Periodic orbits of nonscaling Hamiltonian systems from quantum mechanics

Quantal (E,tau) plots are constructed from the eigenvalues of the quantum system. We demonstrate that these representations display the periodic orbits of the classical system, including bifurcations and the transition from stable to unstable. (c) 1995 American Institute of Physics.     

Closed orbits and semiclassical wavefunctions in two-dimensional Hamiltonian systems

The periodic-orbit theory of the density-of-states and the closed-orbit theory of atomic absorption spectra relate properties of a quantum system to properties of periodic or of closed orbits of a system. In these theories, every return of an orbit to the initial point makes a contribution, so in general for each orbit an infinite number of terms must be computed and summed. We show that the term asising from thenth return of an orbit can be calculated from properties of the orbit on its first return.     

Electronic properties of a class of one-dimensional quasiperiodic systems

Electronic properties of a class of one-dimensional quasiperiodic systems are studied by the extended Kohmoto-Kadanoff-Tang (KKT) renormalization-group method. The employed models are tight-binding diagonal and off-diagonal models. It is showed that the energy spectra of the quasiperiodic systems are Cantor-like, namely the spectra are self-similar and the energy gaps are every-where dense on the realE-line.     

Non-linear stability of singular relative periodic orbits in Hamiltonian systems with symmetry

We generalize the sufficient condition for the stability of relative periodic orbits in symmetric Hamiltonian systems presented in [J.-P. Ortega, T.S. Ratiu, J. Geom. Phys. 32 (1999) 131–159] to the case in which these orbits have non-trivial symmetry. We also describe a block diagonalization, similar in philosophy to the one presented in [J.-P. Ortega, T.S. Ratiu, Nonlinearity 12 (1999) 693–720] for relative equilibria, that facilitates the use of this result in particular examples and shows the relation between the stability of the relative periodic orbit and the orbital stability of the associated singular reduced periodic orbit.     

The potential energy surface and chaos in 2D Hamiltonian systems

We provide a new insight into the relationship between the geometric property of the potential energy surface and chaotic behavior of 2D Hamiltonian dynamical systems, and give an indicator of chaos based on the geometric property of the potential energy surface by defining Mean Convex Index (MCI). We also discuss a model of unstable Hamiltonian in detail, and show our results in good agreement with HBLSL's (Horwitz, Ben Zion, Lewkowicz, Schiffer and Levitan) new Riemannian geometric criterion.     

Perturbation theory for periodic orbits in a class of infinite dimensional Hamiltonian systems

We consider a class of Hamiltonian systems describing an infinite array of coupled anharmonic oscillators, and we study the bifurcation of periodic orbits off the equilibrium point. The family of orbits we construct can be parametrized by their periods which belong to Cantor sets of large measure containing certain periods of the linearized problem as accumulation points. The infinitely many holes forming a dense set on which the existence of a periodic orbit cannot be proven originate from a dense set of resonances that are present in the system. We also have a result concerning the existence of solutions of arbitrarily large amplitude.Address after September 1990     

Distribution of fixed-point energies of a quasiperiodic Hamiltonian flow

Energy distributions rho(+/-)(E) for the elliptic and hyperbolic fixed points of the Hamiltonian H(x,y)= summation operator (k=0) (4) cos [x cos(2pik/5)+y sin(2pik/5)] are calculated as integrals over a one-dimensional manifold M(E) in five-dimensional space. Singular points of M(E) produce three logarithmic singularities of rho(+/-)(E), and vanishing of connected components of M(E) gives rise to three discontinuities. The strengths of the singularities and discontinuities of rho(+/-)(E) are determined analytically, and the distributions are evaluated numerically for representative points in the nonsingular intervals. The calculation provides an explicit realization of general theorems concerning the critical points of infinitely smooth functions defined on an n-dimensional torus and restricted to a k-dimensional linear subset. Formally the calculation resembles the determination of the density of states of a dynamical system with one degree of freedom on a 2-torus, but with important differences due to topology and symmetry.     

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.     

