H. Waalkens

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 non-Hamiltonian 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.

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.

Hamiltonian dynamical systems possessing equilibria of saddle x center x∙∙∙x center stability type display reaction-type 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 2-degree-of-freedom (DoF) systems in the three-dimensional 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.