《理论力学 Theoretical Mechanics——动力学》（双语版）第十七章 拉格朗日方程 Lagrange’s equations

学 两=2a(=12m)(c 代入质点系动力学普遍方程,得: ∑(F-ma1)c=∑F·C-∑ma,1=0(d) F=F①)=2(F,m) +rOi =(X001z,02 ∑O 16

16  =  =   = k j j j i i q i n c q r r 1   ( 1,2, ) ( ) 代入质点系动力学普遍方程，得：    = = = − =  − = n i n i i i i i i n i Fi mi ai ri F r m a r d 1 1 1 ( )   0 ( )         = = = = = = = = =    +   +   =     =     =  k j j j k j j j i i n i j i i j i i k j n i j j i i k j j j i n i i n i i i Q q q q z Z q y Y q x X q q r q F q r F r F 1 1 1 1 1 1 1 1 [ ( )] ( ) ( )     

Dynamic Q=∑ H a +yoi+Z z oqr, dq The @i are called generalized forces mhn∑(-ma)=O刘义ma之听a) =∑(Q2mtcq )∞g=0 Q∑m dt aq =0(j=1,2…,k).(f) generalized inertial forces 7

17 The Qj are called generalized forces. ( ) ( ) 1 e q z Z q y Y q x Q X j i i j i i j i i n i j   +   +   == ( ) 0 Then ( ) ( ) 1 1 1 1 1 1 =   = −     −  = −       = = = = = = j j i k j n i i j i n i j k j j i i i k j j j n i i i i i q q r dt dv Q m q q r F m a r Q q m a     0 ( 1,2 , ). ( ) 1 j k f q r dt dv Q m j i n i i j i = =    −= generalized inertial forces

称Q=>(X1+z2)为广义力(e) 则义(F,一m)=SQA1义ma测) k or k n dy. ar (Q,→2mm)=0 j=1i=1 q J 0,-2mi dt aq, =0(1=12…k)() 广义惯性力 18

18 称 ( ) 为广义力 ( ) 1 e q z Z q y Y q x Q X j i i j i i j i i n i j   +   +   = = ( ) 0 ( ) ( ) 1 1 1 1 1 1 =   = −     −  = −       = = = = = = j j i k j n i i j i n i j k j j i i i k j j j n i i i i i q q r dt dv Q m q q r F m a r Q q m a  则    0 ( 1,2 , ) ( ) 1 j k f q r dt dv Q m j i n i i j i = =    − = 广义惯性力

Dynarnics The generalized inertial force can be obtained from the kinetic energy of the system of particle in the following way ∑m dt ac ∑m1,(v1·) dt ∑m i=1 dt aq For the following calculation we need the two expression or. O 00j an d dg. C at dq, The first expression is obtained by differentiation of both sides of the equation(b)by q 19

19 The generalized inertial force can be obtained from the kinetic energy of the system of particle in the following way: ( ) ( ). 1 1 1 j i n i i i j i i n i i j i n i i i q r dt d m v q r v dt d m q r dt dv m   −    =        = = = For the following calculation we need the two expression: j i j i j i j i q v q r dt d q v q r   =     =   and  The first expression is obtained by differentiation of both sides of the equation (b) by . j q 

力单 广义惯性力可改变为用质点系的动能表示,因此 又、CDF ∑ ∑m,( d or 41i=1 vi al Z dt ac 为简化计算,需要用到以下两个关系式: d or O q O dt aqaq 下面来推导这两个关系式: 第一式只须将(b)式两边对q求偏导数即可得到。 20

20 广义惯性力可改变为用质点系的动能表示 , 因此 ( ) ( ) 1 1 1 j i n i i i j i i n i i j i n i i i q r dt d m v q r v dt d m q r dt dv m   −    =        = = = 为简化计算 , 需要用到以下两个关系式： j i j i j i j i q v q r dt d q v q r   =     =   ;  下面来推导这两个关系式： 第一式只须将(b)式两边对 q  j 求偏导数即可得到