Canonical Action-angle Variables Research Articles

Guiding centre motion for particles in a ponderomotive magnetostatic end plug

The Hamiltonian dynamics of a single particle in a rotating plasma column, interacting with an magnetic multipole is perturbatively solved for up to second order, using the method of Lie transformations. First, the exact Hamiltonian is expressed in terms of canonical action-angle variables, and then an approximate integrable Hamiltonian is introduced, using another set of actions and angles, which describe the centre of oscillation for the particle. The perturbation introduces an effective ponderomotive potential, which to leading order is positive. At the second order, the pseudopotential consists of a sum of terms of the Miller form, and can have either sign. Additionally, at second order, the ponderomotive interaction introduces a modification to the particle effective mass, when considering the motion along the column axis. It is found that particles can be axially confined by the ponderomotive potentials, but acquire radial excursions which scale as the confining potential. The radial excursions of the particle along its trajectory are investigated, and a condition for the minimal rotation frequency for which the particle remains radially confined is derived. Last, we comment on the changes to the aforementioned solution to the pseudopotentials and particle trajectory in the case of resonant motion, that is, a motion which has the same periodicity as the perturbation.

Evaluation of toroidal torque by non-resonant magnetic perturbations in tokamaks for resonant transport regimes using a Hamiltonian approach

Toroidal torque generated by neoclassical viscosity caused by external non-resonant, non-axisymmetric perturbations has a significant influence on toroidal plasma rotation in tokamaks. In this article, a derivation for the expressions of toroidal torque and radial transport in resonant regimes is provided within quasilinear theory in canonical action-angle variables. The proposed approach treats all low-collisional quasilinear resonant neoclassical toroidal viscosity regimes including superbanana-plateau and drift-orbit resonances in a unified way and allows for magnetic drift in all regimes. It is valid for perturbations on toroidally symmetric flux surfaces of the unperturbed equilibrium without specific assumptions on geometry or aspect ratio. The resulting expressions are shown to match the existing analytical results in the large aspect ratio limit. Numerical results from the newly developed code NEO-RT are compared to calculations by the quasilinear version of the code NEO-2 at low collisionalities. The importance of the magnetic shear term in the magnetic drift frequency and a significant effect of the magnetic drift on drift-orbit resonances are demonstrated.

On Tomonaga’s theory of split-anode magnetrons

This article is meant to formulate the equations of motion of an electron in a cavity magnetron using action-angle variables. This means following the electron's path on its way from a cylindrical cathode moving toward a co-axial cylindrical anode in presence of a uniform magnetic field parallel to the common axis. After analyzing the situation without coupling to an external oscillatory electric field, we employ methods of canonical perturbation theory to find the resonance condition between the frequencies of the free theory w_r, w_phi and the applied perturbing oscillatory frequency w. A long-time averaging process will then eliminate the periodic terms in the equation for the now time-dependent action-angle variables. The terms that are no longer periodic will cause secular changes so that the canonical action-angle variables (J, delta) change in a way that the path of the electron will deform gradually so that it can reach the anode. How the ensemble of the initially randomly distributed electrons forms spokes and how their energy is conveyed to the cavity-field oscillation is the main focus of this article. Some remarks concerning the importance of results in QED and the invention of radar theory and application conclude the article.

Numerical Solution to Vinti’s Problem

A practical, calculable solution to Vinti’s problem is offered. Vinti’s problem describes the motion of a satellite orbiting an oblate planet, including the point mass and oblateness terms and differing from the actual geopotential only at the higher harmonics. Vinti’s original solution is extended to cover motion over a rotating Earth. An eccentric-anomaly-like transform removes the singularities in the elliptic integrals, rendering them accessible to numerical calculation. The prediction problem requires the simultaneous solution of two “Kepler’s equations.” Reduction to canonical action angle variables can also be accomplished via numerical techniques. This new realization of Vinti’s solution is free of series expansions that limit accuracy and uses only simple well-understood numerical techniques. Possible use of Vinti’s solution for further efforts is discussed.

