arXiv subjects

Holger R. Dullin

Publications and source records attributed to Holger R. Dullin.

At least 37 records · Page 2Linked to original sources

arXiv2018

Using the Geometric Phase to Optimise Planar Somersaults

We derive the equations of motion for the planar somersault, which consist of two additive terms. The first is the dynamic phase that is proportional to the angular momentum, and the second is the geometric phase that is independent of angular momentum and depends solely on the details of the shape change. Next, we import digitised footage of an elite athlete performing 3.5 forward somersaults off the 3m springboard, and use the data to validate our model. We show that reversing and reordering certain sections of the digitised dive can maximise the geometric phase without affecting the dynamic phase, thereby increasing the overall rotation achieved. Finally, we propose a theoretical planar somersault consisting of four shape changing states, where the optimisation lies in finding the shape change strategy that maximises the overall rotation of the dive. This is achieved by balancing the rotational contributions from the dynamic and geometric phases, in which we show the geometric phase plays a small but important role in the optimisation process.

physics.class-ph↗

arXiv2017

A New Twisting Somersault - 513XD

We present the mathematical framework of an athlete modelled as a system of coupled rigid bodies to simulate platform and springboard diving. Euler's equations of motion are generalised to non-rigid bodies, and are then used to innovate a new dive sequence that in principle can be performed by real world athletes. We begin by assuming shape changes are instantaneous so that the equations of motion simplify enough to be solved analytically, and then use this insight to present a new dive (513XD) consisting of 1.5 somersaults and 5 twists using realistic shape changes. Finally, we demonstrate the phenomenon of converting pure somersaulting motion into pure twisting motion by using a sequence of impulsive shape changes, which may have applications in other fields such as space aeronautics.

physics.class-ph↗

arXiv2016

Instability of Equilibria for the 2D Euler Equations on the torus

We consider the hydrodynamics of an incompressible fluid on a 2D periodic domain. There exists a family of stationary solutions with vorticity given by $Ω^*=α\cos (\mathbf{p} \cdot \mathbf{x} )+β\sin (\mathbf{p} \cdot \mathbf{x} )$. This situation can be approximated as a structure preserving finite dimensional Hamiltonian system by a truncation introduced by Zeitlin (1990,2005) or by the more standard Galerkin style finite element method. We use these two truncations to analyse the linear stability of these solutions and analytical and numerical results are compared. Following the methods used by Li (2000) the problem is divided into subsystems and we prove that most subsystems are linearly stable. We derive a sufficient condition for a subsystem to be linearly unstable and derive an explicit lower bound for the associated real eigenvalues independent of the truncation size $N$. Then we show that the corresponding eigenvectors are in $\ell^2$. This together with known stability results for the 2D periodic Euler equations allows us to conclude that most of these stationary solutions are nonlinearly unstable. We confirm our results with a numerical computation of the spectrum for a large, finite truncation. Finally we discuss the essential spectrum of the full problem as the limit of the truncated problem.

math.DS↗

arXiv2015

Twisting Somersault

A complete description of twisting somersaults is given using a reduction to a time-dependent Euler equation for non-rigid body dynamics. The central idea is that after reduction the twisting motion is apparent in a body frame, while the somersaulting (rotation about the fixed angular momentum vector in space) is recovered by a combination of dynamic and geometric phase. In the simplest "kick-model" the number of somersaults $m$ and the number of twists $n$ are obtained through a rational rotation number $W = m/n$ of a (rigid) Euler top. This rotation number is obtained by a slight modification of Montgomery's formula [9] for how much the rigid body has rotated. Using the full model with shape changes that take a realistic time we then derive the master twisting-somersault formula: An exact formula that relates the airborne time of the diver, the time spent in various stages of the dive, the numbers $m$ and $n$, the energy in the stages, and the angular momentum by extending a geometric phase formula due to Cabrera [3]. Numerical simulations for various dives agree perfectly with this formula where realistic parameters are taken from actual observations.

