generic bifurcation of hamiltonian systems with symmetry we study generic bifurcations of equilibria in one parameter hamiltonian systems with symmetry group on the generalized eigenvalues of the linearized system go through zero theorem 3 3 classifies expected actions of on the generalized eigenspace of this zero eigenvalue generic one degree of freedom symmetric systems is section 4 remarks concerning systems with more degrees of freedom are given in section 5

an integrable normal form for water waves in infinite depth abstract we consider the birkhoff normal form for the water wave problem posed in a fluid of infinite depth with the starting point of our analysis a version of the hamiltonian given by zakharov we verify that in the fourth order normal form the coefficients vanish for all non generic resonant terms and we show that the resulting truncated system is completely integrable in contrast we show that there are resonant fifth order terms with nonvanishing coefficients answering to the negative the conjecture of dyachenko zakharov on the integrability of free surface hydrodynamics we analyse all solutions of the fourth order integrable system in particular the bifurcation structures of standing wave and travelling wave solutions are remarkably explicit due to the normal form

simulating mesoscopic reaction diffusion systems using the gillespie algorithm we examine an application of the gillespie algorithm to simulating spatially inhomogeneous reaction diffusion systems in mesoscopic volumes such as cells and microchambers the method involves discretizing the chamber into elements and modeling the diffusion of chemical species by the movement of molecules between neighboring elements these transitions are expressed in the form of a set of reactions which are added to the chemical system the derivation of the rates of these diffusion reactions is by comparison with a finite volume discretization of the heat equation on an unevenly spaced grid the diffusion coefficient of each species is allowed to be inhomogeneous in space including discontinuities the resulting system is solved by the gillespie algorithm using the fast direct method we show that in an appropriate limit the method reproduces exact solutions of the heat equation for a purely diffusive system and the nonlinear reaction rate equation describing the cubic autocatalytic reaction

periodic solutions of a newtonian equation stability by the third approximation abstract this paper presents a sufficient condition for the stability of periodic solutions of a newtonian equation this condition depends on the third order approximation and does not involve small parameters an application to an equation with cubic potential is given

fluctuations in the lattice gas for burgers equation the langevin equation for the biased random hopping of particles onto unoccupied sites of a lattice is obtained from the master equation for finite systems by invoking a limit theorem due to kurtz for a weakly asymmetric process with diffusive scaling the continuum limit of this equation yields a deterministic equation that is equivalent to burgers equation the stochastic equation for the leading lattice corrections is derived and the approach to the continuum limit is illustrated with simulations

geometry and chaos near resonant equilibria of 3 dof hamiltonian systems in this paper we study the dynamics near resonant elliptic equilibria in three degree of freedom hamiltonian systems the resonances we consider have multiplicity two and the corresponding local normal form for the equilibrium is integrable at cubic order we prove the existence of families of 3 tori and whiskered 2 tori with nearby chaotic dynamics in the quartic normal form the whiskers of the 2 tori intersect in a non trivial way giving rise to multi pulse homoclinic and heteroclinic connections these connections survive in the full system as orbits homoclinic to invariant 3 spheres

classification of perturbations of the hydrogen atom by small static electric and magnetic fields we consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations normalizing with regard to the keplerian sy

unipotent normal forms for symplectic maps abstract a normal form theory for symplectic maps with non diagonalizable linear part is presented the theory is obtained by working directly with the group of symplectic diffeomorphisms using the fact that non diagonalizable linear symplectic maps have a unique semisimple unipotent decomposition the normalization of a symplectic map with respect to its linear part proceeds in two steps first a semisimple normalization for which the theory is well established and second a unipotent normalization which is presented here the role of normal form symmetry and reduction of the normalized map is also considered the theory is applied to two examples of symplectic maps on r 4 first a non semisimple multiple 1 eigenvalue and second the collision of floquet multipliers of opposite sign

