Class Of Hamiltonian Systems Research Articles

Classical dynamical systems fromq-algebras: cluster variables and explicit solutions

A general procedure to get the explicit solution of the equations of motion for N-body classical Hamiltonian systems equipped with coalgebra symmetry is introduced by defining a set of appropriate collective variables which are based on the iterations of the coproduct map on the generators of the algebra. In this way several examples of N-body dynamical systems obtained from q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2) Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of Ruijsenaars type arising from the same (non co-boundary) q-deformation of the (1+1) Poincare' algebra. Also, a unified interpretation of all these systems as different Poisson-Lie dynamics on the same three dimensional solvable Lie group is given.

Chaotic Level Statistics in One-Dimensional Conservative System

Quantum mechanics of one-dimensional time-independent Hamiltonian system whose energy level statistics obeys the Gaussian ensemble is numerically studied by the inverse scattering method. It has been known that the system must be integrable. However, at least numerically, it can be shown that we can construct a potential for the Schrodinger equation that reproduces a finite number of given energy levels of chaotic regime, i.e. given by the random matrix theory (RMT). Contrary to usual classical Hamiltonian systems, there cannot exit the shortest period from the consequence of the semi-classical argument, if the system completely reproduces the chaotic levels of the RMT. If its density of states is unfolded and its average density is independent of energy, the coarse averaged shape of the potential is just like a harmonic oscillator and the potential has peculiar ripples on it. The more energy levels of chaotic regime we take, the more intricate and the finer these ripples become.

Periodic Solutions of Classical Hamiltonian Systems without Palais–Smale Condition

In this paper, we study the existence of periodic solutions for classical Hamiltonian systems without the Palais–Smale condition. We prove that the information of the potential function contained in a finite domain is sufficient for the existence of periodic solutions. Moreover, we establish the existence of infinitely many periodic solutions without any symmetric condition on the potential function V.

Classical Hamiltonian systems with sl(2) coalgebra symmetry and their integrable deformations

Several families of classical integrable systems with two degrees of freedom are derived from phase-space realizations of sl(2) Poisson coalgebras. As a remarkable fact, the existence of the N-dimensional integrable generalization of all these systems is always ensured (by construction) due to their underlying dynamical coalgebra symmetry. By following the same approach, different integrable deformations for such systems are obtained from the q-deformed analogues of sl(2). The well-known Jordan-Schwinger realization is also proven to be related to a (non-coassociative) coalgebra structure on sl(2) and the 2 N dimensional integrable Hamiltonian generated by such Jordan-Schwinger representation is obtained. Finally, the relation between complete integrability and the properties of the initial phase-space realization is elucidated through two more examples based on the Heisenberg-Weyl and so(3,2) Poisson coalgebras.

Stability of Blow-Up Profile and Lower Bounds for Blow-Up Rate for the Critical Generalized KdV Equation

The generalized Korteweg-de Vries equations are a class of Hamiltonian systems in infinite dimension derived from the KdV equation where the quadratic term is replaced by a higher order power term. These equations have two conservation laws in the energy space H 1 (L 2 norm and energy). We consider in this paper the critical generalized KdV equation, which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in H 1 uniform in time for all H 1 solutions (and thus global existence). From [15], there do exist for this equation solutions u(t) such that |u(t)| H1 → +∞ as t ↑ T, where T < +∞ (we call them blow-up solutions). The question is to describe, in a qualitative way, how blow up occurs. For solutions with L 2 mass close to the minimal mass allowing blow up and with decay in L 2 at the right, we prove after rescaling and translation which leave invariant the L 2 norm that the solution converges to a universal profile locally in space at the blow-up time T. From the nature of this profile, we improve the standard lower bound on the blow-up rate for finite time blow-up solutions.

Systemes hamiltoniens k-symplectiques

We study some properties of the k&#8722;symplectic Hamiltonian systems in analogy with the well-known classical Hamiltonian systems. The integrability of k&#8722;symplectic Hamiltonian systems and the relationships with the Nambu’s statistical mechanics are given.

