The three-dimensional hydrogen cyanide/isocyanide isomerization problem is taken as an example to present a general theory for computing the phase space structures which govern classical reaction dynamics in systems with an arbitrary ͑finite͒ number of degrees of freedom. The theory, which is algorithmic in nature, comprises the construction of a dividing surface of minimal flux which is locally a ''surface of no return.'' The theory also allows for the computation of the global phase space transition pathways that trajectories must follow in order to react. The latter are enclosed by the stable and unstable manifolds of a so-called normally hyperbolic invariant manifold ͑NHIM͒. A detailed description of the geometrical structures and the resulting constraints on reaction dynamics is given, with particular emphasis on the three degrees of freedom case. A procedure is given which uses these structures to compute orbits homoclinic to, and heteroclinic between, NHIMs. The role of homoclinic and heteroclinic orbits in global recrossings of dividing surfaces and transport in complex systems is explained. The complete description provided here is inherently one within phase space; it cannot be inferred from a configuration space picture. A complexification of the classical phase space structures to incorporate quantum effects is also discussed. The results presented here call into question certain assumptions routinely made on the global dynamics; this paper provides methods that enable one to understand and quantify the phase space dynamics of reactions without making such assumptions.

It has been thought that the capture of irregular moons--with non-circular orbits--by giant planets occurs by a process in which they are first temporarily trapped by gravity inside the planet's Hill sphere (the region where planetary gravity dominates over solar tides). The capture of the moons is then made permanent by dissipative energy loss (for example, gas drag) or planetary growth. But the observed distributions of orbital inclinations, which now include numerous newly discovered moons, cannot be explained using current models. Here we show that irregular satellites are captured in a thin spatial region where orbits are chaotic, and that the resulting orbit is either prograde or retrograde depending on the initial energy. Dissipation then switches these long-lived chaotic orbits into nearby regular (non-chaotic) zones from which escape is impossible. The chaotic layer therefore dictates the final inclinations of the captured moons. We confirm this with three-dimensional Monte Carlo simulations that include nebular drag, and find good agreement with the observed inclination distributions of irregular moons at Jupiter and Saturn. In particular, Saturn has more prograde irregular moons than Jupiter, which we can explain as a result of the chaotic prograde progenitors being more efficiently swept away from Jupiter by its galilean moons.

A procedure is presented for computing the phase space volume of initial conditions for trajectories that escape or "react" from a multidimensional potential well. The procedure combines a phase space transition state theory, which allows one to construct dividing surfaces that are free of local recrossing and that minimize the directional flux, and a classical spectral theorem. The procedure gives the volume of reactive initial conditions in terms of a sum over each entrance channel of the well of the product of the phase space flux across the dividing surface associated with the channel and the mean residence time in the well of trajectories which enter through the channel. This approach is illustrated for HCN isomerization in three dimensions, for which the method is several orders of magnitude more efficient than standard Monte Carlo sampling.

A computational procedure that allows the detection of a new type of highdimensional chaotic saddle in Hamiltonian systems with three degrees of freedom is presented. The chaotic saddle is associated with a so-called normally hyperbolic invariant manifold (NHIM). The procedure allows to compute appropriate homoclinic orbits to the NHIM from which we can infer the existence a chaotic saddle. NHIMs control the phase space transport across an equilibrium point of saddle-centre-...-centre stability type, which is a fundamental mechanism for chemical reactions, capture and escape, scattering, and, more generally, "transformation" in many different areas of physics. Consequently, the presented methods and results are of broad interest. The procedure is illustrated for the spatial Hill's problem which is a well known model in celestial mechanics and which gained much interest e.g. in the study of the formation of binaries in the Kuiper belt.

We present the formal proof of a procedure to compute the phase-space volume of initial conditions for trajectories that, for a constant energy, escape or 'react' from a multi-dimensional potential well with one or several exit/entrance channels. The procedure relies on a phase-space formulation of transition state theory. It gives the volume of reactive initial conditions as the sum over the exit/entrance channels where each channel contributes by the product of the phase-space flux associated with the channel and the mean residence time in the well of those trajectories which escape through the channel. An example is given to demonstrate the computational efficiency of the procedure.