Area Quantization of the Parameter Space of Riemann Surfaces in Genus Two

We consider a model of compact Riemann surfaces in genus two represented geometrically by two-parametric hyperbolic octagons with an order п/2 automorphism. We compute the generators of the Fuchsian group and give a real-analytic description of the corresponding Teichm¨uller space parametrized by the Fenchel–Nielsen variables in terms of geometric data. We state the structure of the parameter space by computing the Weil–Petersson (WP) symplectictwo-form and analyzing the isoperimetric orbits. Combining these results, the WP area in the parameter space and the canonical action–angle variables for the orbits are found. Using the ideas from the loop quantum gravity, we apply our formalism to the description of the classical geometrodynamics of Riemann surfaces and the WP area quantization. The results of the paper may be interesting due to their applications to the quantum geometry, chaotic systems, and low-dimensional gravity.

The hyperbolic [formula omitted] Sutherland and the rational [formula omitted] Ruijsenaars–Schneider–van Diejen models: Lax matrices and duality

In this paper, we construct canonical action–angle variables for both the hyperbolic B C n Sutherland and the rational B C n Ruijsenaars–Schneider–van Diejen models with three independent coupling constants. As a byproduct of our symplectic reduction approach, we establish the action–angle duality between these many-particle systems. The presented dual reduction picture builds upon the construction of a Lax matrix for the B C n -type rational Ruijsenaars–Schneider–van Diejen model.

Self-oscillations of two bodies in the case of nonlinear friction

The motion of two bodies is studied. One of them is attached to a fixed base by a spring, whereas the other one is attached to a body moving at a constant velocity. The interaction of the bodies is described by a nonlinear friction law with a falling section of the characteristic. The averaged equations of motion are obtained in terms of the canonical action-angle variables. The stationary points corresponding to self-oscillation modes are found and their stability is studied.

Pendulum, elliptic functions, and relative cohomology classes

Revisiting canonical integration of the classical pendulum around its unstable equilibrium, normal hyperbolic canonical coordinates are constructed and an identity between elliptic functions is found whose proof can be based on symplectic geometry and global relative cohomology. Alternatively it can be reduced to a well known identity between elliptic functions. Normal canonical action-angle variables are also constructed around the stable equilibrium and a corresponding identity is exhibited.

Statistical Description of Ray Chaos in an Underwater Acoustic Waveguide

An approximate analytical approach is developed to describe the chaotic behavior of ray trajectories in a deep-water acoustic waveguide up to three to five thousands of kilometers in length. The ray dynamics is investigated using the Hamiltonian formalism expressed in terms of the canonical action-angle variables. A realistic waveguide model is used, with refractive-index fluctuations due to the random field of internal waves. The Fokker-Planck equation is obtained for the action variable, and it is shown that the range dependence of this variable can be approximated by the Wiener random process, which represents the simplest model of diffusion. Formulas are derived for calculating the probability density of the coordinate and other ray characteristics. An approximate expression is found for the smoothed field intensity of a point source. For illustrating and testing the formulas obtained, their predictions are compared with the results of numerical solutions of ray equations and the results of field calculations by the parabolic equation method.

The motion of a body with elastic elements around a fixed point in action-angle variables

The motion of a mechanical system formed by a rigid rotating around a fixed point and carrying elastic rods, which undergo flexural deformation, is investigated. The asymptotic method of constructing approximate equations describing the evolution of the system motion in canonical action-angle variables is employed. The method of separating the motions and using the averaging operator enables the qualitative features of the behaviour of the system to be investigated, since, as a rule, the equations of motion of such systems cannot be integrated in explicit form.

Coherent states for systems with discrete and continuous spectrum

For a system with one degree of freedom, coherent states that are parametrized by classical canonical action-angle variables are introduced. These states also possess continuity of labelling, a resolution of unity, and temporal stability. The insistence on canonical action-angle variables strongly restricts any remaining arbitrariness in the coherent state definition. Such states are introduced for semibounded Hamiltonian operators having either a discrete or a continuous spectrum. Hamiltonians that have both discrete and continuous parts in their spectrum are also discussed.

