Sph\303\244risches PendelErstellt: 28.4.2020
Ersteller: W. Kemmetm\303\274ller
\303\204nderungen: restart;
with(LinearAlgebra):
with(VectorCalculus,Jacobian):
with(VectorCalculus,SpaceCurve):
with(plots):MFNWtKUb<ob<R=MDLCdNNZ^K;PV>D:;braCT;_ZRLCTJcDXoHuuX;cDOqtAIVRME^[TN=R;eRfqDu?p<@utuoWqUM\\keUkWioQarhqwh<rhEpj@pOdJ=\\JN`LJ<yp`MWxKcDl`=YYhr\\ewSamSlVpIMrisyarxLKKHlKHu:>jrI_l^:<KEC>::::BZ[G;<RR;:::B:CVC^J::::>^fF:;CE:::Fjnyh[ay?@xjAh@I:=U<CxTYheudmicU[c=\\j_<Lq@lJ@lEtvIYQt<Qj\\\\Jnl:_uhhewP]R>]bavbAsJCB>qWPasOdvwxjwxr?yjI^]HZ:@p>NfD@kB@amOyn`tpOfE_??h>KVmEiuuxMwtawE[EsP?U:CGGEe[`vBDKCDYqqwWxlQxlb]vFImMAtJATQuTqdwYpj=IoMMppXw]tTquXqUN]TNPq[>h^LYdZ?fZQmoVgHv]x^isQx_AtJAtbyadFd\\`beqtgxjFF]\\?fZ?l`wsAIt;IxQYmHfaugx\\g`;>^fFlh_bFOuSIusWiuWkpPkDPcQp_WO]n>gZPodPynIwMi_UopvFilafpawSYkPf_TGp\\gbIp]wNtZIyvvixayoYoQGtu_mBVCCVChN\\LN\\xUypYeuthhlhMMUTMYTQp]onEwLQp?lNF=mZUV]UNP<ytixeusXhQ=MR<]rlIowao\\uMKYvvUrSytQyPQmoV]olNd>vZrAvI^kWVaPolNPnN`ax_yr_x^`lDfsOWoPgtbHd=Oqsywyxy`HrC`]HvnVNfAPqfAvRCBSS^geEydissSWHAmXmWYqqw_OSOOc]iG_?grkWVAIcgGdYscUt`gfdYhtAykYvwsDQ?uwaS_YJ[Ekn=P[Mr>@wqTUn@WKtuLhxeytQMTLMlTyJH\\MjixoiSwxVOtuCPNQtsY=LKesTUJ:=U\\dwvXYqqwfIumHtuMSN@OKtu<QUTMUOPOOMlTIn[HnQqoWUSwxN`AMganK@w<MR<mOX=q:Qk=<RRM^k_kO?r_IsMx^;hvEn\\f>_ewnXqgEwiyvqeythxruamSQ\\E>^?@hn^_LfvsYxaI`tothhee@nZ@\\YQxiouvHlohnM`cLWrsXhaQr<Qr\\YycI]aXt[Iv[QpGNR;DP?SWEh=sWFAekCCYiRKExuqXEcEWcelgVEqfLYhl=caUEf_vmyvIyHWgeewyv[HQ<LPNqn`pnPfnAkLpcDXrnQxwwo^IyYawO`hn?hZ^_;^^YVyfFlbnj@pjkNi<Pj<PlmAjC^gxfc^XgQwo@^Z>^yp^_SwHKGE]fKSHmShRcIimuvGckSF;MFg]urGXdatcoYxmYtQytYGQmgFgyBycYsIXsHh]ujIv=YWOQgO?r]aFvgCPMv[qwkAwKYf^gII=BZOh>yec_TcOT[OR?OfqaesSX`yhqQylivEqwWYiwcIwcVwSUqoWWUipewOWGQ]RPqIumxfgVEqdkWg=qEwKYfyuywYuqxgICvKxqIIt]eLcHAucMWBQ]w<@p>MUCHPcDjMXlMyWiQPd]tbpxgYuiMunHStHy\\irETQuTq=tJH<MR<M>pj@pYuqxGytQyx;YjADPlitSIuSMkaaZlXkEXkQxlQOv<QlgFPKrLCW_wVIqeoWWQqdWaIl]vBiiuuxL;UxausWXewTioTOeTgeutghNYHumxfIr=wBaisUwTnwGX]iRCESCGTcrDgDEoYiWxaysEOgkSUNaH:ECv;i=IsMGcvMuIoEwKXquXaex;;^T`qQ_\\Ov[YnafVmrxb>OeUp`goeypiWvv<Qnv_l`YrU_qrGhKQn?gb=_b]Ir=xZ_w\\gQuqxgY`dxnvHimuNyWfnsOjD^\\b>f\\OfL@dw@ymOl>Ph_YsqY]j>fwqnGGioax^pm;nppVp`qscotvHqPpppys\\QlgJCQr;lQrYJtPqtxTipUgitexlBhqfETUhTelTf@uyRUqhgUCQkGVgIumXqsHqWeqsWhgUuoTNgd\\EbKYfAUcYOSqiTJ=TrGX]qbNmTxeytiXyqywQfKifEudRixbyWEQSkYYYeiveVWafCuwxgYd[Tf?y]igf_y]isf=disUUqXgarQ=VZ=vXiuumtfomKIr[xRipmfpaBVC]FQmgVMfd]It]xbawSYhisUxXsQxOmMn>AUFatNaXPaoSmTPmTKYQ?=nDek]tRH@yuYyEPToXVAqkKyLraysYpE]t_\\RFisGYLrmKIumvDuvhWeeobUu>Xv^HocHuwxVIpxWTQRyOUmMJ<LJTyNEd:kDPm?EYwQyo@O>dmWhmwdQwdqwhqo\\Pn\\xoixEe\\D_f^GylivM>quqpgWu^`bn>`j>kXy_UIlS@h_G_dOs>yaOghogx[W^@XpDE>gFLybkGhkucbsYturEudhcYIaIeUwRcyLqCEQb?SKI`sjQJBAQr<rADWgYvAykLaSmHpaaMJpXRmPb=t:HT<<RRMNexj_`tBpYStofETmepYiquulj`J:=U<DuDmlfDo\\PnLhOdAmRAwwYJ@\\KJpXRMLP@JBTS^eXdPodhwTipEqxmAPY`WKutIDwYyqyqmwTYlhRgyJGHTkMJ:Mq]AJY\\n=hPemtBPNlG_Mo_X_qbhoghiwQyoqewPiOv_X_ycYtaH[YvZE>Z>h^lP[Gf[GFdPoyqyuyoohpoTPmTh\\d^djAxjAmpfgT@yK``iy;[b_KhhCTJMcdMTtCI>uGD_tyGYmqVkiUYuUyGsY_e=EKBTSBInHemtdwVIqMAsBAwgYuqPpHETPMSm=JGqmwTUGeSGHT<<lKAhY^ncGecfcCV]wOvX@pj@pxiyuyZ?fZGP]oN_CVxUjDQLleR;EbCHkCGqlgvP`_SO^oaZ>^ZnpgWQuCIsc?tZHbkx^Rh:?h>mTnoXgQuHieusSWOIo_TcOTiuuxguppp<dJD\\UsPX_pVGQmFl[>h^jiegQuoxwYyqyOx^Icqv`qPwOYoqywYya[YwPC:=U\\SGLqtPqWUqhgSv:;FMCv:sM:<\\;>:S;:::Z:FfC>:==::::KrL;B:K:::::;DP?JZ:::::>Z<Q\\:>:::::B><J:<;::::<B;BZ:::::<:;j^HTHAmIUs]UwxATiuUKAsKenvpWgQofxN:Nrln^qh\\UOZVwmCpjkAly?[ECJoiRaCE<J<K<;:::JJJJ::::J:Z[;:::ZBJ:\\:::::Z:?E<>::::<J[E;::::JBJB:::::>J>J:::JJhtbTsAYsrqxgYmixoi@yXXOQmOoeXoAMydKHlKpeXoowhqef@ekPd_Pckpt;RPECEAsBAsMufHeWWQq_XSQ`uVau^eytPqthtexTiatcXTApKW\\qwhqAikUtp]urH@nZ@Ncyn=lJF<pqxpaer=<J^yQftmwTYPIqmuVauVaEtP]TN]WPaoSelddTNeTOumvDYlxWYqqMDTfawfQJ<LJ>QJwTYpaLEaOWMqN]pQ]oREtaxSYTTMUdu?Z:`aAhnB_rTipeWkdHdnhlN`lHomvFujHf=GuIwnFGm\\nidIv[IvhqwhngDPpL_lRXaqowNPrL?xlQx\\gr=>ZLv_mi]ZY\\SF]SinEw\\cF_?pdgPe?wcDGnJGn<Wf@akKhunhpohxnaxfouvHqB?dkWtTQuTauVamOv[:>laQsLir?iv`isUPpUH`LPd\\Gf[G^eWyrAyfXeqpsHeu_XcQTR=UROcDOCeeM:mseTScuQ_`Sx`YsQViTVZ=VJdy`@kCEo;MT<<Jl`RRmqvEylLyGAQmySOeToLyGuQytQPa`:o\\DVs<gjDfbyH]V^]VViS@]W`]uQ;KhP=UpsWXaU;ER;mXmIv=ybkwds=xZIRnYeoUwPiVGUB:KhJ=UTKETKuTQuDqWX[Ir=HlKHlkrlsBH[EBoUPmhcUt@gHG<l?PrL=mytYx@oidnpimutXtqxgYw]Yv]duFDKCDUp`WSQx_Ys_Xm`lN`tjItMP]T^MKV=J:Mt_MsCEKNatNQx>ETLMTtAykYvTipeuPgMu^]Wf]OtyUyNeThpAWb=WnQwoXWp`gsLqtPqfoV:_GQkdqywYyurGX]AcKSF`]sRGvbAwrEx\\xoXeqtaxcYTR=UREt\\hRE\\Lb<TepTgTSEUSWeUWmTj>bPf[EixeytAob@o\\f>eJgb?gnC_\\P_>;SwCeluVXYvAyc?yxGUepcYuqxGgTB_fkyxqyVJ<laQsltVHam\\YyGYmqTMqlWvLV>AKKMVdTOeTOcDOcPodPo?ETQuTQtuSUqBxSR=URyUgPUODOL]XFAUuPR@qrF<[l?]PhldfdTIQ\\PHuRCxWkdWgewLhNEmTcPT_XTapSoMwNIV_UR`Us`isitUv`LYuvHqmWqquuLMqT:<SRmnVaKUYnAukfIumxypEPyuYyuVauvOT_jV`mVoXgqtYwQyoSwuJ^qwhqQp_X__oXkPV_ToqvGi@aZ:`]@hfXaqsgb_GyAwfIourHh]Pf?QkqwwXifOXvyhZE^ywqifApVvakOnLPlLVf<>bHVrlwthheMv^H_kB@krYnugisQx_xfIqmVV\\NhrwNinAgqqsWOdUQqCWldi`d?``>ZxW_miwm`]l^fBQl_Vc>WbN^Z]qrGXgYqqwoqvGy\\xsPVmgNwji[YweYyvIymYGqlgfoQwOqeXoeLwqb`_Iimuv\\ai^jFl?_sRPaOOyRhndpntNsh_[=NZ:KsxYyqYcDO?hjEtLHlMv<QxhNmqvGYUv@YkqNGMmF=mjDnAukXLt;DRSQnHHK\\\\Om`o]yrHhUypXWMWqIwUyp:Ej<\\kCFkKHspVrXxayoePqoQwoXOfhQn`pnpNinAgSisIx]yNPagc[F^[VhagSQTUuUYuiiuyxaK:LxSllsMx^IlohUV]ULDYixTipUCQuNTLSmtThTehuWiu_pxBIP]hPodPST\\^mWMqnElGqPjtVHaMSqsBaJrAyraSR@P;MWeT_hqtgX]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>Z:>B::\\:^JbjB:==J:<J<[<===[XPwyAMu<uWfCWKCtC;B:;B:MTKWDKWgJ;eZ3:\"\{\}-%+ANNOTATIONG6'-%)BOUNDS_XG6#$"#5!""-%)BOUNDS_YG6#$""!F*-%-BOUNDS_WIDTHG6#$"%IKF*-%.BOUNDS_HEIGHTG6#$"%]TF*-%)CHILDRENG6"L\303\266sung entsprechend handschriftlichem RechenwegFreiheitsgradeqq:=<theta, varphi>;
qq_p:=<theta_p, varphi_p>;
qq_pp:=<theta_pp, varphi_pp>;Substitutionslisten (Zeitabh\303\244ngigkeit)
Hinweis: Diese Substitutionslisten sind notwendig, da Maple nicht nach der Zeitableitung dx(t)/dt einer Gr\303\266\303\237e ableiten kann.set_q2qt:={seq(s=s(t),s in convert(qq,list))};
set_qt2q:={seq(s(t)=s,s in convert(qq,list))};
set_qpt2qp:={seq(diff(convert(qq,list)[i](t),t)=convert(qq_p,list)[i],i=1..Dimension(qq))};