cotangent bundle reduction and poincare birkhoff normal forms in this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free lie group action the canonical coordinates enable us to compute poincare birkhoff normal forms of relative equilibria using standard algorithms the case of simple mechanical systems with symmetries is studied in detail as examples we compute poincare birkhoff normal forms for a lagrangian equilateral triangle configuration of a three body system with a morse type potential and the stretched out configuration of a double spherical pendulum

unique normal forms for vector fields and hamiltonians on construit des formes canoniques pour tous les champs vectoriels resp hamiltoniens en equilibre a 0 r n relativement au groupe des diffeomorphismes formels resp des transformations canoniques tangentiels a l identite

a stochastic lattice gas for burgers equation a practical study we continue our investigation of stochastic lattice gases as a highly parallel means of simulating given pdes in this case burgers equation in one dimension the lattice dynamics consists of stochastic unidirectional particle displacement and our attention is turned toward the reliability of the model i e its ability to reproduce the unique physical solution of burgers equation lattice gas results are discussed and compared against finite difference calculations and exact solutions in examples which include shocks and rarefaction waves

complete classification of qualitatively different perturbations of the hydrogen atom in weak near orthogonal electric and magnetic fields we consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields in near orthogonal configurations normalization of the keplerian symmetry reveals that in the parameter space such systems belong in a zone of systems close to the 1 1 resonance the latter corresponding to the exactly orthogonal configuration integrable approximations obtained from second normalization of systems in the 1 1 zone are classified into several different qualitative types many of which possess nontrivial monodromy we compute monodromy of the complete three dimensional energy momentum map compare the joint quantum spectrum to classical bifurcation diagrams and show the effect of second normalization to the joint spectrum

monodromy in the hydrogen atom in crossed fields abstract 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

on reduction of two degrees of freedom hamiltonian systems by an s1 action and so0 1 2 as a dynamical group reduction by an s1 action is a method of finding periodic solutions in hamiltonian systems which is known rather as the method of averaging such periodic solutions can be reconstructed as s1 orbits by pulling back the critical points of an associated reduced hamiltonian on a reduced phase space along the reduction for hamiltonian systems of two degrees of freedom a geometric setting of the reduction is already accomplished in the case where the reduced phase space is a two sphere in the euclidean space r3 and the reduced hamilton s equations of motion are euler s equations this article deals with the case where the reduced phase space will be a two hyperboloid in the three minkowski space and the reduced hamilton s equations of motion will be euler s equations with respect to the lorentz metric this reduction is associated with su 1 1 symplectic action on the phase space r4 as a consequence of this association the reduced hamiltonian system proves to admit a dynamical group so0 1 2 a w

algorithm refinement for the stochastic burgers equation in this paper we develop an algorithm refinement ar scheme for an excluded random walk model whose mean field behavior is given by the viscous burgers equation ar hybrids use the adaptive mesh refinement framework to model a system using a molecular algorithm where desired while allowing a computationally faster continuum representation to be used in the remainder of the domain the focus in this paper is the role of fluctuations on the dynamics in particular we demonstrate that it is necessary to include a stochastic forcing term in burgers equation to accurately capture the correct behavior of the system the conclusion we draw from this study is that the fidelity of multiscale methods that couple disparate algorithms depends on the consistent modeling of fluctuations in each algorithm and on a coupling such as algorithm refinement that preserves this consistency

non hamiltonian monodromy abstract it is well known that the energy momentum map of a two degree of freedom integrable hamiltonian system with a focus focus singularity has monodromy in this paper we generalize this result to systems which are not necessarily hamiltonian

normal forms for perturbed keplerian systems abstract perturbations of rotating and nonrotating keplerian systems with two and three degrees of freedom are considered the dominant unperturbed part of the hamilton function is the sum of the two body part and the coriolis term the flows associated to these two components are used to put the whole system into normal form by means of symplectic transformations three different situations are analysed i the two body effect dominates the coriolis part slow rotations ii the coriolis effect is stronger than the two body hamiltonian very fast rotations and iii the two effects are comparable moderate rotations after performing the transformations to normal form the system is reduced by one or two degrees of freedom we describe the reduced phase spaces by calculating the invariants of the symmetry groups associated to the different normal forms reduced systems the technique is applied to the reduction of a hamiltonian system modelling the trapping mechanisms for the electron of a strongly ionized hydrogen atom

