Holger Dullin
University of Sydney
Publications:
Cushman R., Dullin H. R., Hanßmann H., Schmidt S.
The $1:\pm2$ Resonance
2007, vol. 12, no. 6, pp. 642663
Abstract
On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Nonlinear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio $\pm 1/2$ and its unfolding. In particular we show that for the indefinite case ($1:2$) the frequency ratio map in a neighborhood 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 nondegenerate (i.e. the Kolmogorov nondegeneracy condition holds) for every torus in a neighborhood of the equilibrium point. As a by product of our analysis of the frequency map we obtain another proof of fractional monodromy in the $1:2$ resonance.

Dullin H. R., VuNgoc S.
Symplectic Invariants Near HyperbolicHyperbolic Points
2007, vol. 12, no. 6, pp. 689716
Abstract
We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolichyperbolic type. These invariants consist in some signs which determine the topology of the critical Lagrangian fibre, together with several Taylor series which can be computed from the dynamics of the system. We show how these series are related to the singular asymptotics of the action integrals at the critical value of the energymomentum map. This gives general conditions under which the nondegeneracy conditions arising in the KAM theorem (Kolmogorov condition, twist condition) are satisfied. Using this approach, we obtain new asymptotic formulae for the action integrals of the C.~Neumann system. As a corollary, we show that the Arnold twist condition holds for generic frequencies of this system.

Davison C. M., Dullin H. R.
Geodesic Flow on ThreeDimensional Ellipsoids with Equal SemiAxes
2007, vol. 12, no. 2, pp. 172197
Abstract
Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semiaxes, here we study the remaining cases: Ellipsoids with two sets of equal semiaxes with $SO(2) \times SO(2)$ symmetry, ellipsoids with equal larger or smaller semiaxes with $SO(2)$ symmetry, and ellipsoids with three semiaxes coinciding with $SO(3)$ symmetry. All of these cases are Liouvilleintegrable, and reduction of the symmetry leads to singular reduced systems on lowerdimensional ellipsoids. The critical values of the energymomentum 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 nondegenreate. 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 energymomentum map is a convex polyhedron. The case with $SO(3)$ symmetry is noncommutatively integrable, and we show that the fibers over regular points of the energycasimir map are $T^2$ bundles over $S^2$.

Dullin H. R.
Poisson integrator for symmetric rigid bodies
2004, vol. 9, no. 3, pp. 255564
Abstract
We derive an explicit second order reversible Poisson integrator for symmetric rigid bodies in space (i.e. without a fixed point). The integrator is obtained by applying a splitting method to the Hamiltonian after reduction by the $S^1$ body symmetry. In the particular case of a magnetic top in an axisymmetric magnetic field (i.e. the Levitron) this integrator preserves the two momentum integrals. The method is used to calculate the complicated boundary of stability near a linearly stable relative equilibrium of the Levitron with indefinite Hamiltonian.

Dullin H. R., Matveev V. S., Topalov P. I.
On Integrals of the Third Degree in Momenta
1999, vol. 4, no. 3, pp. 3544
Abstract
Consider a Riemannian metric on a surface, and let the geodesic flow of the metric have a second integral that is a third degree polynomial in momenta. Then we can naturally construct a vector field on the surface. We show that the vector field preserves the volume of the surface, and therefore is a Hamiltonian vector field. As examples we treat the Goryachev–Chaplygin top, the Toda lattice and the Calogero–Moser system, and construct their global Hamiltonians. We show that the simpliest choice of Hamiltonian leads to the Toda lattice.

Dullin H. R., Richter P. H., Veselov A. P.
Action variables of the Kovalevskaya top
1998, vol. 3, no. 3, pp. 1831
Abstract
An explicit formula for the action variables of the Kovalevskaya top as Abelian integrals of the third kind on the Kovalevskaya curve is found. The linear system of differential equations of Picard–Fuchs type, describing the dependence of these variables on the integrals of the Kovalevskaya system, is presented in explicit form. The results are based on the formula for the actions derived by S.P.Novikov and A.P.Veselov within the theory of algebrogeometric Poisson brackets on the universal bundle of hyperelliptic Jacobians.

Bolsinov A. V., Dullin H. R.
On Euler Case in Rigid Body Dynamics and Jacobi Problem
1997, vol. 2, no. 1, pp. 1325
Abstract
Using two classical integrable problems, we demonstrate some methods of a new theory of orbital classification for integrable Hamiltonian systems with two degrees of freedom. We show that the Liouville foliations (i.e., decompositions of the phase space into Liouville tori) of the two systems under consideration are diffeomorphic. Moreover, these systems are orbitally topologically equivalent, but this equivalence cannot be made smooth.
