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1、HOWTOCOMPUTEPROPERLYNEWTON'S;EQUATIONOFMOTION?
x1ProblemsandMotivationsThemainthemeofmodernscienticcomputingisthenumericalsolutionofvariousdierentialequationsofmathematicalphysicsbearingthenamessuchasNewton,Euler,Lagrange,Laplace,Navier-Stokes,Maxwell,Boltzmann,Einstein,Schro
2、dinger,Yang-Mills,etc.AtthetopofthelististhemostcelebratedNewton'sequationofmotion.Thehistorical,theoreticalandpracticalimportanceofNewton'sequationhardlyneedsanycomment,soistheimportanceofthenumericalsolutionofsuchequations.Ontheotherhand,startingfromEuler,rightdowntothepresentcomputera
3、ge,agreatwealthofscienticliteratureonnumericalmethodsfordierentialequationshasbeenaccumulated,andagreatvarietyofalgorithms,softwarepackagesandevenexpertsystemshadbeendeveloped.Undersuchcircumstanceswestillfeelmotivatedtoraisethefollowingtwoquestions:Question1:Aretheexistingnumericalmet
4、hodsadequateforcomputingNewton'sequa-tionsofmotion?Question2:WhatistheproperwaytocomputeNewton'sequationsofmotion?Question1seemstohaveneverbeenseriouslyraisedbefore,soquestion2seemstohaveneverbeenstudiedsystematically.WestartwiththecaseofNewton'sequationsofmotioninconservativeforceeldin
5、ncongurationspaceR,whichisphysicallymorebasicandmathematicallymoredicult:Mq=;V(q)(1)qwhereq=(qq)isthepositionvariable,Mistheinertiamatrix,V(q)isthepotential1nenergy.ItiswellknownthattheconservativeNewton'sequationshastwoalternativemathematicallyequivalentformalisms:theLagrangianfo
6、rmalismasavariationalprincipleandtheHamiltonianformalisminwhichtheNewton'ssystemofdierentialequationsofsecondorderincongurationspaceisreducedtoHamilton'ssystemofcanonicalequationsofrstorderinphasespacewithdoubleddimension.Themathematicallyequivalentformalismsexpressthesamephysicallaws
7、,buttheylookverydierentinform,sotheyleadtodierenttechnicalapproachesinproblemsolving,;InYingLA,GuoBY,ed.Procof2ndConfonNumericalMethodsforPartialDierentialEquations.Singapore:WorldScientic,1992.15{22186theyarenotequallyeectiveinpractice.Soajudiciouschoiceatthestartam