rotation forms and local hamiltonian monodromy the monodromy of torus bundles associated with completely integrable systems can be computed using geometric techniques constructing homology cycles or analytic arguments computing discontinuities of abelian integrals in this article we give a general approach to the computation of monodromy that resembles the analytical one reducing the problem to the computation of residues of polar 1 forms we apply our technique to three celebrated examples of systems with monodromy the champagne bottle the spherical pendulum the hydrogen atom and to the case of non degenerate focus focus singularities re obtaining the classical results an advantage of this approach is that the residue like formula can be shown to be local in a neighborhood of a singularity hence allowing the definition of monodromy also in the case of non compact fibers this idea has been introduced in the literature under the name of scattering monodromy we prove the coincidence of the two definitions with the monodromy of an appropr

proof of nishida s conjecture on anharmonic lattices we prove nishida s 1971 conjecture stating that almost all low energetic motions of the anharmonic fermi pasta ulam lattice with fixed endpoints are quasi periodic the proof is based on the formal computations of nishida the kam theorem discrete symmetry considerations and an algebraic trick that considerably simplifies earlier results

reduction of the semisimple 1 1 resonance the method of averaging is often used in hamiltonian systems of two degrees of freedom to find periodic orbits such periodic orbits can be reconstructed from the critical points of an associated reduced hamiltonian on a reduced space this paper details the construction of the reduced space and the reduced hamiltonian for the semisimple 1 1 resonance case the reduced space will be a 2 sphere in r3 and the reduced differential equations will be euler s equations restricted to this sphere the orbit projection from the energy surface in phase space to this sphere will be the hopf map the results of the paper are related to problems in physics on degeneracies due to symmetries of classical two dimensional harmonic oscillators and their quantum analogues for the hydrogen atom

a universal procedure for normalizing n degree of freedom polynomial hamiltonian systems we depart from an n degree of freedom hamiltonian formed by the sum of homogeneous polynomials in n coordinates and n momenta with arbitrary coefficients by extending formally an integral of the principal part of the system to the full hamiltonian and truncating higher order terms we obtain a simplified hamiltonian this normalization procedure can be used to extract qualitative features of the departure system in this paper we present the symbolic routines needed to achieve the normalization the power and generality of the algorithm are exhibited through two examples

normal form and representation theory representation theory of lie algebras is called upon to develop a procedure for normalizing a dynamical system with two degrees of freedom in the neighborhood of an equilibrium when the hamiltonian h x y x y in the coordinates x y and their conjugate momenta x y is of the type h x2 y2 2 v x y x y the potential energy v being a sum of homogeneous polynomials in the phase variables of degree strictly greater than 2 the fact that the resulting potential v is a polynomial in the new coordinates x y and the angular momentum g x y y x implies that the normalization is a rotation in the configuration space from a fixed frame to an ideal frame the technique is intended for normalizing an hamiltonian in equilibrium at the origin when the lie derivative associated with the quadratic part is not semisimple e g the planar restricted problem of three bodies at the equilateral equilibrium l4 when the basic frequencies are equal routh s singular case

nonlinearly coupled oscillators in quantum mechanics a normal form approach quantum mechanical hamiltonians of the form of n harmonic oscillators coupled via an interaction of the form e times a polynomial in the position and momenta variables are studied in a rigorous hilbert space setting in particular normal form theory is used to define the mth approximation to the associated schrodinger initial value problem and it is shown that it deviates in norm from the exact solution by a term of order em 1 t t time provided only that the initial vector is confined to an appropriate dense subspace the main concentration is on the case in which there exist no resonances of order m between the frequencies of the n oscillators but the case of two oscillators in 1 1 resonance is also taken up

