that unidirectional flows that satisfy this condition are linearly unstable. Our main result is an instability theorem that provides a necessary and sufficient condition for the existence of a positive eigenvalue to the operator $L_{B,\mathbf q}$ in terms of equations involving certain continued fractions. Moreover, we are also able to provide a complete characterization of the corresponding eigenvector. The proof is based on the use of continued fractions techniques expanding upon the ideas of Friedlander and Howard.

math.SP↗

arXiv2019

Regularisation for Planar Vector Fields

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 degenerate singularities is studied. The paper is concerned with asymptotic properties of the transition map between a section before and after the singularity. Block regularisation is reviewed before topological and explicit conditions for the $ C^0 $-regularity of the map are given. Computation of the $ C^1 $-regularisation is reduced to summing residues of a rational function. It is shown that the transition map is in general only finitely differentiable and a method of computing the map is conveyed. In particular, a perturbation of a toy example derived from the 4-body problem is shown to be $ C^{4/3} $. The regularisation of all homogeneous quadratic vector fields is computed.

math.DS↗

arXiv2016

Defect in the Joint Spectrum of Hydrogen due to Monodromy

In addition to the well known case of spherical coordinates the hydrogen atom separates in three further coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators. We show that the joint spectrum of the Hamilton operator, and the $z$-components of the angular momentum and quantum Laplace-Runge-Lenz vectors obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. This means that one cannot globally assign quantum numbers to the joint spectrum. Whereas the principal quantum number $n$ and the magnetic quantum number $m$ correspond to the Bohr-Sommerfeld quantization of globally defined classical actions a third quantum number cannot be globally defined because the third action is globally multi valued.

math-ph↗

arXiv2016

Stability Results for Idealised Shear Flows on a Rectangular Periodic Domain

We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain $[0,2π)\times[0,2π/ κ)$ for $κ\in\mathbb{R}^+$, the Euler equations admit a family of stationary solutions given by the vorticity profiles $Ω^*(\mathbf{x})= Γ\cos(p_1x_1+ κp_2x_2)$. We show linear stability for such flows when $p_2=0$ and $κ\geq |p_1|$ (equivalently $p_1=0$ and $κ{|p_2|}\leq{1}$). The classical result due to Arnold is that for $p_1 = 1, p_2 = 0$ and $κ\ge 1$ the stationary flow is {nonlinearly} stable via the energy-Casimir method. We show that for $κ\ge |p_1| \ge 2, p_2 = 0$ the flow is linearly stable, but one cannot expect a similar nonlinear stability result. Finally we prove nonlinear instability for all equilibria satisfying $p_1^2+κ^2{p_2^2}>\frac{3(κ^2+1)}{4(7-4\sqrt{3})}$. The modification and application of a structure-preserving Hamiltonian truncation is discussed for the $κ\neq 1$ case. This leads to an explicit Lie-Poisson integrator for the truncated system.

math.DS↗

arXiv2013

A symplectic integrator for the symmetry reduced and regularised planar 3-body problem with vanishing angular momentum

We construct an explicit reversible symplectic integrator for the planar 3-body problem with zero angular momentum. We start with a Hamiltonian of the planar 3-body problem that is globally regularised and fully symmetry reduced. This Hamiltonian is a sum of 10 polynomials each of which can be integrated exactly, and hence a symplectic integrator is constructed. The performance of the integrator is examined with three numerical examples: The figure eight, the pythagorean orbit, and a periodic collision orbit.

physics.comp-ph↗