Jesús F. Palacián

The geometry of reaction dynamics

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.

Reduction of polynomial Hamiltonians by the construction of formal integrals

We present a technique based on Lie transformations, which allows us to reduce the number of degrees of freedom of a Hamiltonian system of polynomial type, by extending an integral of the quadratic part to the transformed system, up to a certain order of approximation. The procedure is illustrated for four cases in two and three degrees of freedom.

Transition state theory for laser-driven reactions

Recent developments in transition state theory brought about by dynamical systems theory are extended to time-dependent systems such as laser-driven reactions. Using time-dependent normal form theory, the authors construct a reaction coordinate with regular dynamics inside the transition region. The conservation of the associated action enables one to extract time-dependent invariant manifolds that act as separatrices between reactive and nonreactive trajectories and thus make it possible to predict the ultimate fate of a trajectory. They illustrate the power of our approach on a driven Henon-Heiles system, which serves as a simple example of a reactive system with several open channels. The present generalization of transition state theory to driven systems will allow one to study processes such as the control of chemical reactions through laser pulses.

On perturbed oscillators in 1–1–1 resonance: the case of axially symmetric cubic potentials

Abstract Axially symmetric perturbations of the isotropic harmonic oscillator in three dimensions are studied. A normal form transformation introduces a second symmetry, after truncation. The reduction of the two symmetries leads to a one-degree-of-freedom system. To this end we use a special set of action–angle variables, as well as conveniently chosen generators of the ring of invariant functions. Both approaches are compared and their advantages and disadvantages are pointed out. The reduced flow of the normal form yields information on the original system. We analyse the 2-parameter family of (arbitrary) axially symmetric cubic potentials. This family has rich dynamics, displaying all local bifurcations of co-dimension one. With the exception of six ratios of the parameter values, the dynamical behaviour close to the origin turns out to be completely determined by the normal form of order 1. We also lay the ground for a further study at the exceptional ratios.

Reduction of Polynomial Planar Hamiltonians with Quadratic Unperturbed Part

We classify the possible normal forms of quadratic Hamiltonians in 2 dimensions. Then we give a method to reduce by one the number of degrees of freedom of an arbitrary polynomial Hamiltonian whose principal part is quadratic in positions and moments. The reduction procedure is based on the extension of an integral of the unperturbed part to the whole system, up to a certain order. The corresponding reduced phase spaces have dimension 2 and are described by means of the set of invariants associated to the reduction.

Decomposition of functions for elliptic orbits

In the algebraA of functions periodic in the mean anomaly we relate the problem of integrating over the mean anomaly with that of decomposing an element ofA as the direct sum of two functions, one in the kernel of the Lie derivative in the Keplerian flow and one in the image of this Lie derivative. We propose recursive rules amenable to general purpose symbolic processors for accomplishing such decomposition in a wide subclass ofA. We introduce the dilogarithmic function to express in exact terms quadratures involving the equation of the center.

Hill problem analytical theory to the order four: application to the computation of frozen orbits around planetary satellites

Frozen orbits of the Hill problem are determined in the double-averaged problem, where short and long-period terms are removed by means of Lie transforms. Due to the perturbation method we use, the initial conditions of corresponding quasi-periodic solutions in the nonaveraged problem are computed straightforwardly. Moreover, the method provides the explicit equations of the transformation that connects the averaged and nonaveraged models. A fourth-order analytical theory is necessary for the accurate computation of quasi-periodic frozen orbits.

Numerical evaluation of the dilogarithm of complex argument

A numerical algorithm to evaluate the dilogarithmic function of a complex argument is proposed. The use of the dilogarithm in celestial mechanics appears in the exact Delaunay normalization of some functions involving the equation of the centre.

Hamiltonian Oscillators in 1—1—1 Resonance: Normalization and Integrability

Summary. We consider a family of three-degree-of-freedom (3-DOF) Hamiltonian systems defined by a Taylor expansion around an elliptic equilibrium. More precisely, the system is a perturbed harmonic oscillator in 1‐1‐1 resonance. The perturbation is an arbitrary polynomial with cubic and quartic terms in Cartesian coordinates. We obtain the second-order normal form using the invariants of the reduced phase space. This normal form is defined by six quantities that correspond to the interaction terms associated to this resonance. Then, by means of the nodal-Lissajous variables, we obtain relations among the parameters defining the perturbation, which lead to integrable subfamilies. Finally some applications are given.

Geometric Averaging of Hamiltonian Systems: Periodic Solutions, Stability, and KAM Tori

We investigate the dynamics of various problems defined by Hamiltonian systems of two and three degrees of freedom that have in common that they can be reduced by an axial symmetry. Specifically, the systems are either invariant under rotation about the vertical axis or can be made approximately axially symmetric after an averaging process and the corresponding truncation of higher-order terms. Once the systems are reduced we study the existence and stability of relative equilibria on the reduced spaces which are unbounded two- or four-dimensional symplectic manifolds with singular points. We establish the connections between the existence and stability of relative equilibria and the existence and stability of families of periodic solutions of the full problem. We also discuss the existence of KAM tori surrounding the periodic solutions.