fractional monodromy in the 1 2 resonance we give an analytic proof of the fractional monodromy theorem for the 1 2 oscillator system with s 1 symmetry formulated by n n nekhoroshev d a sadovskii and b i zhilinskii in c r acad sci paris ser i 335 2002 985 988 our proof is based on an analytic description of the hamiltonian flow on the fibers of the integral map of this system c 2006 elsevier inc all rights reserved

the geometrical description of the nonlinear dynamics of a multiple pendulum a system of two coupled pendula is an example of a hamiltonian system with two degrees of freedom rott considered such a system in the presence of 1 2 resonance and observed a phase locking phenomenon in this paper this phenomenon is explained by means of duistermaat s method of bringing a hamiltonian system with two degrees of freedom into standard form his theory is based on mather s theory of stability of differentiable mappings a transparent geometrical picture is obtained in the phase space in particular a criterion of the phase locking phenomenon is provided based on this picture

renormalization of stochastic lattice models basic formulation we describe a general method for the multiscale analysis of stochastic lattice models beginning with a lattice langevin formulation of site fluctuations we derive stochastic partial differential equations by regularizing the transition rules of the model subsequent coarse graining is accomplished by calculating renormalization group rg trajectories from initial conditions determined by the regularized atomistic models the rg trajectories correspond to hierarchies of continuum equations describing lattice models over expanding length and time scales these continuum equations retain a quantitative connection over different scales as well as to the underlying atomistic dynamics this provides a systematic method for the derivation of continuum equations from the transition rules of lattice models for any length and time scales as an illustration we consider the one dimensional 1d wolf villain wv model europhys lett 13 389 1990 the rg analysis of this model which we develop in detail is generic and can be applied to a wide range of conservative lattice models the rg trajectory of the 1d wv model shows a complex crossover sequence of linear and nonlinear stochastic differential equations which is in excellent agreement with kinetic monte carlo simulations of this model we conclude by discussing possible applications of the multiscale method described here to other nonequilibrium systems

uncovering fractional monodromy the uncovering of the role of monodromy in integrable hamiltonian fibrations has been one of the major advances in the study of integrable hamiltonian systems in the past few decades on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action angle coordinates while on the other hand it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra fractional monodromy is a generalization of the concept of monodromy instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres in this paper we analyze fractional monodromy for n1 n2 resonant hamiltonian systems with n1 n2 coprime natural numbers we consider in particular systems that for n1 n2 1 contain one parameter families of singular fibres which are curled tori we simplify the geometry of the fibration by passing to an appropriate branched covering in the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration we essentially obtain the same type of generalized monodromy independently of n1 n2 fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system

the effect of pitchfork bifurcations on the spectral statistics of hamiltonian systems we present a quantitative semiclassical treatment of the effects of bifurcations on the spectral rigidity and the spectral form factor of a hamiltonian quantum system defined by two coupled quartic oscillators which on the classical level exhibits mixed phase space dynamics we show that the signature of a pitchfork bifurcation is two fold beside the known effect of an enhanced periodic orbit contribution due to its peculiar hbar dependence at the bifurcation we demonstrate that the orbit pair born at the bifurcation gives rise to distinct deviations from universality slightly above the bifurcation this requires a semiclassical treatment beyond the so called diagonal approximation our semiclassical predictions for both the coarse grained density of states and the spectral rigidity are in excellent agreement with corresponding quantum mechanical results

hamiltonian hopf bifurcation of the hydrogen atom in crossed fields we consider the hydrogen atom in crossed electric and magnetic fields we prove that near the stark and zeeman limits the system goes through two qualitatively different hamiltonian hopf bifurcations we explain in detail the geometry of the bifurcations

