Alexahin, Yu I

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Particle Accelerators, 1996, Vol. 55, pp. [257-262] /11-16Reprints available directly from the publisherPhotocopying permitted by license only© 1996 OPA (Overseas Publishers Association)Amsterdam RV. Published in The Netherlands underlicense by Gordon and Breach Science Publishers SAPrinted in MalaysiaMATHEMATICA APPLICATIONS TONONLINEAR DYNAMICS ANALYSISYURI ALEXAHINJoint Institute for Nuclear Research, Dubna, 141980 Russia(Received 30 January 1996; infinalform 30 January 1996)Various possibilities offered by the Mathematica system for single particle dynamics analysisare discussed: symbolic computation in Hamiltonian perturbation theory, running trackingprograms for calculating the dynamic aperture, post-processing of tracking results.Keywords: Nonlinear dynamics; Mathematica system.1 INTRODUCTIONMathematical is a powerful software system which can be used for• numerical and symbolic calculations;• data analysis and representation;• control of external programs and processes.It has a programming language which permits to implement practically anykind of algorithm.There are a number of instances of Mathematica application to problemsof particle dynamics in accelerators: generating databases of MAD results2 ;running MAD for LEP dynamic aperture calculation3; symbolic compu-tations in beam optics4 ; derivation of analytical formulae in Hamiltonianperturbation theory.5The objective of the present report is to give a brief description of theseapplications and to show on simple examples some of the Mathematicafeatures which seem to be particularly useful.[257]/11[258]/12 Y.ALEXAHIN2 PROCESSING OF A STANDARD PROGRAM OUTPUTIt is quite often that analysis of a large number of runs of a standard programunder varying conditions (e.g. different distributions of errors) is needed.Since the output from the standard program is not always convenientlyorganized for the specific purpose, extracting and processing the relevantinformation may pose a formidable problem.In this case Mathematica proves to be an indispensable tool enabling oneto:• read output of standard programs;• extract necessary data from the output;• accumulate the data in database;• analyze and represent the results.An important feature of Mathematica with this respect is an easyconversion between character strings and expressions. Let for exampletrack. n , n= 1, nr, be nr output files from some tracking program. Toread and analyze them in a cycle over n Mathematica permits to constructthe filenames in the following manner:fname=StringJoin [ II track. II, ToString [n] ].Then at a particular iteration, say n =13, the string f name value will beIItrack.13 11 •As a rule the standard program output consists of intermixed text andnumerical data. The simplest way to deal with such a file is to consider it asa record. The following command reads in a text file and converts it in a listof character strings, one per a line, named mess:mess=ReadList[ fname, Record];.Now each string of mess (i.e. each line of the output) can be testedon occurrence of a substring, converted into expressions, etc. Consider forexample the MAD tracking module output which may contain a number offragments of the type:Particle(s) lost during turn 4, superperiod 1, s =157.182100 by overflow in LS [3]~~cr x ~ y ~ t ~7 1.01550589E+00 1.16274136E-Ol 4.22375450E-Ol -6.02355138E-02-1.77061405E-016.53721162E-06MATHEMATICA APPLICATIONS [259]/13With the help of the auxiliary functioninQ[x_String] :=StringMatchQ[ x, "*Particle(s) lost*"];one can determine the numbers of lines containing the message on loss andput them in the list lines:lines=Flatten[ Position[ Map [ inQ, mess], True]];Now information on the lost particles can be extracted from the subsequentlines and stored; e.g. particle number following the i-th occurrence of themessage can be found asnpart=ToExpression[ First[ ReadList[StringToStream[ mess[[ lines[[i]]+2 ]] ], Word]];This chain of commands analyzes the relevant character string (the wholeline) into a stream of words and then converts the first one (also a string) intoa number.It is far beyond the scope of this report to cover all the possibilities ofMathematica in data handling. A more extensive example of employingMathematica for this purpose can be found in Ref. 2.3 RUNNING EXTERNAL PROGRAMSThe analytical capabilities of Mathematica permit to use it as a supervisorfor guiding external programs. If the code insides are inaccessible forMathLink5 then Mathematica can support communication with the externalprogram by exchanging ordinary text files (the so called unstructuredcommunication). An-·example of operation in this mode was given in Ref. 3where Mathematica was running MAD for dynamic aperture calculation.The basic steps in the unstructered communication are as follows:• preparation of a task for external program;• launching execution of the external program;• reading and analysis of the external program output;• formulation of a new task.In the particular instance Mathematica analyzed the MAD trackingmodule output as described in the previous section, found the maximal stableinitial amplitude