H. Waalkens
Publications:
de Neeling D., Roest D., Seri M., Waalkens H.
Extremal Black Holes as Relativistic Systems with Kepler Dynamics
2024, vol. 29, no. 2, pp. 344368
Abstract
The recent detection of gravitational waves emanating from inspiralling black hole
binaries has triggered a renewed interest in the dynamics of relativistic twobody systems. The
conservative part of the latter are given by Hamiltonian systems obtained from socalled post
Newtonian expansions of the general relativistic description of black hole binaries. In this paper
we study the general question of whether there exist relativistic binaries that display Keplerlike
dynamics with elliptical orbits. We show that an orbital equivalence to the Kepler problem
indeed exists for relativistic systems with a Hamiltonian of a Keplerlike form. This form is
realised by extremal black holes with electric charge and scalar hair to at least first order in
the postNewtonian expansion for arbitrary mass ratios and to all orders in the postNewtonian
expansion in the testmass limit of the binary. Moreover, to fifth postNewtonian order, we
show that Hamiltonians of the Keplerlike form can be related explicitly through a canonical
transformation and time reparametrisation to the Kepler problem, and that all Hamiltonians
conserving a Laplace – Runge – Lenzlike vector are related in this way to Kepler.

Efstathiou K., Lin B., Waalkens H.
Loops of Infinite Order and Toric Foliations
2022, vol. 27, no. 3, pp. 320332
Abstract
In 2005 Dullin et al. proved that the
nonzero vector of Maslov indices is an eigenvector with eigenvalue
$1$ of the monodromy matrices of an integrable Hamiltonian system.
We take a close look at the geometry behind this result and extend
it to the more general context of possibly nonHamiltonian systems.
We construct a bundle morphism defined
on the lattice bundle of an (general) integrable system, which can
be seen as a generalization of the vector of Maslov indices. The nontriviality of this bundle morphism implies the existence of common eigenvectors with eigenvalue $1$
of the monodromy matrices, and gives rise to a corank $1$ toric foliation
refining the original one induced by the integrable system. Furthermore,
we show that, in the case where the system has $2$ degrees of freedom,
this implies the existence of a compatible free $S^{1}$ action on the regular part of the system.

Martynchuk N. N., Waalkens H.
Knauf’s Degree and Monodromy in Planar Potential Scattering
2016, vol. 21, no. 6, pp. 697706
Abstract
We consider Hamiltonian systems on $(T^{*}\mathbb R^2, dq \wedge dp)$ defined by a Hamiltonian function $H$ of the “classical” form $H = p^2/2 + V (q)$. A reasonable decay assumption $V(q) \to 0, \, \q\ \to \infty$, allows one to compare a given distribution of initial conditions at $t = −\infty$ with their final distribution at $t = +\infty$. To describe this Knauf introduced a topological invariant $\text{deg}(E)$, which, for a nontrapping energy $E > 0$, is given by the degree of the scattering map. For rotationally symmetric potentials $V = W(\q\)$, scattering monodromy has been introduced independently as another topological invariant. In the present paper we demonstrate that, in the rotationally symmetric case, Knauf’s degree $\text{deg}(E)$ and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree $\text{deg}(E)$, which appears when the nontrapping energy $E$ goes from low to high values.

Waalkens H., Wiggins S.
Geometrical models of the phase space structures governing reaction dynamics
2010, vol. 15, no. 1, pp. 139
Abstract
Hamiltonian dynamical systems possessing equilibria of saddle x center x∙∙∙x center stability type display reactiontype dynamics for energies close to the energy of such equilibria; entrance and exit from certain regions of the phase space is only possible via narrow bottlenecks created by the influence of the equilibrium points. In this paper we provide a thorough pedagogical description of the phase space structures that are responsible for controlling transport in these problems. Of central importance is the existence of a Normally Hyperbolic Invariant Manifold (NHIM), whose stable and unstable manifolds have sufficient dimensionality to act as separatrices, partitioning energy surfaces into regions of qualitatively distinct behavior. This NHIM forms the natural (dynamical) equator of a (spherical) dividing surface which locally divides an energy surface into two components ("reactants" and "products"), one on either side of the bottleneck. This dividing surface has all the desired properties sought for in transition state theory where reaction rates are computed from the flux through a dividing surface. In fact, the dividing surface that we construct is crossed exactly once by reactive trajectories, and not crossed by nonreactive trajectories, and related to these properties, minimizes the flux upon variation of the dividing surface.
We discuss three presentations of the energy surface and the phase space structures contained in it for 2degreeoffreedom (DoF) systems in the threedimensional space $\mathbb{R}^3$, and two schematic models which capture many of the essential features of the dynamics for $n$DoF systems. In addition, we elucidate the structure of the NHIM. 