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

bifurcations in the slow fluctuation technique in the slow fluctuation method the bifurcations between solutions of a conservative autonomous nonlinear oscillatory system can be found by an essentially algebraic systematic study of constant amplitude motions and root parities this is shown for simple systems with sufficient detail to indicate the extensions needed in the treatment of more complex systems an example from celestial mechanics is given

the probability of the quantum diffusion controlled reaction of a species pair abstract the kinetics of diffusion controlled bimolecular reaction a b p products is investigated in the case when the motion of the diffusing particle is governed by the quantum markovian master equation for a density matrix in the site representation in the framework of the discrete model of random walks definitions are given of the survival and reaction probabilities for a pair of particle species a and b and of the rate constant a method for calculating these quantities is proposed formulae are obtained for the survival and reaction probabilities and the rate constant corresponding to chemical interaction between the particles only when they are at one and the same site

langevin equations for fluctuating surfaces exact langevin equations are derived for the height fluctuations of surfaces driven by the deposition of material from a molecular beam we consider two types of model deposition models where growth proceeds by the deposition and instantaneous local relaxation of particles with no subsequent movement and models with concurrent random deposition and surface diffusion starting from a chapman kolmogorov equation the deposition relaxation and hopping rules of these models are first expressed as transition rates within a master equation for the joint height probability density function the kramers moyal van kampen expansion of the master equation in terms of an appropriate largeness parameter yields according to a limit theorem due to kurtz stoch proc appl 6 223 1978 a fokker planck equation that embodies the statistical properties of the original lattice model the statistical equivalence of this fokker planck equation solved in terms of its associated langevin equation and solutions of the chapman kolmogorov equation as determined by kinetic monte carlo kmc simulations of the lattice transition rules is demonstrated explicitly by comparing the surface roughness and the lateral height correlations obtained from the two formulations for the edwards wilkinson proc roy soc london ser a 381 17 1982 and wolfvillain europhys lett 13 389 1990 deposition models and for a model with random deposition and surface diffusion in each case as the largeness parameter is increased the langevin equation converges to the surface roughness and lateral height correlations produced by kmc simulations for all times including the crossover between different scaling regimes we conclude by examining some of the wider implications of these results including applications to heteroepitaxial systems and the passage to the continuum limit

fluctuations in the weakly asymmetric exclusion process with open boundary conditions we investigate the fluctuations around the average density profile in the weakly asymmetric exclusion process with open boundaries in the steady state we show that these fluctuations are given in the macroscopic limit by a centered gaussian field and we compute explicitly its covariance function we use two approaches the first method is dynamical and based on fluctuations around the hydrodynamic limit we prove that the density fluctuations evolve macroscopically according to an autonomous stochastic equation and we search for the stationary distribution of this evolution the second approach which is based on a representation of the steady state as a sum over paths allows one to write the density fluctuations in the steady state as a sum over two independent processes one of which is the derivative of a brownian motion the other one being related to a random path in a potential

homoclinic and heteroclinic orbits of reversible vectorfields under perturbation on demontre l existence de liens homocline heterocline entre les orbites periodiques hyperboliques d une classe de systemes hamiltoniens admettant un nombre suffisant de symetries sous une reduction isoenergetique le probleme se reduit a un probleme comprenant des champs vectoriels reversibles avec perturbations dependantes du temps admettant les memes symetries de renversement

singular manifolds and quasi periodic solutions of hamiltonians for vortex lattices abstract a general method for establishing the existence of quasi periodic solutions of hamiltonian systems for vortex lattices is illustrated in a simple example involving two degrees of freedom the geometry of intersecting singular manifolds of the hamiltonians introduces suitable canonical transformations which put the hamiltonian into the form of singular weakly coupled oscillators as by products of this procedure additional integrals of motion are found for the leading term in the transformed hamiltonian these extra integrals are approximate invariants for the full hamiltonians

quantum markovian master equation theory of particle migration in a stochastic medium abstract the problem of quasi particle motion in a stochastic medium is investigated under the supposition that it is a markovian process the phenomenological approach to this problem starts from the well known quantum markovian master equation the paper is devoted to the construction of the so called dissipative terms of the master equation a special form of dissipative terms is suggested the quasi particle performing random walks from site to site in a crystal lattice and the diffusing particle in a stochastic continuum are discussed as particular cases of an open quantum system an evolution of which may be described by means of a markovian master equation containing the dissipative terms constructed with the help of the suggested method

