New invariant relations for the generalized two-field gyrostat

In the paper, we consider a completely integrable Hamiltonian system with three degrees of freedom found by V.V. Sokolov and A.V. Tsiganov. This system is known as the generalized two-field gyrostat. For the case of only gyroscopic forces present, we find new invariant four-dimensional submanifolds such that the induced dynamical systems are almost everywhere Hamiltonian with two degrees of freedom.     

Classical mechanics on the GL(n, ℝ) group and Euler-Calogero-Sutherland model

Relations between free motion on the GL ⁺(n, ?) group manifold and the dynamics of an n-particle system with spin degrees of freedom on a line interacting with a pairwise 1/sinh² x “potential” (Euler-Calogero-Sutherland model) are discussed within a Hamiltonian reduction. Two kinds of reductions of the degrees of freedom are considered: that which is due to continuous invariance and that which is due to discrete symmetry. It is shown that, upon projecting onto the corresponding invariant manifolds, the resulting Hamiltonian system represents the Euler-Calogero-Sutherland model in both cases.     

Renormalization-group analysis for the transition to chaos in Hamiltonian systems

We study the stability of Hamiltonian systems in classical mechanics with two degrees of freedom by renormalization-group methods. One of the key mechanisms of the transition to chaos is the break-up of invariant tori, which plays an essential role in the large scale and long-term behavior. The aim is to determine the threshold of break-up of invariant tori and its mechanism. The idea is to construct a renormalization transformation as a canonical change of coordinates, which deals with the dominant resonances leading to qualitative changes in the dynamics. Numerical results show that this transformation is an efficient tool for the determination of the threshold of the break-up of invariant tori for Hamiltonian systems with two degrees of freedom. The analysis of this transformation indicates that the break-up of invariant tori is a universal mechanism. The properties of invariant tori are described by the renormalization flow. A trivial attractive set of the renormalization transformation characterizes the Hamiltonians that have a smooth invariant torus. The set of Hamiltonians that have a non-smooth invariant torus is a fractal surface. This critical surface is the stable manifold of a single strange set encompassing all irrational frequencies. This hyperbolic strange set characterizes the Hamiltonians that have an invariant torus at the threshold of the break-up. From the critical strange set, one can deduce the critical properties of the tori (self-similarity, universality classes).     

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.     

Reducing or enhancing chaos using periodic orbits

A method to reduce or enhance chaos in Hamiltonian flows with two degrees of freedom is discussed. This method is based on finding a suitable perturbation of the system such that the stability of a set of periodic orbits changes (local bifurcations). Depending on the values of the residues, reflecting their linear stability properties, a set of invariant tori is destroyed or created in the neighborhood of the chosen periodic orbits. An application on a paradigmatic system, a forced pendulum, illustrates the method.     

Jump of the adiabatic invariant at a separatrix crossing: Degenerate cases

An expression for the quasi-random jump of the adiabatic invariant at a separatrix crossing is obtained for a slow-fast Hamiltonian system with two degrees of freedom in the case when the separatrix passes through a degenerate saddle point in the phase plane of the fast variables. The general case with an arbitrary degree of degeneracy was considered, and this degree is assumed to remain fixed in the process of evolution of the slow variables. The typical value of the jump is larger than in the non-degenerate case studied earlier. Though strongly degenerate, such a setting can be relevant for physical problems. The influence of the asymmetry of a phase portrait on the magnitude of adiabatic invariant jumps was considered as well. An example of this kind is studied, namely the motion of ions in current sheets with complex inner structure.     

Pseudo-billiards

A new class of Hamiltonian dynamical systems with two degrees of freedom and kinetic energy of the form T = c₁|p₁| + c₂|p₂| (called “pseudo-billiards”) is studied. For any kind of interaction, the canonical equations can always be integrated on sequential time intervals; i.e. in principle all the trajectories can be found explicitly.