Existence of periodic solutions to second-order Hamiltonian systems with potential indefinite in sign

We have presented some new results for periodic solutions to a class of Hamiltonian systems with potential indefinite in sign. These results improve and correct some known results in the literature.

Energy of internal modes of nonlinear waves and complex frequencies due to symmetry breaking.

Considering a class of Hamiltonian systems, it is demonstrated that energy of the internal modes with real frequencies supported by nonlinear waves and appearing due to perturbations breaking a continuous symmetry has its sign determined by the symmetry itself, independently of the nature of the perturbations. In particular, it is shown that negative energy modes emerge as a result of the breaking of the phase symmetry in the perturbed nonlinear-Schrödinger equation. An expression for energy of the Vakhitov-Kolokolov internal modes is also derived. Comparative analysis of the energy signs of the internal modes in these two cases explains the ubiquity of instabilities with complex frequencies of solitary and continuous waves in systems with broken phase symmetry.

On mixing and metaequilibrium in nonextensive systems

The analytical and computational studies of various isolated classical Hamiltonian systems including long-range interactions suggest that the N→∞ and t→∞ limits do not commute for entire classes of initial conditions. This is, for instance, the case for inertial planar rotators whenever the time evolution is started with the so-called waterbag distribution for velocities and full parallelism for the angles. For fixed N, after a transient, a long and robust anomalous plateau can emerge as time goes on whose velocity distribution is not the Maxwellian one; at later times, the system eventually crosses over onto the usual, Maxwellian distribution. The duration of the plateau diverges with N. This plateau can be considered as a metaequilibrium (or metastable) state, and its description might be in the realm of nonextensive statistical mechanics (for which the entropic index q≠1), whereas at later times the description is well done by the usual Boltzmann–Gibbs statistical mechanics ( q=1). The purpose of these lines is to present a scenario for the mixing properties (i.e., sensitivity to the initial conditions) which is consistent with the observations just mentioned.

Quantum chaos in atom optics

I shall discuss the quantum and classical dynamics of a class of nonlinear Hamiltonian systems. The discussion will be restricted to systems with one degree of freedom. Such systems cannot exhibit chaos, unless the Hamiltonians are time dependent. Thus we shall consider systems with a potential function that has a higher than quadratic dependence on the position and, furthermore, we shall allow the potential function to be a periodic function of time. This is the simplest class of Hamiltonian system that can exhibit chaotic dynamics. I shall show how such systems can be realized in atom optics, where very cord atoms interact with optical dipole potentials of a far-off resonance laser. Such systems are ideal for quantum chaos studies as (i) the energy of the atom is small and action scales are of the order of Planck's constant, (ii) the systems are almost perfectly isolated from the decohering effects of the environment and (iii) optical methods enable exquisite time dependent control of the mechanical potentials seen by the atoms.

Solutions of minimal period for even classical Hamiltonian systems

Solutions of minimal period for even classical Hamiltonian systems

Design of Dynamical Controllers for Electromechanical Hamiltonian Systems

Linear Hamiltonian control systems with collocation of sensor and actuator are considered. Based on a frequency domain approach a controller design algorithm is stated. The design leads to a controller with internal dynamics which uses the output of the system and its first time derivative. The presence of internal dynamics in the controller is an extension of the usual PD-control law and a main result of the work. The design is based on the special properties of the proposed class of systems. In particular, this class of Hamiltonian systems is passive. It is shown that the design leads to strictly passive controllers for a certain choice of the design parameters. This is another significant result and offers a way for robust L2-stabilization even in the case of infinite dimensional systems. Some features of the controller design are discussed with respect to an application, the control of an infinite dimensional system.

Monodromy in the hydrogen atom in crossed fields

We show that the hydrogen atom in orthogonal electric and magnetic fields has a special property of certain integrable classical Hamiltonian systems known as monodromy. The strength of the fields is assumed to be small enough to validate the use of a truncated normal form H snf which is obtained from a two step normalization of the original system. We consider the level sets of H snf on the second reduced phase space. For an open set of field parameters we show that there is a special dynamically invariant set which is a “doubly pinched 2-torus”. This implies that the integrable Hamiltonian H snf has monodromy. Manifestation of monodromy in quantum mechanics is also discussed.

