双摆系统运动理论与实验研究

igenvalues,ie.theLyapunovexponents,isalsozero.Thephysicalimplicationofthisistheinvarianceofphasespacevolumeforaconservativesystem.Foradissipativesystem,thiswouldnotbetrue.ealgebraicallymessy.Inthiscase,wewillonlybeinterestedinobtainingnumericalsolutions.TheJacobianmatrixforthisproblemisa4×4array:j11jjj121314jjjjJ21222324j31jjj323334j41jjj424344Bydefiningthefollowingintermediatequantities,putedfairlyeasily.(Seepreviousparameterdefinitions.)ra12b12cos212saar1122taprp2212urpap1112tubsinq1221s2tubcostpupb22sin4rtub22sindq12211212211221ss23Theindividualmatrixelementsaregivenby:sp2aubsinj222122111s2aj2212ssp2rtbsinj2122113s2rj14sj21cdq1cos1j22j11j23dqj24j33j31j33j32j14UniversityofMichigan10DepartmentofPhysics