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The universal description of orbits in the domain swept by a slowly varying separatrix is provided through a symplectic map derived by means of an extension of classical adiabatic theory. This map connects action-angle-like variables of an orbit when far from the instantaneous separatrix to time-energy variables at a reference point of the orbit very close to the corresponding separatrix. map with WKB theory to obtain a description of the structure underlying chaos: the homoclinic tangle related to the hyperbolic fixed point whose separatrix is pulsating. For each extremum of the area within the pulsating separatrix, an initial branch of length 0 ( 1 / ~) of the stable manifold is explicitly constructed, and makes 0 ( 1 / ~) transverse homoclinic intersections with a similar branch of the unstable manifold. These intersections define parallelograms whose O ( E ) area provides an upper bound to that of any island possibly trapped in the tangle. The area of the homoclinic lobe enclosed by the constructed branches is almost equal to that swept by the separatrix since the preceding extremum.The paper is divided into two parts: our results are first presented on a simple model, emphasizing their physical and pictorial aspects; full mathematical statements and proofs for the general case follow.

Classical mechanics provides the intuitive and unified description of spontaneous emission, Landau growth and damping of Langmuir waves, the cold beam–plasma instability, and van Kampen modes. This is done by studying the interaction between M weak modes of a plasma without resonant particles and N quasiresonant particles, which leads to an exactly solvable high-dimensional Floquet problem. Growth corresponds to an eigenmode of the system, whereas damping requires statistical averaging. Both imply synchronization of near-resonant particles with waves, and the corresponding force on individual particles is computed explicitly.

The one-dimensional (1-D) spatially periodic system of N classical particles, interacting via a Coulomb-like repulsive long-range force, is studied using classical mechanics. The usual Bohm–Gross dispersion relation for the collective modes is obtained in the absence of quasiresonant particles. In the presence of R quasiresonant particles, the evolution equations for M long-wavelength modes are coupled to the particles’ motion through a self-consistent wave–particle Hamiltonian. The wave–particle Lagrangian is derived from the full N-body Lagrangian. The derivation relies on an explicit scale separation argument and avoids the use of kinetic theory and continuous medium formalism.

The validity of quasilinear (QL) theory describing the weak warm beam-plasma instability has been a controversial topic for several decades. This issue is tackled anew, both analytically and by numerical simulations which benefit from the power of modern computers and from the development in the last decade of Vlasov codes endowed with both accuracy and weak numerical diffusion. Self-consistent numerical simulations within the Vlasov-wave description show that QL theory remains valid in the strong chaotic diffusion regime. However, there is a non-QL regime before saturation, which confirms previous analytical work and numerical simulation, but contradicts another analytical work. We show analytically the absence of mode coupling in the saturation regime of the instability where a plateau is present in the tail of the particle distribution function. This invalidates several analytical works trying to prove or to contradict the validity of QL theory in the strongly nonlinear regime of the weak warm beam-plasma instability.