Correlation properties of dynamical chaos in Hamiltonian systems

The structure of chaos border in phase space and its impact on the correlation and other statistical properties of the chaotic motion are considered. We conjecture that such a structure is described by a chaotic renormalization group. The effects of an external noise and of dissipation are discussed.     

Geometric Quantization of Relativistic Hamiltonian Mechanics

A relativistic Hamiltonian mechanical system is seen as a conservative Dirac constraint system on the cotangent bundle of a pseudo-Riemannian manifold. We provide geometric quantization of this cotangent bundle where the quantum constraint serves as a relativistic quantum equation.     

Real Hamiltonian forms of Hamiltonian systems

We introduce the notion of a real form of a Hamiltonian dynamical system in analogy with the notion of real forms for simple Lie algebras. This is done by restricting the complexified initial dynamical system to the fixed point set of a given involution. The resulting subspace is isomorphic (but not symplectomorphic) to the initial phase space. Thus to each real Hamiltonian system we are able to associate another nonequivalent (real) ones. A crucial role in this construction is played by the assumed analyticity and the invariance of the Hamiltonian under the involution. We show that if the initial system is Liouville integrable, then its complexification and its real forms will be integrable again and this provides a method of finding new integrable systems starting from known ones. We demonstrate our construction by finding real forms of dynamics for the Toda chain and a family of Calogero-Moser models. For these models we also show that the involution of the complexified phase space induces a Cartan-like involution of their Lax representations.Received: 8 October 2003, Published online: 8 June 2004PACS: 02.30.Ik Integrable systems - 45.20.Jj Lagrangian and Hamiltonian mechanics     

Irrational mode locking in quasiperiodic systems

A model for ac-driven systems, based on the Tang-Wiesenfeld-Bak-Coppersmith-Littlewood automaton for an elastic medium, exhibits mode-locked steps with frequencies that are irrational multiples of the drive frequency, when the pinning is spatially quasiperiodic. Detailed numerical evidence is presented for the large-system-size convergence of such a mode-locked step. The irrational mode locking is stable to small thermal noise and weak disorder. Continuous-time models with irrational mode locking and possible experimental realizations are discussed.     

Photonic properties of metallic-mean quasiperiodic chains

The light propagation through a stack of two media with different refractive indices, which are aligned according to different quasiperiodic sequences determined by metallic means, is studied using the transfer matrix method. The focus lies on the investigation of the influence of the underlying quasiperiodic sequence as well as the dependence of the transmission on the frequency, the incidence angle of the light wave and different ratios of the refractive indices. In contrast to a periodically aligned stack we find complete transmission for the quasiperiodic systems for a wide range of different refractive indices for small incidence angles. Additional bands of moderate transmission occur for frequencies in the range of the photonic band gaps of the periodic system. Further, for fixed indices of refraction we find a range of almost perfect transmission for angles close to the angle of total reflection, which is caused by the bending of photonic transmission bands towards higher frequencies for increasing incidence angles. Comparing with the results of a periodic stack the quasiperiodicity seems to have only an influence in the region around the midgap frequency of a periodic stack.     

On the absence of stable periodic orbits in domains of separatrix crossings in nonsymmetric slow-fast Hamiltonian systems

We consider a two degree of freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. We assume that at frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under rather general conditions, we prove that there are no stable periodic trajectories of any prescribed period inside the domain of separatrix crossings, except maybe for periodic trajectories passing anomalously close to the saddle point.     

Reduction of quantum analogs of Hamiltonian systems described by Lie algebras to orbits in a coadjoint representation

A study is made of the possibility of reducing quantum analogs of Hamiltonian systems to Lie algebras. The procedure of reducing classical systems to orbits in a coadjoint representation based on Lie algebra is well-known. An analog of this procedure for quantum systems described by linear differential equations (LDEs) in partial derivatives is proposed here on the basis of the method of noncommutative integration of LDEs. As an example illustrating the procedure, an examination is made of nontrivial systems that cannot be integrated by separation of variables: the Gryachev-Chaplygin hydrostat and the Kovalevskii gyroscope. In both cases, the problem is reduced to a system with a smaller number of variables. Tomsk University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 69–74, May, 1998.