statistical theory of systems of created reactive species abstract a statistical theory is developed for systems in which the creation of identical species a in a condensed medium is accompanied by their decay as a result of the reaction a a p products many species distribution functions which describe the processes of creation and decay of species are defined on the basis of statistical ensembles equations for these functions are obtained the theory is a statistical validation of the author s earlier kinetic theories of reactive species accumulation and bimolecular reactions in condensed media

pulsation and precession of the resonant swinging spring when the frequencies of the elastic and pendular oscillations of an elastic pendulum or swinging spring are in the ratio 2 1 there is a regular exchange of energy between the two modes of oscillation we refer to this phenomenon as pulsation between the horizontal excursions or pulses the spring undergoes a change of azimuth which we call the precession angle the pulsation and stepwise precession are the characteristic features of the dynamics of the swinging spring the modulation equations for the small amplitude resonant motion of the system are the well known three wave equations we use hamiltonian reduction to determine a complete analytical solution the amplitudes and phases are expressed in terms of both weierstrass and jacobi elliptic functions the strength of the pulsation may be computed from the invariants of the equations several analytical formulas are found for the precession angle we deduce simplified approximate expressions in terms of elementary functions for the pulsation amplitude and precession angle and demonstrate their high accuracy by numerical experiments thus for given initial conditions we can describe the envelope dynamics without solving the equations conversely given the parameters which determine the envelope we can specify initial conditions which to a high level of accuracy yield this envelope

symbolic dynamics for the henon heiles hamiltonian on the critical level abstract we present a computer assisted proof of the existence of a rich symbolic dynamic structure for the henon heiles hamiltonian system at the critical energy e 1 6

on averaging reduction and symmetry in hamiltonian systems abstract the existence of periodic orbits for hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged hamiltonian on an associated reduced space alternatively in classical kinetic plus potential energy hamiltonians the existence of such orbits can often be established by elementary geometrical arguments the present paper unifies the two approaches by exploiting discrete symmetries including reversing diffeomorphisms that occur in a given system the symmetries are used to locate the periodic orbits in the averaged hamiltonian and thence in the original hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries in applications to the henon heiles hamiltonian it is illustrated how higher order averaging can sometimes be used to overcome degeneracies encountered at first order

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

lattice gases large deviations and the incompressible navier stokes equations we study the incompressible limit for a class of stochastic particle systems on the cubic lattice zd d 3 for initial distributions corresponding to arbitrary macroscopic l2 initial data the distributions of the evolving empirical momentum densities are shown to have a weak limit supported entirely on global weak solutions of the incompressible navier stokes equations furthermore explicit exponential rates for the convergence large deviations are obtained the probability to violate the divergence free condition decays at rate at least exp e d l while the probability to violate the momentum conservation equation decays at rate exp fe d 2 with an explicit rate function given by an hi1 norm

time dependent normal form theory and schrodinger initial value problems time dependent normal form theory is introduced and studied in the context of schrodinger initial value problems associated with hamiltonians having time dependent perturbations to arrive at approximations that deviate in norm from the exact solution by a term of order em 1 1 t provided the initial vector is restricted to an appropriate dense subspace in addition to the abstract theory within a rigorous hilbert space formalism it is applied to the quantum mechanical hamiltonians of the form n harmonic oscillators coupled via an interaction of the form e times sum of products of bounded continuous functions with polynomials in the position and momenta variables satisfying a nonresonance condition in particular first order approximation is explicitly constructed for time evolution of a system describing anharmonic oscillator which changes trigonometrically with time

