Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom

This paper concerns Hamiltonian and non-Hamiltonian perturbations of integrable two degree of freedom Hamiltonian systems which contain homoclinic and periodic orbits. Our main example concerns perturbations of the uncoupled system consisting of the simple pendulum and the harmonic oscillator. We show that small coupling perturbations with, possibly, the addition of positive and negative damping breaks the integrability by introducing horseshoes into the dynamics.Research partially supported by ARO Contract DAAG-29-79-C-0086 and by NSF Grants ENG 78-02891 and MCS-78-06718     

Degenerate resonances in Hamiltonian systems with 3/2 degrees of freedom

Hamiltonian systems with 3/2 degrees of freedom close to autonomous systems are considered. Special attention is focused on the case of degenerate resonances. In this case, an averaged system in the first approximation reduces to an area-preserving mapping of a cylinder whose rotation number is a nonmonotonic function of the action variable. Behavior of the trajectories of such a map is similar to that of the trajectories of a Poincare map. Three regions: B(+/-) in the upper and lower parts of the cylinder and an additional region A which contains separatrices of fixed points for the corresponding resonance are distinguished on the cylinder. It is shown that there is a nonempty set of initial points corresponding to walking trajectories in B(+/-) and, hence, there are no closed invariant curves that are homotopically nontrivial on the cylinder. Cells limited by a "stochastic network" can exist in region A. The number of cells is the greater the higher the order of degeneration of the resonance. Possible types of orbit behavior in region A are described. (c) 2002 American Institute of Physics.     

A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom

We study the propagation of lattice vibrations in models of disordered, classical anharmonic crystals. Using classical perturbation theory with an optimally chosen remainder term (i.e. a Nekhoroshev-type scheme), we are able to show that vibrations corresponding to localized initial conditions do essentially not propagate through the crystal up to times larger than any inverse power of the strength of the anharmonic couplings.     

On the periodic orbits of perturbed Hooke Hamiltonian systems with three degrees of freedom

We study periodic orbits of Hamiltonian differential systems with three degrees of freedom using the averaging theory. We have chosen the classical integrable Hamiltonian system with the Hooke potential and we study periodic orbits which bifurcate from the periodic orbits of the integrable system perturbed with a non-autonomous potential.     

Quantization of a Hamiltonian system with an infinite number of degrees of freedom

We propose a method of quantization of a discrete Hamiltonian system with an infinite number of degrees of freedom. Our approach is analogous to the usual finite-dimensional quantum mechanics. We construct an infinite-dimensional Schrödinger equation. We show that it is possible to pass from the finite-dimensional quantum mechanics to our construction in the limit when the number of particles tends to infinity. Rigorous mathematical methods are used.     

On chaotic dynamics in "pseudobilliard" Hamiltonian systems with two degrees of freedom

A new class of Hamiltonian dynamical systems with two degrees of freedom is studied, for which the Hamiltonian function is a linear form with respect to moduli of both momenta. For different potentials such systems can be either completely integrable or behave just as normal nonintegrable Hamiltonian systems with two degrees of freedom: one observes many of the phenomena characteristic of the latter ones, such as a breakdown of invariant tori as soon as the integrability is violated; a formation of stochastic layers around destroyed separatrices; bifurcations of periodic orbits, etc. At the same time, the equations of motion are simply integrated on subsequent adjacent time intervals, as in billiard systems; i.e., all the trajectories can be calculated explicitly: Given an initial data, the state of the system is uniquely determined for any moment. This feature of systems in interest makes them very attractive models for a study of nonlinear phenomena in finite-dimensional Hamiltonian systems. A simple representative model of this class (a model with quadratic potential), whose dynamics is typical, is studied in detail. (c) 1997 American Institute of Physics.     

Renormalization method for computing the threshold of the large-scale stochastic instability in two degrees of freedom Hamiltonian systems

An approximate renormalization procedure is derived for the HamiltonianH(v,x,t)=v²/2–M cosx–P cosk(x–t). It gives an estimate of the large scale stochastic instability threshold which agrees within 5–10% with the results obtained from direct numerical integration of the canonical equations. It shows that this instability is related to the destruction of KAM tori between the two resonances and makes the connection with KAM theory. Possible improvements of the method are proposed. The results obtained forH allow us to estimate the threshold for a large class of Hamiltonian systems with two degrees of freedom.     

Hamiltonian formulation of Riemann ellipsoid dynamics

A Riemann ellipsoid is a classical fluid with an ellipsoidal boundary whose motion depends linearly on position. The Riemann ellipsoid Newtonian equations of motion are proven to form a Hamiltonian dynamical system. The co-adjoint orbits of a Lie group GCM(3) on which the inertia tensor is positive-definite are the reduced phase spaces of Riemann ellipsoids for which conservation of circulation has been exploited fully.     