Depending on the potential, a dynamical system of this class can either be completely integrable or behave just as a usual non-integrable Hamiltonian system with two degrees of freedom: in its phase space there exist invariant tori, stochastic layers, domains of global chaos, etc. Pseudo-billiard models of both the types are considered.

If a potential of a pseudo-billiard system has critical points (equilibria), then trajectories close to these points (“loops”) can exist; they can be treated as images of self-localized objects with finite duration. Such a model (with quartic potential) is also studied.     

Models of relativistic hamiltonian interactions

A model of a relativistically invariant Hamiltonian 2-particle interaction is given. It is classical in the sense of having 6 degrees of freedom. This model shows that the author's earlier general definition of such systems is not vacuous. In this model the forces of interaction die away as the particles are removed from each other.     

Lagrange multiplier modified Ho?ava–Lifshitz gravity

We consider RFDiff invariant Hořava–Lifshitz gravity action with additional Lagrange multiplier term that is a function of scalar curvature. We find its Hamiltonian formulation and we show that the constraint structure implies the same number of physical degrees of freedom as in General Relativity.     

Explicit determination of certain periodic motions of a generalized two-field gyrostat

The case of motion of a generalized two-field gyrostat found by V. V. Sokolov and A.V. Tsiganov is known as a Liouville integrable Hamiltonian system with three degrees of freedom. For this system, we find some special periodic motions at which the momentum mapping has rank 1. For such motions, all phase variables can be expressed in terms of algebraic functions of a single auxiliary variable and a set of constants. This auxiliary variable satisfies a differential equation which can be integrated in elliptic functions of time. As an application, the explicit formulas of characteristic exponents for determining the Williamson type of the special periodic motions are obtained.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

Magneto-metric Hamiltonians on quantum surfaces in the configuration space

The paper deals with the Schrödinger operator on a generic Riemann surface in a generic magnetic field. On a class of nano-momentum states, this operator is reduced to a system with one degree of freedom whose (quantized) phase space is the given configuration surface. The Hamiltonian of the reduced system is a joint geometric invariant of metric and magnetic tensors. Its spectrum can be computed semiclassically, which provides the asymptotics of eigenvalue clusters for the original Schrödinger operator near the ground level.     

Hamiltonian Dynamics for Einstein’s Action in <Emphasis Type="Italic">G</Emphasis>→0 Limit

The Hamiltonian analysis for the Einstein’s action in G→0 limit is performed. Considering the original configuration space without involve the usual ADM variables we show that the version G→0 for Einstein’s action is devoid of physical degrees of freedom. In addition, we will identify the relevant symmetries of the theory such as the extended action, the extended Hamiltonian, the gauge transformations and the algebra of the constraints. As complement part of this work, we develop the covariant canonical formalism where will be constructed a closed and gauge invariant symplectic form. In particular, using the geometric form we will obtain by means of other way the same symmetries that we found using the Hamiltonian analysis.     

An Approximate KAM-Renormalization-Group Scheme for Hamiltonian Systems

We construct an approximate renormalization scheme for Hamiltonian systems with two degrees of freedom. This scheme is a combination of Kolmogorov–Arnold–Moser (KAM) theory and renormalization-group techniques. It makes the connection between the approximate renormalization procedure derived by Escande and Doveil and a systematic expansion of the transformation. In particular, we show that the two main approximations, consisting in keeping only the quadratic terms in the actions and the two main resonances, keep the essential information on the threshold of the breakup of invariant tori.     

On unconstrained SU(2) gluodynamics with theta angle

The Hamiltonian reduction of classical SU(2) Yang–Mills field theory to the equivalent unconstrained theory of gauge invariant local dynamical variables is generalized to the case of nonvanishing -angle. It is shown that for any -angle the elimination of the pure gauge degrees of freedom leads to a corresponding unconstrained non-local theory of self-interacting second rank symmetric tensor fields, and that the obtained classical unconstrained gluodynamics with different -angles are canonically equivalent as on the original constrained level. Received: 16 November 2001 / Published online: 5 April 2002