physics.class-ph↗

arXiv2015

The Diver with a Rotor

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 devise explicit analytical formulas whilst still maintaining sufficient complexity to preserve the shape-changing dynamics essential for twisting somersaults in springboard and platform diving. With `rotor on' and with `rotor off' the corresponding Euler-type equations can be solved, and the essential quantities characterising the dynamics, such as the periods and rotation numbers, can be computed in terms of complete elliptic integrals. Thus we arrive at explicit formulas for how to achieve a dive with m somersaults and n twists in a given total time. This can be thought of as a special case of a geometric phase formula due to Cabrera 2007.

math.DS↗

arXiv2015

Syzygies in the two center problem

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) of the three bodies as symbols. We show that motion without collision on regular tori of the regularised integrable system are given by so called Sturmian sequences. Sturmian sequences were introduced by Morse and Hedlund in 1940. Our main theorem is that the periodic Sturmian sequences are in one to one correspondence with the periodic orbits of the two center problem. Similarly, finite Sturmian sequences correspond to collision-collision orbits.

math.DS↗

arXiv2015

Generating hyperbolic singularities in completely integrable systems

Let $(M,Ω)$ be a connected symplectic 4-manifold and let $F=(J,H) : M \to \mathbb{R}^2$ be a completely integrable system on $M$ with only non-degenerate singularities and for which $J : M \to \mathbb{R}$ is a proper map. Assume that $F$ does not have singularities with hyperbolic blocks and that $p_1,...,p_n$ are the focus-focus singularities of $F$. For each subset $S=\{i_1,...,i_j\}$ we will show how to modify $F$ locally around any $p_i, i \in S$, in order to create a new integrable system $\tilde{F}=(J, \tilde{H}) : M \to \mathbb{R}^2$ such that its classical spectrum $\tilde{F}(M)$ contains $j$ smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of $\tilde{F}$. Moreover the focus-focus singularities of $\tilde{F}$ are precisely $p_i$, $i \in \{1,...,n\} \setminus S$, and each of these $p_i$ is non-degenerate. The proof is based on Eliasson's linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.

math-ph↗

arXiv2013

Semi-global symplectic invariants of the Euler top

We compute the semi-global symplectic invariants near the hyperbolic equilibrium points of the Euler top. The Birkhoff normal form at the hyperbolic point is computed using Lie series. The actions near the hyperbolic point are found using Frobenius expansion of its Picard-Fuchs equation. We show that the Birkhoff normal form can also be found by inverting the regular solution of the Picard-Fuchs equation. Composition of the singular action integral with the Birkhoff normal form gives the semi-global symplectic invariant. Finally, we discuss the convergence of these invariants and show that in a neighbourhood of the separatrix the pendulum is not symplectically equivalent to any Euler top.

math.SG↗

arXiv2012

The Equilateral Pentagon at Zero Angular Momentum: Maximal Rotation Through Optimal Deformation

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 question of how much the pentagon can rotate with zero angular momentum. We show that the shape space of the equilateral pentagon has genus 4 and find a fundamental region by discrete symmetry reduction with respect to symmetry group D_5. The amount of rotation Δθfor a loop in shape space at zero angular momentum is interpreted as a geometric phase and is obtained as an integral of a function B over the region of shape space enclosed by the loop. With a simple variational argument we determine locally optimal loops as the zero contours of the function B. The resulting shape change is represented as a Fourier series, and the global maximum of Δθ\approx 45\degree is found for a loop around the regular pentagram. We also show that restricting allowed shapes to convex pentagons the optimal loop is the boundary of the convex region and gives Δθ\approx 19\degree.

math.DS↗

arXiv2012

The Lie-Poisson structure of the reduced n-body problem

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 result we obtain a reduced system with a Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.

math.DS↗

arXiv2012

The degenerate C. Neumann system I: symmetry reduction and convexity

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 eigenvalues with multiplicity. Each group of m_σequal eigenvalues gives rise to an O(m_σ)-symmetry in configuration space. The combined symmetry group G is a direct product of l+1 such factors, and its cotangent lift has an Ad^*-equivariant Momentum mapping. Regular reduction leads to the Rosochatius system on S^l, which has the same form as the Neumann system albeit for an additional effective potential. To understand how the reduced systems fit together we use singular reduction to construct an embedding of the reduced Poisson space T^*{S^n}/G into R^{3l+3}$. The global geometry is described, in particular the bundle structure that appears as a result of the superintegrability of the system. We show how the reduced Neumann system separates in elliptical-spherical co-ordinates. We derive the action variables and frequencies as complete hyperelliptic integrals of genus l. Finally we prove a convexity result for the image of the Casimir mapping restricted to the energy surface.

