理论力学—拉格朗日方程（课堂PPT）

1、.1.2.3), 2 , 1(0Nnimiiii aFF), 2 , 1(nii r), 2 , 1(0)(Nnimiiiiii raFF.4), 2 , 1(0Nniiii rF), 2 , 1(0)(nimiiiii raF), 2 , 1(0)(Nnimiiiiii raFF.5nizzmFyymFxxmFiiiziiiiyiiiiixi, 2 , 10)()()( ), 2 , 1(0)(nimiiiii raF.6.7CmgaCCmaFIRCICJM0sinRxMxF-xmgICIRsin32gaCRamRJCC,212.82IIsinmlFFBA.90211IICBABBAAygm

2、ygmygmxFxFcos2cossincossinlylylxlylxCBBAAsin2sincossincoslylylxlylxCBBAA.100211IICBABBAAygmygmygmxFxFsin2sincossincoslylylxlylxCBBAA0sin2sin2cossin22121glmglmllmcos)(1212lmgmm .11.1211I1amF 12I2eamFr2I2ramF22I2JM22221RmJ00，x00，x.130cossin22r2Ie2I2JRFRFRgm0)23cos(1sinr1aag00，x11I1amF 12I2eamFr2I2ramF

3、22I2JM22221RmJ2rRa .14cos)(2121mammar00，x0cos)(r2Ie2I1IxFxFF11I1amF 12I2eamFr2I2ramF22I2JM22221RmJ2rRa .15222121r222121cos2)3()(sin2cos2)3(sin2mmmmmgammmgmacos)(2121mammar0)23cos(1sinr1aag.16),(21nFFFF ),(21nrrrr)(21Nqqqq, ),(21tqqqNii rr011iniiiiniimrarFNkkkiniiqQ11rF.17Nkkkiiqq1rriniiimra 10)(1111

4、kkiniiiNkkiniiiiniiqqmQmrrrarF ), 2 , 1(01NkqmQnikiiik rrkkiniiiNkkNkkiniiiqqrmqqrm)(1111rr .18)()(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tqqqqtkkkNkkiiidd1rrr.19kNkkiiiqqt1rrrkNkkjijijiqqqtqq122rrrkNk

5、kjijijiqqqtqqt122ddrrrkiqr kiqtrdd.20nikiiininikiiikiiiqdtdmqdtdmqm111)()(rrrrrr nikiiinikiinikiiinikiiiqmqmdtdqmqdtdm1111)(rrrrrrrrknikiikniiikniiiiknikiiqTqmqTvmqmqqm11211)(21)(21rrrrrrkknikiiiqTqTdtdqm 1rrkikiqqrrkikiqqdtdrr.21), 2 , 1(01NkqmQnikiiik rrkknikiiiqTqTdtdqm 1rr), 2 , 1(NkQqTqTdtdkkkk

6、kqVQ.22kkkqVqTqTt)(dd), 2 , 1( , 0),(21NkqVqqqtVVkN 0)()(ddkkkkqVqTqVqTt0)(ddkkqLqLt.23.24OARr2222212121AAAOJvmJT2221)(92(121rRmmrrRrvrRvAAA)(.25OARr2222212121AAAOJvmJT2221)(92(121rRmmMMWQFQTTt)(ddMrRmm 221)(92(61221)(92(6rRmmM 002221)(92(3ttrRmmM.26CROxx22221212121CCJvmxmT2221RmJRxvCC222222143)(21R

7、mxRmxmmgfmmFxxFFxWQsFx)()(2100，x00，x0FWQ.27xQxTxTt)(dd RmxmmxTdtdRmxmmxT221221)()(gfmmFRmxmm)()(21221 QTTt)(dd 2222222323RmxRmxTRmxRmT023222 RmxRm3)(2121mmgmmfFa解得：解得：222222143)(21RmxRmxmmT0QgfmmFQx)(21.28.29222121CCJmvTcos2sin2lylyCCsin2cos2lxxlxxCC)31sin(2121)cos21()sin2(21222222llxxmJlxmTC)cos2(2

8、12lxmgkxV.30)cos2(212lxmgkxVTL)31sin(21222llxxm2121,2 , 1,qxqqxqkqk广义坐标广义坐标0)(ddkkqLqLt.31mgxkxxLsin21lmxmxLcos21sin21)(dd2lmlmxmxLt )cos2(212lxmgkx)31sin(21222llxxmVTL0)(ddxLxLt0cos21sin212mgkxmlmlxm .320)(ddLLtcos21sin213)(dd2lxmxmlmlLt sin21cos21mgllxmLsin21312xmlmlL0sin21sin213gxl 0cos21sin212mg

9、kxmlmlxm )cos2(212lxmgkx)31sin(21222llxxmVTL.33OACkxxy.34OACkxxy22222121212121CCOOJvmJxmTsin2cos2lylyCCcos2sin2lxxlxxCCcos212122glmkxVcos2161)23(412222221x lmlmxmmT.35OACkxxy0)(ddkkqLqLtVTLcos2161)23(412222221x lmlmxmmcos212122glmkx2121,2 , 1,qxqqxqkqk广广义义坐坐标标.36kxxLcos21)23(21221lmxmmxLsin21cos21)2

10、3(21)(dd22221 lmlmxmmxLt0)(ddxLxLtcos2161)23(412222221x lmlmxmmVTLcos212122glmkx0sincos)23(22221kxlmlmxmm .37sin21sin2122glmxmLcos2131222x lmlmLsin21cos2131)(dd2222 x lmx lmlmxLt0)(ddLLtcos2161)23(412222221x lmlmxmmVTLcos212122glmkx0sincos32gxl 0sincos)23(22221kxlmlmxmm .38.39222222112121212121OOOOO

11、OJmvJmvT11,3OORv232,3222RvvRvRvrOErOrOErcos1249cos22222221222122RRRvvvvvrOOrOOO.40222222112121212121OOOOOOJmvJmvT11,3OORv2322RvvrOErOcos1249cos22222221222122RRRvvvvvrOOrOOO)cos1 (641822222mRmRmRcos2mgRV.410)(ddkkqLqLtVTL2121,2 , 1,qqqqkqk广义坐标广义坐标)cos1 (641822222mRmRmRcos2mgR.420L)cos1(63622mRmRLsin6

12、)cos1 (636)(dd2222 mRmRmRLt0)(ddLLt0sin)cos1 (62 cos2)cos1 (641822222mgRmRmRmRVTL.43sin2sin62mgRmRL)cos1(6822mRmRLsin6)cos1 (68)(dd222 mRmRmRLt0)(ddLLt0sin)cos1 (34Rg cos2)cos1 (641822222mgRmRmRmRVTL0sin)cos1 (62 .44广广义义动动量量jjjjpconstpqTqLEVT.45)cos1 (641822222mRmRmRTcos2mgRVcos2)cos1 (641822222mgRmRmRmRVTLconstgRcos2)cos1 (3922constLL