set_qp2qpt:={seq(convert(qq_p,list)[i]=diff(convert(qq,list)[i](t),t),i=1..Dimension(qq))};
set_qppt2qpp:={seq(diff(convert(qq,list)[i](t),t$2)=convert(qq_pp,list)[i],i=1..Dimension(qq))};
set_qpp2qppt:={seq(convert(qq_pp,list)[i]=diff(convert(qq,list)[i](t),t$2),i=1..Dimension(qq))};Vektor pp vom Ursprung des Koordinatensystems zur Massepp:=<l*sin(theta)*cos(varphi),-l*cos(theta),l*sin(theta)*sin(varphi)>;Geschwindigkeitpp_p:=Jacobian(pp,convert(qq,list)).qq_p;Kinetische EnergieT:=simplify(1/2*m*pp_p^+.pp_p);Potentielle EnergieV:=m*g*(pp[2]-eval(pp[2],theta=0));Externe Kraftff_e:=<-fx_e,0,0>;Vektor der verallgemeinerte Kr\303\244fteftheta:=(ff_e^+.Jacobian(pp,[theta]))[1];
fvarphi:=(ff_e^+.Jacobian(pp,[varphi]))[1];Lagrange-FunktionL:=T-V;Differentialgleichung f\303\274r theta und varphi:diff(eval(diff(L,theta_p),set_qp2qpt union set_q2qt),t)-eval(diff(L,theta),set_q2qt union set_qp2qpt)=eval(ftheta,set_q2qt);
diff(eval(diff(L,varphi_p),set_qp2qpt union set_q2qt),t)-eval(diff(L,varphi),set_q2qt)=eval(fvarphi,set_q2qt);L\303\266sung mittels homogener TransformationenDefinition der elementaren DrehmatrixRR_z:=phi-><<cos(phi),sin(phi),0>|<-sin(phi),cos(phi),0>|<0,0,1>>;
RR_y:=phi-><<cos(phi),0,-sin(phi)>|<0,1,0>|<sin(phi),0,cos(phi)>>;
RR_x:=phi-><<1,0,0>|<0,cos(phi),sin(phi)>|<0,-sin(phi),cos(phi)>>;DrehmatrizenRR_0_1:=RR_y(-varphi);
RR_1_2:=RR_z(theta);
Verschiebungsvektoren der Koordinatensystemedd_0_1:=<0,0,0>;
dd_1_2:=<0,0,0>;Vektoren zur Punktmasse im Koordinatensystem 2pp_2_m:=<0,-l,0>;Homogene TransformationenHH:=(RR,dd)-> <<RR|dd>,<0|0|0|1>>;
PP:=pp-> <pp,1>;HH_0_1:=HH(RR_0_1,dd_0_1);
HH_1_2:=HH(RR_1_2,dd_1_2);Position der Masse im InertialsystemPP_2_m:=PP(pp_2_m);
PP_0_m:=HH_0_1.HH_1_2.PP_2_m;Jacobi Matrix und MassenmatrixJJv_0_m:=Jacobian(convert(PP_0_m[1..3],Vector),convert(qq,list));MM:=simplify(JJv_0_m^+.JJv_0_m*m);Kinetische EnergieT:=simplify(1/2*qq_p^+. MM.qq_p);Die restlichen Schritte sind analog zur obigen L\303\266sung und werden daher nicht wiederholt.