symmetry and resonance in periodic fpu chains the symmetry and resonance properties of the fermi pasta ulam chain with periodic boundary conditions are exploited to construct a near identity transformation bringing this hamiltonian system into a particularly simple form this birkhoff gustavson normal form retains the symmetries of the original system and we show that in most cases this allows us to view the periodic fpu hamiltonian as a perturbation of a nondegenerate liouville integrable hamiltonian according to the kam theorem this proves the existence of many invariant tori on which motion is quasiperiodic experiments confirm this qualitative behaviour we note that one can not expect this in lower order resonant hamiltonian systems so the periodic fpu chain is an exception and its special features are caused by a combination of special resonances and symmetries

long time approximation to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics a simple normal form approach is used to obtain uniform long time approximations to the evolution of a resonant or nonresonant anharmonic oscillator system governed by a hamiltonian he which is a self adjoint operator acting in the hilbert space h l2 r 2 and is given formally by h0 ev here h0 denotes the hamiltonian of one dimensional harmonic oscillators whose coupling is represented by ev where v is an operator of multiplication by a smooth function of at most polynomial growth at infinity and e 0 is a small parameter we consider the general situation in which 1 of the frequencies of these oscillators are rationally independent imposing a standard diophantine condition on the independent frequencies if 2 under these assumptions which are stated in a mathematically precise way in the paper an nth order approximant n t e to the exact solution t e of the schrodinger equation id t e dt he t e satisfying the initial condition 0 e 0 is constructed inductively 0 being an arbi

monodromy in the resonant swing spring abstract in this paper it is shown that an integrable approximation of the spring pendulum when tuned to be in 1 1 2 resonance has monodromy the stepwise precession angle of the swing plane of the resonant spring pendulum is shown to be a rotation number of the integrable approximation due to the monodromy this rotation number is not a globally defined function of the integrals in fact at lowest order it is given by arg i where and are functions of the integrals the resonant swing spring is therefore a system where monodromy has easily observed physical consequences

reduction of polynomial planar hamiltonians with quadratic unperturbed part we classify the possible normal forms of quadratic hamiltonians in 2 dimensions then we give a method to reduce by one the number of degrees of freedom of an arbitrary polynomial hamiltonian whose principal part is quadratic in positions and moments the reduction procedure is based on the extension of an integral of the unperturbed part to the whole system up to a certain order the corresponding reduced phase spaces have dimension 2 and are described by means of the set of invariants associated to the reduction

hamiltonian systems with a discrete symmetry abstract if m m is an antisymplectic involution of a symplectic manifold m then the fixed set of is a lagrangian submanifold l m moreover there exist cotangent bundle coordinates in a neighborhood of l in m such that in these coordinates maps a covector into its negative thus classical examples which have a discrete symmetry such as the restricted three body problems are locally like a reversible system

lie series approach to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics a version of a normal form method of kummer and gompa analogous to the lie series methods of classical mechanics is used to rigorously construct uniform long time approximations to the quantum evolution of a system of resonant or nonresonant anharmonic oscillators the hamiltonian he governing the system is a self adjoint operator in the hilbert space h l2 r 2 given formally by h0 ev here h0 denotes the hamiltonian of one dimensional simple harmonic oscillators and ev their anharmonic coupling e 0 being a small parameter and v an operator of multiplication by a smooth function of polynomial growth at infinity we treat the case in which an arbitrary number 1 of the frequencies of these oscillators are rationally independent imposing a standard diophantine condition on the independent frequencies if 2 under these assumptions stated precisely in the paper an nth order approximant n t e to the exact solution t e of the schrodinger equation id t e dt he t e satisfying the initi

normal forms and quasi integrals for the hamiltonians of magnetic bottles the well known birkhoff gustavson normal form theory suffers from the restraint that the quadratic part of the hamiltonian must be of the harmonic oscillator type in this paper we describe a generalized normal form concept which can be applied to any polynomial hamiltonian thus rendering the above restriction to harmonic oscillators unnecessary as in the classical theory we can derive an asymptotic expression for a second integral of motion the truncated formal integral the quasi integral exhibits good convergence properties in regions of phase space where the dynamics is regular whereas in chaotic regions the convergence deteriorates in order to exemplify these findings we apply the theory to a hamiltonian describing a particular type of magnetic bottle which cannot be analysed using the birkhoff gustavson normal form we calculate the quasi integrals up to and including the 14th order and analyse their convergence properties