Perturbation theory for the Benjamin-Ono equation

We develop a perturbation theory for the Benjamin-Ono (BO) equation. This perturbation theory is based on the inverse scattering transform for the BO equation, which was originally developed by Fokas and Ablowitz and recently refined by Kaup and Matsuno. We find the expressions for the variations of the scattering data with respect to the potential, as well as the dual expression for the variation of the potential in terms of the variations of the scattering data. This allows us to introduce the squared eigenfunctions for the BO equation, whose completeness and orthogonality in both x- and -spaces we also establish. We consider the two most important applications of the developed machinery. First, we present an explicit first-order solution of the BO equation driven by a small perturbation. Second, we introduce the Poisson bracket and a set of the canonical action-angle variables for the BO equation, and thus demonstrate its complete integrability as a Hamiltonian dynamical system.

Asymptotic expansion of ship capsizing in random sea waves—II. Second-order approximation

This is a continuation of Part I [ Int. J. Non-Linear Mech. 30, 727–740 (1995)] on the first-passage problem of ship roll oscillations in a random sea. A new coordinate system based on the canonical action-angle variables is used to express ship roll motion in terms of a set of Ito stochastic differential equations. The Pontryagin equation which describes the mean exit time of the system is solved using the method of asymptotic expansion. In the present analysis, a second-order approximation is carried out. The solution includes the contribution of the boundary layer, which compensates for the residual in the boundary condition at the barrier. It is found that the second-order approximation solution yields better results and takes into account such effects as the mean value of the excitation and other additional non-linearities which were not accounted for in the first-order approximation.

Self-consistent chaos in the beam-plasma instability

The effect of self-consistency on Hamiltonian systems with a large number of degrees of freedom is investigated for the beam-plasma instability using the single-wave model of O'Neil, Winfrey, and Malmberg. The single-wave model is reviewed and then rederived within the Hamiltonian context, which leads naturally to canonical action-angle variables. Simulations are performed with a large (10 4) number of beam particles interacting with the single wave. It is observed that the system relaxes into a time asymptotic periodic state where only a few collective degrees are active; namely, a clump of trapped particles oscillating in a modulated wave, within a uniform chaotic sea with oscillating phase space boundaries. Thus self-consistency is seen to effectively reduce the number of degrees of freedom. A simple low degree-of-freedom model is derived that treats the clump as a single macroparticle, interacting with the wave and chaotic sea. The uniform chaotic sea is modeled by a fluid waterbag, where the waterbag boundaries correspond approximately to invariant tori. This low degree-of-freedom model is seen to compare well with the simulation.

Hamiltonian theory of the ion cyclotron minority heating dynamics in tokamak plasmas

The question of heating a tokamak plasma by means of electromagnetic waves in the ion cyclotron range of frequencies (ICRF) is considered in the perspective of large rf powers and in the low collisionality regime. In such a case, the quasilinear theory (QLT) is validated by the Hamiltonian dynamics of the wave–particle interaction which exceeds the threshold of the intrinsic stochasticity. The Hamiltonian dynamics is represented by the evolution of a set of three canonical action angle variables well adapted to the tokamak magnetic configuration. This approach allows derivation of the rf diffusion coefficient with very few assumptions. The distribution function of the resonant ions is written as a Fokker–Planck equation but the emphasis is put on the QL diffusion instead of on the usual diffusion induced by collisions. The Fokker–Planck equation is then given a variational form from which a solution is derived in the form of a semianalytical trial function of three parameters: the percentage of resonant particles contained in the tail, an isotropic width ΔT, and an anisotropic width ΔP. This solution is successfully tested against real experimental observations. It is shown that in the case of the JET tokamak [Plasma Phys. Controlled Fusion 30, 1467 (1988)] the distribution function is influenced by adiabatic barriers which in turn limit the Hamiltonian stochasticity domain within energy values typically in the MeV range. Consequently and for a given ICRF power, the tail energy excursion is lower and its concentration higher than that from a bounce-averaged prediction. This may actually be an advantage for machines like JET [Plasma Phys. Controlled Fusion 30, 1467 (1988)] considering the energy range required to simulate the α-particle behavior in a relevant fusion reactor.