This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Regular and Chaotic Dynamics

Quantum defect theory (QDT) is concerned with the properties of an electron in the field of a positive ion and, in particular, with expressing those properties in terms of analytical functions of the energy. It provides a unified theory of bound states, including series perturbations, autoionisation and electron-ion scattering, both elastic and inelastic. The main emphasis of the review is on the foundations of the theory. Properties of Coulomb functions are discussed in some detail and outline sketches are given of relevant topics in collision theory and radiative theory. One-channel and many-channel QDT are discussed separately. Applications to the following problems are considered: resonances, atomic collision calculations, systems with two energy levels of the ion core, helium, other rare gases, alkaline earths and other atomic systems, molecular hydrogen, dielectronic recombination.

https://iopscience.iop.org/article/10.1088/0034-4885/46/2/002/pdf

Quantum defect theory

Presentation du probleme et des methodes theoriques de calcul des intensites. Application aux systemes moleculaires XY 2

Overtone frequencies and intensities in the local mode picture

This paper concerns heteroclinic connections and resonance transitions in the planar circular
restricted 3-body problem, with applications to the dynamics of comets and asteroids and the design
of space missions such as the Genesis Discovery Mission and low energy Earth to Moon transfers.
The existence of a heteroclinic connection between pairs of equal energy periodic orbits around two of
the libration points is shown numerically. This is applied to resonance transition and the construction
of orbits with prescribed itineraries. Invariant manifold structures are relevant for transport between
the interior and exterior Hill’s regions, and other resonant phenomena throughout the solar system.

/pdf/dynamical-systems-the-three-body-problem-and-space-mission-48b7chyrqv.pdf

Dynamical Systems, the Three-Body Problem and Space Mission Design

Review: emphasis on processes that are essential in terms of photovoltaic and photocatlytic cells; 655 refs.

Theoretical Insights into Photoinduced Charge Transfer and Catalysis at Oxide Interfaces

The geometrical structures which regulate transformations in dynamical systems with three or more degrees of freedom (DOFs) form the subject of this paper. Our treatment focuses on the (2n − 3)-dimensional normally hyperbolic invariant manifold (NHIM) in nDOF systems associated with a centre × · ·· ×centre × saddle in the phase space flow in the (2n − 1)dimensional energy surface. The NHIM bounds a (2n − 2)-dimensional surface, called a ‘transition state’ (TS) in chemical reaction dynamics, which partitions the energy surface into volumes characterized as ‘before’ and ‘after’ the transformation. This surface is the long-sought momentum-dependent TS beyond two DOFs. The (2n − 2)-dimensional stable and unstable manifolds associated with the (2n − 3)-dimensional NHIM are impenetrable barriers with the topology of multidimensional spherical cylinders. The phase flow interior to these spherical cylinders passes through the TS as the system undergoes its transformation. The phase flow exterior to these spherical cylinders is directed away from the TS and, consequently, will never undergo the transition. The explicit forms of these phase space barriers can be evaluated using normal form theory. Our treatment has the advantage of supplying a practical algorithm, and we demonstrate its use on a strongly coupled nonlinear Hamiltonian, the hydrogen atom in crossed electric and magnetic fields.

https://iopscience.iop.org/article/10.1088/0951-7715/15/4/301/pdf

The geometry of reaction dynamics

Dynamical systems theory is used to construct a general phase-space version of transition state theory. Special multidimensional separatrices are found which act as impenetrable barriers in phase-space between reacting and nonreacting trajectories. The elusive momentum-dependent transition state between reactants and products is thereby characterized. A practical algorithm is presented and applied to a strongly coupled Hamiltonian.

Impenetrable Barriers in Phase-Space

Normal and local mode behavior in molecular systems and the transition between them is explored using nonlinear mechanics techniques. Significant insight into this behavior and into the structure of phase space results from a generalized definition of local and normal modes and the associated identification of normal modes as a (1:1) resonance between local zeroth order oscillators. In addition to qualitative insight, this approach yields a simple formula [Eq. (28)] for the level of excitation at which local modes become possible. Applications to H2O and to the overtone spectroscopy of the dihalomethanes, benzene, and TMS are provided.

Local and normal modes: A classical perspective

Transition states in phase space are identified and shown to regulate the rate of escape of asteroids temporarily captured in circumplanetary orbits. The transition states, similar to those occurring in chemical reaction dynamics, are then used to develop a statistical semianalytical theory for the rate of escape of asteroids temporarily captured by Mars. Theory and numerical simulations are found to agree to better than 1%. These calculations suggest that further development of transition state theory in celestial mechanics, as an alternative to large-scale numerical simulations, will be a fruitful approach to mass transport calculations.

/pdf/statistical-theory-of-asteroid-escape-rates-1iscqbqr4y.pdf

Statistical theory of asteroid escape rates.

We present the first application of transition state theory to a system that evolves from an initial to a final state without time-reversal symmetry. The problem studied is the chaotic ionization of a hydrogen atom in crossed electric and magnetic fields. The stable manifolds of the transition state reveal a fractal tiling which connects the geometrical properties of the tiling to the ionization rate, leading to a theoretical explanation for the computational and experimental observation of "prompt" and "delayed" electrons in this problem.

Charles Jaffé

Papers

The geometry of reaction dynamics

Impenetrable Barriers in Phase-Space

Local and normal modes: A classical perspective

Statistical theory of asteroid escape rates.

Transition state theory without time-reversal symmetry: chaotic ionization of the hydrogen atom