a jacobson morozov lemma for sp 2n r a variation of the lemma of jacobson morozov on the imbedding of a nonzero nilpotent element of the real symplectic algebra into the split simple three dimensional lie algebra is proved the proof is algorithmic and relies on our earlier work on the theory of normal forms for the real symplectic algebra

classical bethe ansatz and normal forms in an integrable version of the dicke model abstract the dicke jaynes cummings gaudin model describes a collection of n spins coupled to a harmonic oscillator it is known to be integrable so one can define a moment map which associates to each point in phase space the list of values of the n 1 conserved hamiltonians we identify all the critical points of this map and we compute the corresponding quadratic normal forms using the lax matrix representation of the model the normal coordinates are constructed by a procedure which appears as a classical version of the bethe ansatz used to solve the quantum model we show that only elliptic or focus focus singularities are present in this model which provides an interesting example of a symplectic toric action with singularities to explore these we study in detail the degeneracies of the spectral curves for the n 1 and n 2 cases this gives a complete picture for the image of the momentum map imm and the associated bifurcation diagram for n 2 we found in particular some lines of rank 1 which lie for one part on the boundary of the imm where they behave like an edge separating two faces and which go for another part inside the imm

maslov indices and monodromy we prove that for a hamiltonian system on a cotangent bundle that is liouville integrable and has monodromy the vector of maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1 as a corollary the resulting restrictions on the monodromy matrix are derived

the algebraic quantisation of the birkhoff gustavson normal form the author develops an algebraic quantisation method for the birkhoff gustavson normal form for this purpose the weyl quantisation rule is used the method developed here for multidimensional systems allows one to calculate the energy levels and the transition probabilities the author gives a brief review of the normal form derives some of its general properties and finds a general analytic solution for the fourth degree normal form for hamiltonians of two degrees of freedom in particular this includes the henon heiles system the author compares the results of specific examples with other works the question of canonically invariant quantisation the relation to the quantum mechanical perturbation theory and the question of chaotic behaviour and quantum stochasticity are discussed the author shows that the operators corresponding to the formal integrals of the motion are also quantum mechanical integrals if the normal form accidentally terminates so that the classical system is integrable then this implies quantum integrability of the normal form hamiltonian

fluctuations in the motions of mass and of patterns in one dimensional driven diffusive systems the stochastic spreading of mass fluctuations in systems described by a fluctuating burgers equation increases ast2 3 with time as a consequence the stochastic motion of a mass front a point through which no excess mass current is flowing is shown to increase ast1 3 the same is true for the stochastic displacement of mass points and shock fronts with respect to their average drift provided the initial configuration is fixed an additional average over the stationary distribution of the initial configuration yields stochastic displacements increasing with time ast1 2

burger s model of turbulence as a stochastic process a recently proposed mesoscopic description of fluid dynamics leads to a new approach to turbulence in contrast to the classical statistical theory of turbulence the new approach introduces a probabilistic time evolution of the random velocity governed by a master equation the mesoscopic approach is explained by means of the 1 1 dimensional burger s model of turbulence by a continuous time stochastic simulation realizations of turbulent velocity fields are generated correlation functions and energy spectra are evaluated from appropriate ensemble averages

fokker planck equation for lattice deposition models an asymptotically exact fokker planck equation for the height fluctuations of lattice deposition models is derived from a van kampen expansion of the master equation using an edwards wilkinson type model as an example the solution of the equivalent langevin equation reproduces the surface roughness and lateral height correlations obtained with kinetic monte carlo kmc simulations our discrete equations of motion thereby provide an exact analytic and computational alternative to kmc simulations of these models