61 research outputs found
A simple piston problem in one dimension
We study a heavy piston that separates finitely many ideal gas particles
moving inside a one-dimensional gas chamber. Using averaging techniques, we
prove precise rates of convergence of the actual motions of the piston to its
averaged behavior. The convergence is uniform over all initial conditions in a
compact set. The results extend earlier work by Sinai and Neishtadt, who
determined that the averaged behavior is periodic oscillation. In addition, we
investigate the piston system when the particle interactions have been
smoothed. The convergence to the averaged behavior again takes place uniformly,
both over initial conditions and over the amount of smoothing.Comment: Accepted by Nonlinearity. 27 pages, 2 figure
Normal Form and Nekhoroshev stability for nearly-integrable Hamiltonian systems with unconditionally slow aperiodic time dependence
The aim of this paper is to extend the results of Giorgilli and Zehnder for
aperiodic time dependent systems to a case of general nearly-integrable convex
analytic Hamiltonians. The existence of a normal form and then a stability
result are shown in the case of a slow aperiodic time dependence that, under
some smallness conditions, is independent on the size of the perturbation.Comment: Corrected typo in the title and statement of Lemma 3.
Gravitational Ionization: A Chaotic Net in the Kepler System
The long term nonlinear dynamics of a Keplerian binary system under the
combined influences of gravitational radiation damping and external tidal
perturbations is analyzed. Gravitational radiation reaction leads the binary
system towards eventual collapse, while the external periodic perturbations
could lead to the ionization of the system via Arnold diffusion. When these two
opposing tendencies nearly balance each other, interesting chaotic behavior
occurs that is briefly studied in this paper. It is possible to show that
periodic orbits can exist in this system for sufficiently small damping.
Moreover, we employ the method of averaging to investigate the phenomenon of
capture into resonance.Comment: REVTEX Style, Submitte
Square-tiled cyclic covers
A cyclic cover of the complex projective line branched at four appropriate
points has a natural structure of a square-tiled surface. We describe the
combinatorics of such a square-tiled surface, the geometry of the corresponding
Teichm\"uller curve, and compute the Lyapunov exponents of the determinant
bundle over the Teichm\"uller curve with respect to the geodesic flow. This
paper includes a new example (announced by G. Forni and C. Matheus in
\cite{Forni:Matheus}) of a Teichm\"uller curve of a square-tiled cyclic cover
in a stratum of Abelian differentials in genus four with a maximally degenerate
Kontsevich--Zorich spectrum (the only known example found previously by Forni
in genus three also corresponds to a square-tiled cyclic cover
\cite{ForniSurvey}).
We present several new examples of Teichm\"uller curves in strata of
holomorphic and meromorphic quadratic differentials with maximally degenerate
Kontsevich--Zorich spectrum. Presumably, these examples cover all possible
Teichm\"uller curves with maximally degenerate spectrum. We prove that this is
indeed the case within the class of square-tiled cyclic covers.Comment: 34 pages, 6 figures. Final version incorporating referees comments.
