拉格朗日方程复习与例题

1、), 2 , 1(0nnimiiii aff), 2 , 1(nii r), 2 , 1(0)(nnimiiiiii raff), 2 , 1(0nniiii rf), 2 , 1(0)(nimiiiii raf), 2 , 1(0)(nnimiiiiii raffnizzmfyymfxxmfiiiziiiiyiiiiixi, 2 , 10)()()( ), 2 , 1(0)(nimiiiii rafcmgaccmafircicjm0sinrxmxf-xmgicirsin32gacramrjcc,2122iisinmlffba0211iicbabbaaygmygmygmxfxfcos2coss

2、incossinlylylxlylxcbbaasin2sincossincoslylylxlylxcbbaa0211iicbabbaaygmygmygmxfxfsin2sincossincoslylylxlylxcbbaa0sin2sin2cossin22121glmglmllmcos)(1212lmgmm 11i1amf 12i2eamfr2i2ramf22i2jm22221rmj00，x00，x0cossin22r2ie2i2jrfrfrgm0)23cos(1sinr1aag00，x11i1amf 12i2eamfr2i2ramf22i2jm22221rmj2rra cos)(2121ma

3、mmar00，x0cos)(r2ie2i1ixfxff11i1amf 12i2eamfr2i2ramf22i2jm22221rmj2rra 222121r222121cos2)3()(sin2cos2)3(sin2mmmmmgammmgmacos)(2121mammar0)23cos(1sinr1aag),(21nffff ),(21nrrrr)(21nqqqq, ),(21tqqqnii rr011iniiiiniimrarfnkkkiniiqq11rfnkkkiiqq1rriniiimra 10)(1111kkiniiinkkiniiiiniiqqmqmrrrarf ), 2 , 1(01

4、nkqmqnikiiik rrkkiniiinkknkkiniiiqqrmqqrm)(1111rr )()(111kiniiikiiniinikiiiqdtdmqdtdmqmrrrrrr ( (消消点点) )第第一一个个拉拉格格朗朗日日关关系系式式kikiqqrr函数，函数，仅为时间和广义坐标的仅为时间和广义坐标的和和jiiqtrr无无关关与与广广义义速速度度jq 广义速度广义速度tqqqqtkkknkkiiidd1rrrknkkiiiqqt1rrrknkkjijijiqqqtqq122rrrknkkjijijiqqqtqqt122ddrrrkiqr kiqtrddnikiiininikiii

5、kiiiqdtdmqdtdmqm111)()(rrrrrr nikiiinikiinikiiinikiiiqmqmdtdqmqdtdm1111)(rrrrrrrrknikiikniiikniiiiknikiiqtqmqtvmqmqqm11211)(21)(21rrrrrrkknikiiiqtqtdtdqm 1rrkikiqqrrkikiqqdtdrr), 2 , 1(01nkqmqnikiiik rrkknikiiiqtqtdtdqm 1rrkkqvq), 2 , 1(nkqqtqtdtdkkkkkkqvqtqtt)(dd), 2 , 1( , 0),(21nkqvqqqtvvkn 0)()(

6、ddkkkkqvqtqvqtt0)(ddkkqlqltoarr2222212121aaaojvmjt2221)(92(121rrmmrrrrvrrvaaa)(oarr2222212121aaaojvmjt2221)(92(121rrmmmmwqfqttt)(ddmrrmm 221)(92(61221)(92(6rrmmm 002221)(92(3ttrrmmmcroxx22221212121ccjvmxmt2221rmjrxvcc222222143)(21rmxrmxmmgfmmfxxffxwqsfx)()(2100，x00，x0fwqxqxtxtt)(dd rmxmmxtdtdrmxmmxt

7、221221)()(gfmmfrmxmm)()(21221 qttt)(dd 2222222323rmxrmxtrmxrmt023222 rmxrm3)(2121mmgmmffa解得：解得：222222143)(21rmxrmxmmt0qgfmmfqx)(21222121ccjmvtcos2sin2lylyccsin2cos2lxxlxxcc)31sin(2121)cos21()sin2(21222222llxxmjlxmtc)cos2(212lxmgkxv)cos2(212lxmgkxvtl)31sin(21222llxxm2121,2 , 1,qxqqxqkqk广广义义坐坐标标0)(ddk

8、kqlqltmgxkxxlsin21lmxmxlcos21sin21)(dd2lmlmxmxlt )cos2(212lxmgkx)31sin(21222llxxmvtl0)(ddxlxlt0cos21sin212mgkxmlmlxm 0)(ddlltcos21sin213)(dd2lxmxmlmllt sin21cos21mgllxmlsin21312xmlmll0sin21sin213gxl 0cos21sin212mgkxmlmlxm )cos2(212lxmgkx)31sin(21222llxxmvtloackxxyoackxxy22222121212121ccoojvmjxmtsin2

9、cos2lylycccos2sin2lxxlxxcccos212122glmkxvcos2161)23(412222221x lmlmxmmtoackxxy0)(ddkkqlqltvtlcos2161)23(412222221x lmlmxmmcos212122glmkx2121,2 , 1,qxqqxqkqk广义坐标广义坐标kxxlcos21)23(21221lmxmmxlsin21cos21)23(21)(dd22221 lmlmxmmxlt0)(ddxlxltcos2161)23(412222221x lmlmxmmvtlcos212122glmkx0sincos)23(22221kxl

10、mlmxmm sin21sin2122glmxmlcos2131222x lmlmlsin21cos2131)(dd2222 x lmx lmlmxlt0)(ddlltcos2161)23(412222221x lmlmxmmvtlcos212122glmkx0sincos32gxl 0sincos)23(22221kxlmlmxmm 222222112121212121oooooojmvjmvt11,3oorv232,3222rvvrvrvroeroroercos1249cos22222221222122rrrvvvvvroorooo222222112121212121oooooojmvjm

11、vt11,3oorv2322rvvroerocos1249cos22222221222122rrrvvvvvroorooo)cos1 (641822222mrmrmrcos2mgrv0)(ddkkqlqltvtl2121,2 , 1,qqqqkqk广广义义坐坐标标)cos1 (641822222mrmrmrcos2mgr0l)cos1(63622mrmrlsin6)cos1 (636)(dd2222 mrmrmrlt0)(ddllt0sin)cos1 (62 cos2)cos1 (641822222mgrmrmrmrvtlsin2sin62mgrmrl)cos1(6822mrmrlsin6)cos1 (68)(dd222 mrmrmrlt0)(ddllt0sin)cos1 (34rg cos2)cos1 (641822222mgrmrmrmrvtl0sin)cos1 (62 广广义义动动量量jjjjpconstpqtqlevt)cos1 (641822222mrmrmrtcos2mgrvcos2)cos1 (641822222mgrmrmrmrvtlconstgrcos2)cos