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Construction of symplectic full-turn maps by application of an arbitrary tracking code

Description: A map to describe propagation of particles through any section of a nonlinear lattice may be represented as a Taylor expansion about the origin in phase space. Although the technique to compute the Taylor coefficients has been improved recently, the expansion may fail to provide adequate accuracy in regions where nonlinear effects are substantial. A representation of the map in angle-action coordinates, with the angle dependence given by a Fourier series, and the action dependence by polynomials in I/sup 1/2/, may be more successful. Maps of this form are easily constructed by taking Fourier transforms of results from an arbitrary symplectic tracking code. Examples are given of one-turn and two turn maps for the SLC North Damping Ring in a strongly nonlinear region. Results for accuracy and speed of evaluation of the maps are quite encouraging. It seems feasible to make accurate maps for the SSC by this method. 9 refs., 1 tab.
Date: March 1, 1989
Creator: Warnock, R.L.
Partner: UNT Libraries Government Documents Department

Coherent Synchrotron Radiation and Space Charge for a 1-D Bunch on an Arbitrary Planar Orbit

Description: Realistic modeling of coherent synchrotron radiation (CSR) and the space charge force in single-pass systems and rings usually requires at least a two-dimensional (2-D) description of the charge/current density of the bunch. Since that leads to costly computations, one often resorts to a 1-D model of the bunch for first explorations. This paper provides several improvements to previous 1-D theories, eliminating unnecessary approximations and physical restrictions.
Date: January 8, 2008
Creator: Warnock, R. L.
Partner: UNT Libraries Government Documents Department

Joint probabilities of noncommuting observables and the Einstein-Podolsky-Rosen question in Wiener-Siegel quantum theory

Description: Ordinary quantum theory is a statistical theory without an underlying probability space. The Wiener-Siegel theory provides a probability space, defined in terms of the usual wave function and its ``stochastic coordinates``; i.e., projections of its components onto differentials of complex Wiener processes. The usual probabilities of quantum theory emerge as measures of subspaces defined by inequalities on stochastic coordinates. Since each point {alpha} of the probability space is assigned values (or arbitrarily small intervals) of all observables, the theory gives a pseudo-classical or ``hidden-variable`` view in which normally forbidden concepts are allowed. Joint probabilities for values of noncommuting variables are well-defined. This paper gives a brief description of the theory, including a new generalization to incorporate spin, and reports the first concrete calculation of a joint probability for noncommuting components of spin of a single particle. Bohm`s form of the Einstein-Podolsky-Rosen Gedankenexperiment is discussed along the lines of Carlen`s paper at this Congress. It would seem that the ``EPR Paradox`` is avoided, since to each {alpha} the theory assigns opposite values for spin components of two particles in a singlet state, along any axis. In accordance with Bell`s ideas, the price to pay for this attempt at greater theoretical detail is a disagreement with usual quantum predictions. The disagreement is computed and found to be large.
Date: February 1, 1996
Creator: Warnock, R.L.
Partner: UNT Libraries Government Documents Department

Shielded coherent synchrotron radiation and its possible effect in the next linear collider

Description: Shielded coherent synchrotron radiation is discussed in two cases: (1) a beam following a curved path in a plane midway between two parallel, perfectly conducting plates, and (2) a beam circulating in a toroidal chamber with resistive walls. Wake fields and the radiated energy are computed with parameters for the high-energy bunch compressor of the Next Linear Collider. 5 refs., 4 figs., 1 tab.
Date: May 1, 1991
Creator: Warnock, R.L.
Partner: UNT Libraries Government Documents Department

Study of invariant surfaces and their break-up by the Hamilton-Jacobi method

Description: A method is described to compute invariant tori in phase space for calssical non-integrable Hamiltonian systems. Our procedure is to solve the Hamilton-Jacobi equation stated as a system of equations for Fourier coefficients of the generating function. The system is truncated to a finite number of Fourier modes and solved numerically by Newton's method. The resulting canonical transformation serves to reduce greatly the non-integrable part of the Hamiltonian. In examples studied to date the convergence properties of the method are excellent, even near chaotic regions and on the separatrices of isolated broad resonances. We propose a criterion for breakup of invariant surfaces, namely the vanishing of the Jacobian of the canonical transformation to new angle variables. By comparison with results from tracking, we find in an example with two nearly overlapping resonances that this criterion can be implemented with sufficient accuracy to determine critical parameters for the breakup ('transition to chaos') to an accuracy of 5 to 10%.
Date: August 1, 1986
Creator: Warnock, R.L. & Ruth, R.D.
Partner: UNT Libraries Government Documents Department