In particular, a gap in the previous version was corrected. This file uses
the journal's class file (jmd.cls), so that it is very similar to published
versio
Quasiperiodic spin-orbit motion and spin tunes in storage rings
We present an in-depth analysis of the concept of spin precession frequency
for integrable orbital motion in storage rings. Spin motion on the periodic
closed orbit of a storage ring can be analyzed in terms of the Floquet theorem
for equations of motion with periodic parameters and a spin precession
frequency emerges in a Floquet exponent as an additional frequency of the
system. To define a spin precession frequency on nonperiodic synchro-betatron
orbits we exploit the important concept of quasiperiodicity. This allows a
generalization of the Floquet theorem so that a spin precession frequency can
be defined in this case too. This frequency appears in a Floquet-like exponent
as an additional frequency in the system in analogy with the case of motion on
the closed orbit. These circumstances lead naturally to the definition of the
uniform precession rate and a definition of spin tune. A spin tune is a uniform
precession rate obtained when certain conditions are fulfilled. Having defined
spin tune we define spin-orbit resonance on synchro--betatron orbits and
examine its consequences. We give conditions for the existence of uniform
precession rates and spin tunes (e.g. where small divisors are controlled by
applying a Diophantine condition) and illustrate the various aspects of our
description with several examples. The formalism also suggests the use of
spectral analysis to ``measure'' spin tune during computer simulations of spin
motion on synchro-betatron orbits.Comment: 62 pages, 1 figure. A slight extension of the published versio
Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio
We study the exponentially small splitting of invariant manifolds of
whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable
Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a
torus whose frequency ratio is the silver number . We show
that the Poincar\'e-Melnikov method can be applied to establish the existence
of 4 transverse homoclinic orbits to the whiskered torus, and provide
asymptotic estimates for the tranversality of the splitting whose dependence on
the perturbation parameter satisfies a periodicity property. We
also prove the continuation of the transversality of the homoclinic orbits for
all the sufficiently small values of , generalizing the results
previously known for the golden number.Comment: 17 pages, 2 figure
Dynamics of Quantum Collapse in Energy Measurements
The influence of continuous measurements of energy with a finite accuracy is
studied in various quantum systems through a restriction of the Feynman
path-integrals around the measurement result. The method, which is equivalent
to consider an effective Schr\"odinger equation with a non-Hermitian
Hamiltonian, allows one to study the dynamics of the wavefunction collapse. A
numerical algorithm for solving the effective Schr\"odinger equation is
developed and checked in the case of a harmonic oscillator. The situations, of
physical interest, of a two-level system and of a metastable quantum-well are
then discussed. In the first case the Zeno inhibition observed in quantum
optics experiments is recovered and extended to nonresonant transitions, in the
second one we propose to observe inhibition of spontaneous decay in mesoscopic
heterostructures. In all the considered examples the effect of the continuous
measurement of energy is a freezing of the evolution of the system proportional
to the accuracy of the measurement itself.Comment: 20 pages with figures, compressed and uuencoded ps fil
Strong and weak chaos in weakly nonintegrable many-body Hamiltonian systems
We study properties of chaos in generic one-dimensional nonlinear Hamiltonian
lattices comprised of weakly coupled nonlinear oscillators, by numerical
simulations of continuous-time systems and symplectic maps. For small coupling,
the measure of chaos is found to be proportional to the coupling strength and
lattice length, with the typical maximal Lyapunov exponent being proportional
to the square root of coupling. This strong chaos appears as a result of
triplet resonances between nearby modes. In addition to strong chaos we observe
a weakly chaotic component having much smaller Lyapunov exponent, the measure
of which drops approximately as a square of the coupling strength down to
smallest couplings we were able to reach. We argue that this weak chaos is
linked to the regime of fast Arnold diffusion discussed by Chirikov and
Vecheslavov. In disordered lattices of large size we find a subdiffusive
spreading of initially localized wave packets over larger and larger number of
modes. The relations between the exponent of this spreading and the exponent in
the dependence of the fast Arnold diffusion on coupling strength are analyzed.
We also trace parallels between the slow spreading of chaos and deterministic
rheology.Comment: 15 pages, 14 figure
Progress in Classical and Quantum Variational Principles
We review the development and practical uses of a generalized Maupertuis
least action principle in classical mechanics, in which the action is varied
under the constraint of fixed mean energy for the trial trajectory. The
original Maupertuis (Euler-Lagrange) principle constrains the energy at every
point along the trajectory. The generalized Maupertuis principle is equivalent
to Hamilton's principle. Reciprocal principles are also derived for both the
generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis
Principle is the classical limit of Schr\"{o}dinger's variational principle of
wave mechanics, and is also very useful to solve practical problems in both
classical and semiclassical mechanics, in complete analogy with the quantum
Rayleigh-Ritz method. Classical, semiclassical and quantum variational
calculations are carried out for a number of systems, and the results are
compared. Pedagogical as well as research problems are used as examples, which
include nonconservative as well as relativistic systems
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