Bifurcations of periodic trajectories in non-integrable Hamiltonian systems with two degrees of freedom: Numerical and analytical results

Numerical and analytical studies of the types of period n-upling bifurcations undergone by classsical periodic trajectories of non-intergrable Hamiltonians with two degrees of freedom are made. The Hamiltonians studied possess time reversal and reflexion symmetries and we found that these symmetries give rise to additional types of period n-upling bifurcations. The analytical study explains most of the numerically observed bifurcations.     

Superfrustration of charge degrees of freedom

We review recent results, obtained with P. Fendley, on frustration of quantum charges in lattice models for itinerant fermions with strong repulsive interactions. A judicious tuning of kinetic and interaction terms leads to models possessing supersymmetry. In such models frustration takes the form of what we call superfrustration: an extensive degeneracy of supersymmetric ground states. We present a gallery of examples of superfrustration on a variety of 2D lattices.     

Collectivity and single-particle degrees of freedom

Intruder rotational bands in ⁴⁵Sc and ⁴⁵Ti have been investigated up to the maximum aligned angular momentum by means of EUROBALL IV and the Recoil Filter Detector (RFD). The use of the RFD allowed for a reduction of the -line Doppler broadening and, moreover, for a determination of very short level lifetimes. In the studied nuclei, the estimated deformation shows a gradual disappearance of the collectivity at the highest available spins.Received: 19 November 2002, Revised: 14 March 2003, Published online: 2 March 2004PACS: 21.10.Tg Lifetimes - 21.60.Cs Shell model - 27.40. + z P. Bednarczyk: Present address: GSI, Darmstadt, GermanyM.B. Smith: Present address: TRIUMF, Vancouver, Canada     

Superstrings with tensor degrees of freedom

We add antisymmetric tensor degrees of freedom to the usual superstring coordinates. We show that super and kappa symmetries are only achieved for the spacetime dimensionD=4. We also address problems related to the quantization of the model and discuss the influences of this extended spacetime in the usual quantum field theory.     

Reversible maps in two-degrees of freedom Hamiltonian systems

It has been shown that a sub-class of two-degrees of freedom Hamiltonian systems possesses a reversing symmetry discovered by Birkhoff in the restricted problem of three bodies. This mixed space-time reversing symmetry, which is different from the classical time reversal symmetry, can be shared by time-reversible as well as time-irreversible systems. Examples of time-irreversible systems which possess this reversing symmetry are the restricted problem of three bodies as shown by Birkhoff in 1915, and a special case of the motion of a rigid body with a fixed point discussed in this paper. If a Hamiltonian system possesses this Birkhoff reversing symmetry, then there exists a surface of section for which the corresponding Poincare map is Birkhoff-reversible. The Birkhoff-reversibility of this map may be used to study its global dynamics such as the locations and the distribution of the stable and unstable periodic points, the distribution of stable and chaotic regions, and the identification of the scattering regions. (c) 2002 American Institute of Physics.     

On pionic degrees of freedom in atomic nuclei

The possibility of explicit pionic degrees of freedom in atomic nuclei and their association with pionic reactions involving fragment emission is discussed. Although no present direct evidence for such degrees of freedom is available they could be searched for in doorway states — other than the anticipated N¹ doorway — in the excitation function in the pion-induced partial reaction cross sections. Such a search is underway. A framework is discussed in terms of which such new pion doorways might be described.     

Non-nucleonic degrees of freedom in weak nuclear currents

Some of the key issues in the study of exchange currents in nuclei are surveyed. Particular emphasis is placed on the important connection between exchange-current effects and the effects of chiral symmetry in nuclei, a topic which is investigated with great intensity in the current literature.Supported in part by the National Science Foundation (USA), Grant No. PHYS-9006844.     

On the interaction of collective degrees of freedom in nuclei

The structure of the interaction of giant dipole and giant quadrupole oscillations with nuclear surface vibrations is established. The various coupling parameters are calculated in the hydrodynamic model. The significance of the interaction terms is discussed.     

On pseudodynamics of internal degrees of freedom

The results concerning the study of the model formulated within the Grassmann algebra G₆ are presented. It is shown under what assumptions the model reproduces basic features of leptons as they follow from the model of the electroweak interactions. Orbital degrees of freedom are ignored throughout the paper.     

Space-time and degrees of freedom of the elementary particle

First, a general property of Lie groups is used in the case of the Poincaré group in order to define the one particle phase space. It is eight-dimensional in the general case and six-dimensional for a spinless or massless particle.Embedding the Poincaré group into the similitude group of space-time permits us to interpret the dilatation operator as a dynamical variable. The connection between the similitude group and field equations is discussed. Lurçat's ideas on a possible dynamical role of spin and mass-spin spectra of particles (Regge trajectories) are discussed under the point of view of the degrees of freedom.This work constitutes a completed version of a preprint entitled Classical Hamiltonian Formalism for Spin, Argonne, September, 1966.On leave from Université de Marseille, France. Work supported in part by the National Science Foundation.