Asymptotic Hamiltonian dynamics: the Toda lattice, the three-wave interaction and the non-holonomic Chaplygin sleigh

In this paper we discuss asymptotic stability in energy-preserving systems which have an almost Poisson structure. In particular we consider a class of Poisson systems which includes the Toda lattice. In standard Hamiltonian systems one of course does not expect asymptotic stability. The key here is the structure of the phase space of the Poisson or almost Poisson systems and the nature of the equilibria. Our systems are in some sense generalizations of the integrable Toda lattice system but are not integrable in general. As a particular example we point out an interesting connection between three mechanical two degree-of-freedom systems that exhibit asymptotic stability. Two of them are classical Hamiltonian systems while the third is a non-holonomic system. Non-holonomic systems are generally energy-preserving but not Hamiltonian, but the system analyzed here turns out to have a phase space which is the union of Hamiltonian ones. We also discuss various higher-dimensional examples.

A collective-variable theory for nonlinear coherent excitations in classical Hamiltonian systems

We derive a collective-variable theory for nonlinear coherent excitations that is based on Dirac’s formalism of constrained Hamiltonian systems. No approximations are involved in this theory in the sense that we are able to formulate equations of motion for a new set of variables, including the collective variables themselves, which are equivalent to the original set of Hamilton’s equation of motion. We show that the dynamics of collective excitations has to be classified according to the rank of the so-called gyromatrix . Two extreme cases are particularly important: in the first case (zero gyromatrix) collective excitations behave very much like Newtonian particles of classical mechanics, while in the second case (regular gyromatrix) collective excitations are similar to charged, massless particles in an external magnetic field.

Quasi-integrable and superintegrable systems from a `deformed-mass' two-photon algebra

A quantum deformation of the two-photon (or Schrodinger) Lie algebra is introduced in order to construct newn-dimensional classical Hamiltonian systems which have (n−2) functionally independent integrals of motion in involution; we say that such Hamiltonians define quasi-integrable systems. Furthermore, Hopf subalgebras of this quantum two-photon algebra (quantum extended Galilei and harmonic oscillator algebras) provide another set of (n−1) integrals of motion for Hamiltonians defined on these Hopf subalgebras, so that they lead to superintegrable systems.

Variational procedure and generalized Lanczos recursion for small-amplitude classical oscillations

Variational procedure is developed that yields lowest frequencies of small-amplitude oscillations of classical Hamiltonian systems. Genuine Lanczos recursion is generalized to treat related non-Hermitian eigenvalue problems.

Isochronicity for Several Classes of Hamiltonian Systems

In this paper we study isochronous centers of analytic Hamiltonian systems giving special attention to the polynomial case. We first revisit the potential systems and we show the connection between isochronicity and involutions. We then study a more general system, namely the ones associated to Hamiltonians of the form H(x, y)=A(x)+B(x)y+C(x)y2. As an application we classify the cubic polynomial Hamiltonian isochronous centers and we give examples of nontrivial and nonglobal polynomial Hamiltonian isochronous centers.

Existence and uniqueness of hyperbolic periodic solution for a class Hamiltonian systems

Using the component Lyapunov functions and critical point theory, the theorem of exisence and uniqueness of hyperbolic periodic solution for a class of Hamiltonian systems is given.

Complex Hamiltonian evolution equations and field theory

The form of the complex Hamiltonian evolution equations (linear and nonlinear) is deduced here from that of the classical Hamiltonian systems. Their properties and those of the associated continuous Poisson bracket are studied. It is shown that if any space-localized solution of the above type equations is expanded in any Hilbert space basis the expansion coefficients are canonically conjugate variables and obey Hamiltonian systems of complex equations. The last is proved also for sets of “canonically conjugate” functionals of general type. As an application, commutation relations which are equivalent in form to Dirac’s quantization rules (the constant in place of ℏ is undetermined) are obtained. The question of how a nonlinear field theory may contain quantum mechanics as a special case is discussed in view of this paper’s results.