Bounds on nonlinear motion for a finite time

Description: Recent improvements in numerical methods to compute canonical transformations make it feasible to set interesting bounds on the motion of nonlinear Hamiltonian systems over a finite interval of time. 7 refs.
Date: June 1, 1989
Creator: Warnock, R.L. & Ruth, R.D.
Partner: UNT Libraries Government Documents Department

Invariant surfaces and tracking by the Hamilton-Jacobi method

Description: The Hamilton-Jacobi method is described for a model of betatron motion in one degree of freedom, namely, a harmonic oscillator perturbed by a lattice of sextupoles. The Hamilton-Jacobi equation is given in terms of Fourier amplitudes. Invariant surfaces have been obtained in phase space, and finite time symplectic maps were obtained for tracking of single particles. (LEW)
Date: September 1, 1986
Creator: Warnock, R.L. & Ruth, R.D.
Partner: UNT Libraries Government Documents Department

Symplectic maps for accelerator lattices

Description: We describe a method for numerical construction of a symplectic map for particle propagation in a general accelerator lattice. The generating function of the map is obtained by integrating the Hamilton-Jacobi equation as an initial-value problem on a finite time interval. Given the generating function, the map is put in explicit form by means of a Fourier inversion technique. We give an example which suggests that the method has promise. 9 refs., 9 figs.
Date: May 1, 1988
Creator: Warnock, R.L.; Ruth, R. & Gabella, W.
Partner: UNT Libraries Government Documents Department

A code to compute the action-angle transformation for a particle in an abritrary potential well

Description: For a Vlasov treatment of longitudinal stability under an arbitrary wake field, with the solution of the Haiessinski equation as the unperturbed distribution, it is important to have the action-angle transformation for the distorted potential well in a convenient form. The authors have written a code that gives the transformation q,p {yields} J, {phi}, with q(J,{phi}) as a Fourier series in {phi}, the Fourier coefficients and the Hamiltonian H(J) being spline functions of J in C{sup 2} (having continuous second derivatives).
Date: June 1, 1995
Creator: Berg, J.S. & Warnock, R.L.
Partner: UNT Libraries Government Documents Department

Coherent synchrotron radiation and stability of a short bunch in a compact storage ring

Description: It should be possible to observe coherent synchrotron radiation at millimeter wavelengths in a compact electron storage ring, provided that the bunch can be made sufficiently short. On the other hand, for a short bunch the radiation reaction is so strong that it could cause a longitudinal instability if the current exceeded some threshold. This might cause bunch lengthening, and cut off or reduce the coherent radiation. Using wake fields from simple models of the vacuum chamber, the authors estimate the threshold current for a proposed upgrade of the Brookhaven small x-ray light source, SXLS-Phase 1.
Date: June 1, 1995
Creator: Warnock, R.L. & Bane, K.
Partner: UNT Libraries Government Documents Department

Numerical construction of the Poincare map, with application to accelerators

Description: We show how to construct a symplectic approximation to the Poincar{prime}e map, using data from a symplectic integrator. We illustrate by producing a full-turn map for a realistic model of the Large Hadron Collider. Mapping of one turn is typically faster by a factor of 60 than direct integration. This allows one to follow orbits over times comparable to the required storage time of the beam, on a workstation computer. Fast mapping also allows the construction of quasi-invariant actions, which aid in estimates of long-term stability.
Date: October 1, 1995
Creator: Warnock, R.L. & Berg, J.S.
Partner: UNT Libraries Government Documents Department

