David L. Rod

On averaging, reduction, and symmetry in Hamiltonian systems

Abstract The existence of periodic orbits for Hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged Hamiltonian on an associated “reduced space.” Alternatively, in classical (kinetic plus potential energy) Hamiltonians the existence of such orbits can often be established by elementary geometrical arguments. The present paper unifies the two approaches by exploiting discrete symmetries, including reversing diffeomorphisms, that occur in a given system. The symmetries are used to locate the periodic orbits in the averaged Hamiltonian, and thence in the original Hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries. In applications to the Henon-Heiles Hamiltonian, it is illustrated how “higher order” averaging can sometimes be used to overcome degeneracies encountered at first order.

Reduction of the semisimple 1:1 resonance

The method of “averaging” is often used in Hamiltonian systems of two degrees of freedom to find periodic orbits. Such periodic orbits can be reconstructed from the critical points of an associated “reduced” Hamiltonian on a “reduced space”. This paper details the construction of the reduced space and the reduced Hamiltonian for the semisimple 1:1 resonance case. The reduced space will be a 2-sphere in R3, and the reduced differential equations will be Eulers equations restricted to this sphere. The orbit projection from the energy surface in phase space to this sphere will be the Hopf map. The results of the paper are related to problems in physics on “degeneracies” due to symmetries of classical two-dimensional harmonic oscillators and their quantum analogues for the hydrogen atom.

Pathology in dynamical systems. III. Analytic Hamiltonians

This paper presents practical hypotheses for proving the existence of nondegenerate (i.e., transverse) homoclinic and heteroclinic orbits to hyperbolic periodic orbits, and the attendant embeddings of the Smale horseshoe mapping, in real analytic Hamiltonian systems with two degrees of freedom. The results are applied to the H&on-Heiles Hamiltonian in Section 5. The paper extends results of [l, 21, w h ere analyticity was not assumed. Moreover, due to difficulties in proving the hyperbolicity of the relevant periodic orbits, subsequently overcome in [16], these earlier papers only investigated approximate embeddings of a “topological” horseshoe mapping, and at the expense of a considerable amount of technical detail. Here the topological technicalities have been substantially reduced, and the true embedding is obtained. With the exception of the verification of certain hypotheses in the example of Section 5, this work can be read independently of [l, 21. Section 1 discusses how nondegenerate homoclinic orbits can be obtained from “topologically” nondegenerate (seminondegenerate) ones, and provides a complete proof of an assertion in [7]. Section 2 then details conditions for proving the existence of topologically nondegenerate heteroclinic and homoclinic orbits. In Section 3 the results of the first two sections are examined in the context of real analytic Hamiltonians. In particular, a simplification of a set of hypotheses of [2] is given, which hypotheses must be verified when applying

Pathology of invariant sets in the monkey saddle

This paper examines the pathological behavior of the invariant set of bounded orbits for the differential equation f = -(%)’ where z = x1 + ha is a complex variable. The dots denote time differentiation and the bar denotes complex conjugation. Let x = (X 1 , x2), y = (yl , y2), be vectors in the plane, ( , ) the usual planar Euclidean inner product, and 11 * :I = ( , )l12. Then this differential equation is equivalent to the analytic Hamiltonian system of two degrees of freedom

On the determination of Ziglin monodromy groups

The monodromy group of a second-order linear differential equation with rational coefficients is called Ziglin if it preserves a nonconstant rational function. The determination of which monodromy groups are Ziglin is essential in integrability questions for complex analytic Hamiltonian systems. In this paper the problem is solved completely for the Fuchsian case by using the Kovacic algorithm to determine the differential Galois group of that second-order equation and then relating this to the monodromy group. Applications are given to Hamiltonian systems.

Group-theoretic obstructions to integrability

Let V be a 4-dimensional complex symplectic vector space. This paper classiies those connected linear algebraic subgroups of the sym-plectic group Sp(V) that admit two independent rational invariants. As an application we show the non-integrability of a three degree of freedom Hamiltonian system.

Stability transitions for periodic orbits in hamiltonian systems

The paper considers one-parameter families of periodic solutions of real analytic Hamiltonian systems with two degrees of freedom, the parameter being the energy h. Conditions are given which guarantee that this family will undergo infinitely many changes in stability status as h tends to some finite value h0. First considered is the case of a critical point (with eigenvalues ±α, ±iβ, α and β>0) of the Hamiltonian at energy h0 with the property that the family limits to a homoclinic orbit asymptotic to this point. Some generalizations of this case are given, and applications are made to examples such as the Hénon-Heiles Hamiltonian. We obtain an infinite sequence of distinct energy intervals converging to h0 on which the periodic orbits are elliptic. Requirements for the elliptic stability of the orbits are then given. The additional conditions for an infinite sequence of distinct energy intervals converging to h0, on which the orbits are hyperbolic, involve the “coexistence problem” for an associated Hills equation that appears when the relevant Poincaré maps along the orbits are computed in coordinates. The results are compared to the case where the critical point has eigenvalues (±α±iβ), α and β>0, investigated by Henrard and Devaney.

Monodromy and nonintegrability in complex Hamiltonian systems

This paper investigates the monodromy representation of the normal variational equation along a phase curve of a two-dimensional complex analytic Hamiltonian system. Geometrical conditions are presented which guarantee reducibility, together with additional hypotheses to ensure complete reducibility. Symmetries in the equations are treated in detail. Applications to establishing the nonintegrability of specific systems are presented.

Geometrical aspects of Ziglin's non-integrability theorem for complex Hamiltonian systems

On place la methode de Yoshida dans un cadre geometrique conceptuel et on etend son applicabilite

Pathology in dynamical systems I: General theory

In Hamiltonian systems the existence of nondegenerate heteroclinic orbits connecting distinct hyperbolic periodic orbits leads to the Smale horseshoe map and symbolic dynamics [lo]. Th e complex behavior of nearby orbits implies the nonexistence of second analytic integrals for the flow [lo]. Unfortunately, proving the transversal intersection of the stable and unstable manifolds of the periodic orbits, i.e., the nondegeneracy of the heteroclinic orbits, is nontrivial in most problems. This paper presents definitions of “transversal” and “nondegenerate” which are more readily verified in applications, and which still lead to topological results on the pathology of orbits analogous to those obtained in [13]. In Part II we will give examples of Hamiltonian systems in which our definitions apply; in particular the H&on-Heiles potential [8] will be examined. Part I is done completely without smoothness assumptions. The hyperbolic periodic orbits are replaced by invariant sets which admit surfaces of section, and which are “isolated” in the sense that they are the maximal invariant sets within some neighborhood of themselves. This work generalizes several of the results of [12], and is similar in spirit to [5].