Conservation of energy and momentum in the classical theory of radiating electrons has been a challenging problem since its inception. We propose a formulation of classical electrodynamics in Hamiltonian form that satisfies the Maxwell equations and the Lorentz force. The radiated field is represented with eigenfunctions using the Gel'fand β-transform. The electron Hamiltonian is the standard one coupling the particles with the propagating fields. The dynamics conserves energy and excludes self-acceleration. A complete Hamiltonian formulation results from adding electrostatic action-at-a-distance coupling between electrons. PACS numbers: 84.40.Fe (microwave tubes) 52.35.Fp (Plasma: electrostatic waves and oscillations) 52.40.Mj (particle beam interaction in plasmas) 52.20.Dq (particle orbits) Keywords : wave-particle interaction, traveling wave tube, β-transform, Floquet boundary conditionTo model consistently the interaction between electrons and waves in devices such as traveling wave tubes, free electron lasers or synchrotrons, we are presently left with two options. The first [6,11,17] is to consider the flow of electrons as a distributed charge and current density coupled with the field through the Maxwell equations. Since it generates diverging singularities, the particle nature of electrons is intentionally overlooked until the question of determining the trajectories of the flow is raised. For the latter, the only possibility is to return to a particle description in which the Lorentz force applies. This change of model for the flow precludes the description of the wave-electron system in Hamiltonian form. One reason is that a procedure is needed to distribute the electron charge and current into a finite volume. This procedure, usually based on meshing space, can only be arbitrary. The second option [8,10,13] is to consistently consider electrons as particles and to determine the field they radiate starting from the Liénard-Wiechert potentials. Dirac's [4] sharp analysis indeed provides an accurate determination of the reaction from the radiated field in the limit of a point electron, yet this approach leads to serious difficulties [4,10,13,16,18]; among these an infinite rest mass of the electron, self-acceleration and acausality, also appear incompatible with the existence of a well-posed Hamiltonian.Hamiltonians are essential to consistently define energy and momentum. They also have great practical usefulness to find approximate solutions in complex systems or to control errors in numerical integration schemes [7]. So their absence in the case of the classical wave-electron interaction is both theoretically and practically unsatisfactory. While the final solution to these problems may involve (upgraded, regular) quantum electrodynamics, one could at least hope for a consistent classical approximation, which would be the classical limit of its quantum counterpart. For instance, in the limit where the electron radiates a large number of photons, each of them having a small energy compared with its kin...

A reduced Hamiltonian formulation to reproduce the saturated regime of a Single Pass Free Electron Laser, around perfect tuning, is here discussed. Asymptotically, Nm particles are found to organize in a dense cluster, that evolves as an individual massive unit. The remaining particles fill the surrounding uniform sea, spanning a finite portion of phase space, approximately delimited by the average momenta ω+ and ω−. These quantities enter the model as external parameters, which can be self-consistently determined within the proposed theoretical framework. To this aim, we make use of a statistical mechanics treatment of the Vlasov equation, that governs the initial amplification process. Simulations of the reduced dynamics are shown to successfully capture the oscillating regime observed within the original N -body picture. I. GENERAL BACKGROUNDFree-Electron Lasers (FELs) are coherent and tunable radiation sources, which differ from conventional lasers in using a relativistic electron beam as their lasing medium, hence the term free-electron.The physical mechanism responsible for the light emission and amplification is the interaction between the relativistic electron beam, a magnetostatic periodic field generated in the undulator and an optical wave copropagating with the electrons. Due to the effect of the magnetic field, the electrons are forced to follow sinusoidal trajectories, thus emitting synchrotron radiation. This spontaneous emission is then amplified along the undulator until the laser effect is reached. Among different schemes, single-pass high-gain FELs are currently attracting growing interest, as they are promising sources of powerful and coherent light in the UV and X ranges. Besides the Self Amplified Spontaneous Emission (SASE) setting [1], seeding schemes may be adopted where a small laser signal is injected at the entrance of the undulator and guides the subsequent amplification process [2]. In the following we shall refer to the latter case. Basic features of the system dynamics are successfully captured by a simple one-dimensional Hamiltonian model [15] introduced by Bonifacio and collaborators in [3]. Remarkably, this simplified formulation applies to other physical systems, provided a formal translation of the variables involved is performed. As an example, focus on kinetic plasma turbulence, e.g. the electron beam-plasma instability. When a weak electron beam is injected into a thermal plasma, electrostatic modes at the plasma frequency (Langmuir modes) are destabilized. The interaction of the Langmuir waves and the electrons constituting the beam can be studied in the framework of a self-consistent Hamiltonian picture [4], formally equivalent to the one in [3]. In a recent paper [5] we established a bridge between these two areas of investigation (FEL and plasma), and exploited the connection to derive a reduced Hamiltonian model to characterize the saturated dynamics of the laser. According to this scenario, N m particles are trapped in the resonance, i.e. experience a bouncing motion...