and gave MAD a new set of particle start positions for[260]/14 Y.ALEXAHINtracking. Mathematica prepared input for MAD from prototype files byreplacing dummy parameters with actual parameters using transformationrules. To demonstrate this mechanism let proto be such a file which isstored as a list and contains among others the following two linesstart, fx=&fxOEach line is represented by one element (character string) of the list. Now letMathematica decide to start tracking with some numerical value nv of theinitial amplitude which it assigns to variable ax. Then substitutiontask=proto /. IIfxO Il ->axwill produce a file with the numerical value nv instead of fxO. Not onlynumerical but also logical and symbolic values may be substituted in a similarway.The prepared input file can be fed into MAD by command task» II !mad II4 SYMBOLIC COMPUTATIONS IN PERTURBATION THEORYSymbolic computations have already been extensively used in acceleratorphysics, e.g. for generating Lie algebraic maps (see Ref. 7). Mathematicahad introduced some new concepts (pure functions, upvalues etc.) whichproved to be useful in both linear optics4 and nonlinear dynamics analysis.5In Ref. 5 Deprit's algorithm for normalization of 2DoF nonlinear Hamilto-nian was implemented as Mathematica notebook. With the present versionof Mathematica for HP (which has an intrinsic limitation on memory in use)it was possible to obtain in symbolic form normalized Hamiltonian terms offourth order in sextupole strength.The basic idea of Deprit's algorithm is to consider transition from theunperturbed (linear) dynamics to the nonlinear one as a Lie transformationwith perturbation parameter as the independent variable. This leads to awell-structured set of differential equations (Deprit's equations) which linksexpansion coefficients of the original Hamiltonian, the normalized oneand the Lie transformation generator. This set can be easily integrated inaction-angle variables.An important concept in Mathematica employed at this step is the notionof upvalues which permits to define operations involving a function withoutMATHEMATICA APPLICATIONS [261]/15explicitly introducing it. In the particular case it helps to obtain the startingHamiltonian expansion in terms of basis functions of action-angle variablesJ [i], phi [ i], i =1 , 2u[e_List, m_List]:=Product[ J[i]A(e[[i]]/2), {i,2}]*Exp[ I Sum [ phi[i]*m[[i]], {i,2}]];without invocation of this explicit definition. It is sufficient to express coor-dinates through basis functions and then introduce the rule of multiplicationas the upvalue for uu[el_, ml_]*u[e2_, m2_] A:= u[el+e2, ml+m2];This allows to avoid explicit introduction of action-angle variables and makesunnecessary a large amount of mathematical operations.Generally the solution of Deprit's equations is found in terms of integraloperator which is inverse to directional (Lie) derivative. In the thin lensapproximation it is reduced to a sum over nonlinear elements. The followingfeature of Mathematica facilitates to implement this reduction. Let f [a,b , ..., z] be any listable mathematical operation defined on symbolicvariables a, b , ..., z. Substitution of the listsa={a1, a2, ... , aN}; b={b1, b2, ... , bN}; ... z={z1, z2, ... , zN};into f will produce a list{f [a1, b1, ... , z1] , f [a2, b2, ... , z2] , ... , f [aN, bN, ... , zN]}.In this way integration of f is essentially reduced to summation of the listelements.5 FUTURE DEVELOPMENTSThere are interesting possibilities ofMathematica application to some of theproblems discussed at this Workshop, e.g. compatibility of lattice descriptionin different standard programs.8 Mathematica can easily translate from oneinput language to another performing simple calculations in the process(e.g. finding bending angle from field strength and energy) and even consultexternal programs in more complicated cases.Another example is the Numerical Accelerator Project.9 Mathematicacan run in parallel with Sixtrack ("peer-to-peer" mode6) performing expressanalysis and deciding whether to launch another accompanying particle, aborttracking, etc.[262]/16ReferencesY.ALEXAHIN[1] S. Wolfram: "Mathematica: A System for Doing Mathematics by Computer", 2nd ed.(Addison-Wesley Publishing Co., 1993).[2] J.M. Jowett: "Generating Mathematica databases of MAD results, imperfect LEP2 as anexample", in CERN PS/AR Note 94-17, Geneva, 1994.[3] Y. Alexahin: "A Study of the Low Emittance Lattice for LEP2", CERN SL/94-46, Geneva,1994.[4] B. Autin, E. Wildner: "Beam Optics, a Program for Symbolic Beam Optics", CERNPS/DLlNote 95-18, Geneva, 1995.[5] Y. Alexahin: "Improving the Dynamic Aperture of LEP2", CERN SL-95-110, Geneva,1995.[6] MathLink Reference Guide. Version 2.2. Wolfram Research, Inc. Technical Report, 1993.[7] A. Dragt: "Lectures on Nonlinear Orbit Dynamics", 1981 Summer School on High EnergyParticle Accelerators, FNAL, AlP 1982.[8] R. Talman: "A Proposed Flat Accelerator Lattice Description", this Workshop.[9] E. McIntosh, T. Pettersson, F. Schmidt: "Proposal for a Numerical Accelerator Pilot Project",this Workshop.

Mathematica Applications To Nonlinear Dynamics Analysis

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