Fast symplectic mapping, quasi-invariants, and long-term stability in the LHC

Description: A systematic program to explore stability of orbits in hadron storage rings is based on the following steps: (a) beginning with a symplectic tracking code, construct the mixed-variable generator of the full-turn map in a Fourier-spline basis; (b) use the resulting fast mapping to follow long orbits and estimate the long-term dynamic aperture; (c) contruct quasi-invariants and examine their variation in time to set long-term bounds on the motion for any initial condition in a specified region. First results from an application of the program to the Large Hadron Collider (LHC) are reported. Maps can be constructed in a few hours and evaluated at a speed 60 times greater than that of one-turn tracking, on a workstation computer. Orbits of 10{sup 7} turns take 3.6 hours. The value of a ``stroboscopic`` view of the synchro-betatron motion is emphasized. On a Poincare section at multiples of the synchrotron period, one can study resonances and invariant surfaces in two dimensions, thereby taking advantage of techniques that have proved effective in treating pure betatron motion.
Date: January 1, 1996
Creator: Warnock, R.L. & Berg, J.S.
Partner: UNT Libraries Government Documents Department

Invariant tori of the Poincare return map as solutions of functional difference equations

Description: Functional difference equations characterize the invariant surfaces of the Poincare return map of a general Hamiltonian system. Two different functional equations are derived. The first is analogous to the Hamilton-Jacobi equation and the second is a generalization of Moser's equation. Some properties of the equations, and schemes for solving them numerically, are discussed. 7 refs., 1 fig.
Date: January 1, 1991
Creator: Warnock, R.L.
Partner: UNT Libraries Government Documents Department

Shielded coherent synchrotron radiation and its effect on very short bunches

Description: Shielded coherent synchrotron radiation is discussed for two cases: a beam following a circular path midway between two parallel conducting plates, and a beam circulating in a toroidal chamber. Wake fields and the energy radiated are computed for both cases. Under conditions like those of the high-energy bunch compressor of the Next Linear Collider (NLC), in which bunches as short as 40 microns are contemplated, the shielded coherent radiated power is estimated to be small compared to the incoherent power, but can still amount to a few hundred keV over the compressor arc. 7 refs., 6 figs., 1 tab.
Date: November 1, 1990
Creator: Warnock, R.L.
Partner: UNT Libraries Government Documents Department

Superconvergent tracking and invariant surfaces in phase space

Description: The question of long term beam stability in very large storage rings presents an extraordinary challenge in nonlinear dynamics. Since current computational methods seem less than adequate on the long time scales involved, we have undertaken a program of evaluating several methods that either are new or have not been tried in accelerator problems heretofore. The methods we investigate fall into two categories: (1) iteration of maps describing concatenated machine elements, for tracking of single particles, and (2) infinite-time methods for direct computation of invariant surfaces in phase space.
Date: April 1, 1985
Creator: Ruth, R.D.; Raubenheimer, T. & Warnock, R.L.
Partner: UNT Libraries Government Documents Department

Construction and Fourier analysis of invariant surfaces from tracking data

Description: We study invariant surfaces in phase space by application of a symplectic tracking code. For motion in two degrees of freedom we use the code to compute I(s), /Phi/(s) for s = 0,C,2C...nC, where I = (I/sub 1/,I/sub 2/), /Phi/ = (/phi//sub 1/,/phi//sub 2/) are action-angle coordinates of points on a single orbit, and C is the circumference of the reference orbit. As a test to see whether the orbit lies on an invariant surface (i.e., to test for regular and nonresonant motion) we fit the points to a smooth, piece-wise polynomial surface I = /cflx I/(/phi//sub 1/,/phi//sub 2/). We then compute additional points on the same orbit, and test for their closeness to /cflx I/. We find that data from a few thousand turns are sufficient to construct accurate approximations to an invariant surface, even in cases with strong nonlinearities. Two-dimensional Fourier analysis of the surface leads to information on the strength of nonlinear resonances, and provides the generator of a canonical transformation as a Fourier series in angle variables. The generator can be used in a program to derive rigorous bounds on the motion for a finite time T. 6 refs., 2 figs., 1 tab.
Date: March 1, 1989
Creator: Warnock, R.L.; Ruth, R.D. & Ecklund, K.
Partner: UNT Libraries Government Documents Department

Beam dynamics with the Hamilton-Jacobi equation

Description: We describe a non-perturbative method to solve the Hamilton-Jacobi equation for invariant surfaces in phase space. The problem is formulated in action-angle variables with a general nonlinear perturbation. The solution of the Hamilton-Jacobi equation is regarded as the fixed point of a map on the Fourier coefficients of the generating function. Periodicity of the generator in the independent variable is enforced with a shooting method. We present two methods for finding the fixed point and hence the invariant surface. A solution by plain iteration is economical but has a restricted domain of convergence. The Newton iteration is costly but yields solutions up to the dynamic aperture. Examples of lattices with sextupoles for chromatic correction are discussed. 10 refs., 5 figs., 1 tab.
Date: March 1, 1989
Creator: Gabella, W.E.; Ruth, R.D. & Warnock, R.L.
Partner: UNT Libraries Government Documents Department