math.DS↗

arXiv2011

Semi-global symplectic invariants of the spherical pendulum

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 the expansion of the Hamiltonian near the unstable equilibrium point in Eliasson-variables. Combining this with explicit formulas for the action we find the semi-global symplectic invariants near the focus-focus point introduced by Vu Ngoc 2003. We also show that the Birkhoff normal form is the inverse of a complete elliptic integral over a vanishing cycle. To our knowledge this is the first time that semi-global symplectic invariants near a focus-focus point have been computed explicitly. We close with some remarks about the pendulum, for which the invariants can be related to theta functions in a beautiful way.

math.DS↗

arXiv2009

Dynamics near the p : -q Resonance

We study the dynamics near the truncated p : +/- q resonant Hamiltonian equilibrium for p, q coprime. The critical values of the momentum map of the Liouville integrable system are found. The three basic objects reduced period, rotation number, and non-trivial action for the leading order dynamics are computed in terms of complete hyperelliptic integrals. A relation between the three functions that can be interpreted as a decomposition of the rotation number into geometric and dynamic phase is found. Using this relation we show that the p : -q resonance has fractional monodromy. Finally we prove that near the origin of the 1 : -q resonance the twist vanishes.

nlin.SI↗

arXiv2008

Non-uniqueness of phase shift in central scattering due to monodromy

Scattering at a central potential is completely characterized by the phase shifts which are the differences in phase between outgoing scattered and unscattered partial waves. In this letter it is shown that, for 2D scattering at a repulsive central potential, the phase shift cannot be uniquely defined due to a topological obstruction which is similar to monodromy in bound systems.

math-ph↗

arXiv2008

Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R^3. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and also is the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with intersecting two-dimensional stable and unstable manifolds that bound a spherical ``vortex-bubble''. We show that this occurs near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at least in its normal form, an elliptic invariant circle. We develop a simple algorithm to accurately compute these elliptic invariant circles and their longitudinal and transverse rotation numbers and use it to study their bifurcations, classifying them by the resonances between the rotation numbers. In particular, rational values of the longitudinal rotation number are shown to give rise to a string of pearls that creates multiple copies of the original spherical structure for an iterate of the map.

nlin.CD↗

arXiv2008

A Poincaré section for the general heavy rigid body

A general recipe is developed for the study of rigid body dynamics in terms of Poincaré surfaces of section. A section condition is chosen which captures every trajectory on a given energy surface. The possible topological types of the corresponding surfaces of section are determined, and their 1:1 projection to a conveniently defined torus is proposed for graphical rendering.

nlin.CD↗

arXiv2008

The 1:+2 / 1:-2 resonance

On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio 1/2 and its unfolding. In particular we show that for the indefinite case (1:-2) the frequency ratio map in a neighbourhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is non-degenerate (i.e. the Kolmogorov non-degeneracy condition holds) for every torus in a neighbourhood of the equilibrium point. As a byproduct we are able to obtain another proof of fractional monodromy in the 1:-2 resonance.

nlin.SI↗

arXiv2006

Geodesic flow on three dimensional ellipsoids with equal semi-axes

Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semi-axes, here we study the remaining cases: Ellipsoids with two sets of equal semi-axes with $SO(2) \times SO(2)$ symmetry, ellipsoids with equal larger or smaller semi-axes with SO(2) symmetry, and ellipsoids with three semi-axes coinciding with SO(3) symmetry. All of these cases are Liouville-integrable, and reduction of the symmetry leads to singular reduced systems on lower-dimensional ellipsoids. The critical values of the energy-momentum maps and their singular fibers are completely classified. In the cases with SO(2) symmetry there are corank 1 degenerate critical points; all other critical points are non-degenreate. We show that in the case with $SO(2) \times SO(2)$ symmetry three global action variables exist and the image of the energy surface under the energy-momentum map is a convex polyhedron. The case with SO(3) symmetry is non-commutatively integrable, and we show that the fibers over regular points of the energy-casimir map are $T^2$ bundles over $S^2$.

math-ph↗