Patricia Yanguas

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.

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.

Perturbations of the isochrone model

The isochrone is a particular case of a central potential. This paper presents an approach to analyse the qualitative behaviour of perturbations of this model. We define a set of variables which allows us to reduce the system by normalization. In fact, this three-degree-of-freedom Hamiltonian system is reduced by extending the integral of the energy of the unperturbed part to the whole system. Then, we define the reduced phase space with the corresponding invariants. This is the setting needed to analyse the reduced flow. We also consider the possibility of a second reduction by means of the extension of another integral of the unperturbed system to the whole perturbation. In the absence of resonances this reduction is possible and for these cases we calculate the corresponding invariants and reduced phase spaces. Finally, this approach is illustrated with four examples.

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.

Transition state geometry near higher-rank saddles in phase space

We present a detailed analysis of invariant phase space structures near higher-rank saddles of Hamiltonian systems. Using the theory of pseudo-hyperbolic invariant surfaces, we show the existence of codimension-one normally hyperbolic invariant manifolds that govern transport near the higher-rank saddle points. Such saddles occur in a number of problems in celestial mechanics, chemical reactions, and atomic physics. As an example, we consider the problem of double ionization of helium in an external electric field, a basis of many modern ionization experiments. In this example, we illustrate our main results on the geometry and transport properties near a rank-two saddle.

Approximating the Invariant Sets of a Finite Straight Segment near Its Collinear Equilibria

This paper reviews a method for computing explicitly the asymptotic expressions of some invariant manifolds in the vicinity of equilibrium points corresponding to Hamiltonian systems of m degrees of freedom. The method is based on the use of generalized normal forms. Our goal is to show the power of this technique by applying it to the study of one of the colinear equilibrium points corresponding to the Hamiltonian system defined by the motion of a particle orbiting around a finite straight segment. We make use of three different Lie transformations: (i) we calculate the Hamilton function corresponding to the center manifold of the colinear equilibrium; (ii) we determine the Hamiltonian related to the stable-unstable direction; and (iii) we obtain the Hamiltonian related to one of the two stable-stable directions. By means of (i), we are able to parameterize the center, the stable, and the unstable manifolds of the original system using the direct changes of coordinates of the two transformations. We also...

Invariant Manifolds of Spatial Restricted Three-Body Problems: the Lunar Case

New periodic orbits of three-dimensional restricted three-body problems are computed using a technique based on normal forms calculations. The system is formulated as a Hamiltonian perturbation of the two-body problem. Up to a certain order of approximation, the departure Hamiltonian is transformed into simpler ones, by extending the integrals of its principal part to the whole systems using different Lie transformations. Therefore, the resulting normal forms are reduced through invariant theory and the corresponding relative equilibria are determined. Finally, the transformations are inverted to recover the associated higher-dimensional invariant sets of the initial Hamiltonian.