Periodic solutions of the Hamilton-Jacobi equation by the shooting method: A technique for beam dynamics

Description: Periodic solutions of the Hamilton-Jacobi equation determine invariant tori in phase space. The Fourier spectrum of a torus with respect to angular coordinates gives useful information about nonlinear resonances and their potential for causing instabilities. We describe a method to solve the Hamilton-Jacobi equation for an arbitrary accelerator lattice. The method works with Fourier modes of the generating functions, and imposes periodicity in the machine azimuth by a shooting method. We give examples leading to three-dimensional plots in a surface of section. It is expected that the technique will be useful in lattice optimization. 14 refs., 6 figs., 1 tab.
Date: May 1, 1988
Creator: Gabella, W.E.; Ruth, R.D. & Warnock, R.L.
Partner: UNT Libraries Government Documents Department

Methods of stability analysis in nonlinear mechanics

Description: We review our recent work on methods to study stability in nonlinear mechanics, especially for the problems of particle accelerators, and compare our ideals to those of other authors. We emphasize methods that (1) show promise as practical design tools, (2) are effective when the nonlinearity is large, and (3) have a strong theoretical basis. 24 refs., 2 figs., 2 tabs.
Date: January 1, 1989
Creator: Warnock, R.L.; Ruth, R.D.; Gabella, W. & Ecklund, K.
Partner: UNT Libraries Government Documents Department

Progress on a Vlasov Treatment of Coherent Synchrotron Radiation From Arbitrary Planar Orbits

Description: We report on our progress in the development of a fully self-consistent Vlasov treatment of coherent synchrotron radiation (CSR) effects on particle bunches traveling on arbitrary planar orbits. First we outline our Vlasov approach and the approximation we are currently studying. Then we discuss recent numerical results for a benchmark model studied extensively with codes by several authors.
Date: February 10, 2006
Creator: Bassi, G.; Ellison, J.A.; U., /New Mexico; Warnock, R.L. & /SLAC
Partner: UNT Libraries Government Documents Department

Fast symplectic mapping and long-term stability near broad resonances

Description: Fast symplectic mapping, based on a canonical generator of the full-turn map in polar coordinates (I, {Phi}), is a powerful tool to study long-term stability in large hadron storage rings. Accurate maps for realistic lattices can be constructed in a few hours on a workstation computer, and can be iterated to follow orbits for 10{sup 7} turns in less than 4 hours. Orbits of the map can also be used in a non-perturbative construction of quasi-invariant actions. By bounding the small changes in quasi-invariants along many short orbits, one can derive conservative estimates of survival time for long orbits, for any initial condition in a region of phase space. A first quasi-invariant vector, J, arises from a canonical transformation (1, {Phi}) {r_arrow} (J, {Psi}), based on interpolation of invariant tori surrounding the origin. The variation of J is relatively large near a broad resonance. In such a region a second canonical transformation, associated with pendulum-like motion in appropriate variables, is required to produce a good quasi-invariant. This quasi-invariant is used to set a long-term bound on motion near a broad, large-amplitude resonance in a realistic model of the Large Hadron Collider (LHC). Interesting general properties of the pseudo-pendulum motion are studied.
Date: April 1, 1997
Creator: Warnock, R.L. & Berg, J.S.
Partner: UNT Libraries Government Documents Department

Convergence of a Fourier-spline representation for the full-turn map generator

Description: Single-turn data from a symplectic tracking code can be used to construct a canonical generator for a full-turn symplectic map. This construction has been carried out numerically in canonical polar coordinates, the generator being obtained as a Fourier series in angle coordinates with coefficients that are spline functions of action coordinates. Here the authors provide a mathematical basis for the procedure, finding sufficient conditions for the existence of the generator and convergence of the Fourier-spline expansion. The analysis gives insight concerning analytic properties of the generator, showing that in general there are branch points as a function of angle and inverse square root singularities at the origin as a function of action.
Date: April 1, 1997
Creator: Warnock, R. L. & Ellison, J. A.
Partner: UNT Libraries Government Documents Department