Paul J. Channell

Integrators for Lie-Poisson or dynamical systems

Abstract We reformulate the Ge-Marsden (1988) Lie-Poisson integration algorithm in terms of algebra variables alone and show how to implement it for regular quadratic Lie algebras to arbitrary order and for arbitrary Lie algebras to first order. We also describe two generalizations of Ruths method (1983) that can be used to generate very fast algorithms.

Systematic solution of the Vlasov–Poisson equations for charged particle beams

Rapidly oscillating solutions of the Vlasov–Poisson equations are required when external forces are rapidly oscillating, as occurs in focusing systems for charged particle beams. Such systems are treated using a Hamiltonian averaging technique carried out to third order. As a result one can deal with the problems of matching, stability, and transient time evolution using the well-established techniques previously used for simpler, time-independent systems.

Laser cooling of heavy-ion beams

We present a new technique for cooling heavy‐ion beams. The discrete absorption spectrum of the ions and the transverse Doppler shift distinguish the transverse direction of velocity and cool the ions by photon absorption. We evaluate the total energy and time requirements for a particular example relevant to heavy‐ion fusion.

Isodynamic magnetohydrodynamic equilibria

The class of isodynamic magnetohydrodynamic (MHD) equilibria in three dimensions is studied. A reduction of the problem to that of finding a force‐free equilibrium is carried out, and the resulting force‐free problem transformed, by means of a three‐dimensional hodograph technique, to a simple and elegant form. The flux surface equation in a potentially very useful form is split off in a further representation.

Explicit suspensions of diffeomorphisms—An inverse problem in classical dynamics

Presented in this paper is a set of explicit prescriptions for associating with a given map of Rn, which is C2‐isotopic to the identity, a time‐dependent vector field whose time‐1 map is the given one. Also shown is how to apply additional restrictions to the vector field including that it be (1) periodic in time, (2) Hamiltonian, and (3) of potential form; several examples show numerical verification of the theory.

The longitudinal stability of intense nonrelativistic particle bunches in resistive structures

LBL-12107 c. ~ u Preprlnt Lawrence Berkeley Laboratory UN IVERSITY OF CALIFORNIA r RECEIVt:.D I.AWf1EN(;; Accelerator & Fusion Research Division To be submitted for publication MAR 5 -oS< LIBRARY AND DOCL,lENTS SECT ION THE LONGITUDINAL STABILITY OF INTENSE NON-RELATIVISTIC PARTICLE BUNCHES IN RESISTIVE STRUCTURES Paul J. Channell, Andrew M. Sessler and Jonathan S. Wurtele February 1981 TWO-WEEK LOAN COpy This is a Library Circulating Copy which may be borrowed for two weeks. For a personal retention copy} call Tech. Info. Division} Ext. 6782. Prepared for the U.S. Department of Energy under Contract W-7405-ENG-48

Hamiltonian suspensions of symplectomorphisms: an alternative approach to design problems

Abstract We solve the problem that often arises in optics and charged particle transport design tasks of finding a time-dependent Hamiltonian system that produces a given symplectic map. In addition, we show how, given a particular Hamiltonian, to produce many Hamiltonians that have the same time-one map so that one has a large class of Hamiltonians from which to choose the ‘best’ design. We also show how to produce many symplectic maps that are given explicitly rather than implicitly so that the design problem is more tractable.

The Vlasov equation and Poisson maps induced by non-symplectic particle maps

Abstract It is shown that almost symplectic particle maps induce Poisson maps of the truncated moment coordinatizations of solutions of the Vlasov equation, allowing one to use various tools to study the realistic evolution of distribution functions.

Transverse electron-proton two-stream instability in a bunched beam

This paper is an analytical investigation of the transverse electron-proton (e-p) two-stream instability in a proton bunch propagating through a stationary electron background. The equations of motion, including the effect of damping, are derived for the centroids of the proton beam and the electron cloud. An approach is developed to solve the coupled linear centroid equations in the time domain describing the e-p instability in proton bunches with nonuniform line densities. Examples are presented for proton line densities corresponding to uniform and parabolic profiles.

Stable symplectic maps near arbitrary lattice maps

Given a symplectic map for an accelerator lattice or sequence of elements, we show how to generate a nonlinearly stable symplectic map that has the same linear terms and the same lowest‐order nonlinear terms as the given map. Techniques are presented for producing stable maps both for the case in which the map is given by a generating function and in the case in which the map is given by a Lie map.