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¿¼²éÏÂÁзºº¯: S(q)=Integrate[L(q(t),q'(t),t), (t,t1,t2)] S¾ÍÊǹØÓÚº¯Êýq(t)µÄ·ºº¯,ÒòΪ»ý·ÖµÄ½á¹ûÍêÈ«ÒÀÀµÓÚº¯Êýq(t)µÄÐÎʽ. LÊÇÒ»¸öÈÎÒâ µÄÈýÔªº¯Êý.¹À¼ÆÄãµÄÇéÐбÈÕâÓ¦¼òµ¥. Ëùν·ºº¯Î¢·ÖÊÇÖ¸µ±q(t)µÄÐÍʽÓÐС±ä»¯Ê±SȡֵµÄ¸Ä±ä. ¼ÙÉèh(t)ÊÇÒ»¸öСº¯Êý,¼´ ÔÚ¶¨ÒåÓòÄÚÖµ½Ó½üÁã.ÁíÍâ¼ÙÉèµ¼Êýh'(t)Ò²ÊÇСº¯Êý. ¿¼²é: S(q+h)=Integrate[L(q(t)+h(t),q'(t)+h'(t),t), {t,t1,t2}] ÓÉÓÚh(t)(h(t)³ÆÎªÒ»¸ö±ä·Ö)ºÍh'(t)ºÜС,¿É¶ÔL×÷TalorÕ¹¿ª: L(q(t)+h(t),q'(t)+h'(t),t)=L(q(t),q'(t),t)+D[L,q]*h(t)+D[L,q']*h'(t) ËùÒÔ: \Delta S=S(q+h)-S(q)=Integrate[L(q(t)+h(t),q'(t)+h'(t),t) -L(q(t),q'(t),t), {t,t1,t2}] =Integrate[D[L,q]*h(t)+D[L,q'(t)]*h'(t), {t,t1,t2}] ¹À¼ÆÄãÃÇÊÇÒªÇó¼«ÖµÎÊÌâ,Õâ¾ÍÒªÇó\Delta S=0. => Integrate[D[L,q]*h(t)+D[L,q'(t)]*h'(t), {t,t1,t2}]=0 ¿¼Âǵ½: Integrate[D[L,q'(t)]*h'(t) dt,{t1,t2}]=Integrate[D[L,q'(t)] dh(t),{h(t1),h(t2) ] =h(t2)*D[L,q'(t2)]-h(t1)*D[L,q'(t1)]-Integrate[h(t)*(d/dt)D[L,q'(t)] dt,{t1,t2]. (ÒÔÉÏÓõÄÊÇ·Ö²¿»ý·Ö·¨) ÓÉÓÚÎÊÌâÖлý·Ö±ß½çÉÏÒ»°ãÓÐÏÞÖÆ: ±ß½ç¹Ì¶¨. ËùÒÔ: h(t1)=h(t2)=0.ÓÚÊÇ: Integrate[D[L,q'(t)]*h'(t) dt,{t1,t2}] =-Integrate[h(t)*(d/dt)D[L,q'(t)] dt,{t1,t2] Hence: Integrate[D[L,q]*h(t)+D[L,q'(t)]*h'(t), {t,t1,t2}]=0=> Integrate[D[L,q]*h(t)-h(t)*(d/dt)D[L,q'(t)] dt, {t,t1,t2}]=0=> ÓÉÓÚÎÊÌâÖлý·Ö±ß½çÉÏÒ»°ãÓÐÏÞÖÆ: ±ß½ç¹Ì¶¨. ËùÒÔ: h(t1)=h(t2)=0.ÓÚÊÇ: Integrate[D[L,q'(t)]*h'(t) dt,{t1,t2}] =-Integrate[h(t)*(d/dt)D[L,q'(t)] dt,{t1,t2] Hence: Integrate[D[L,q]*h(t)+D[L,q'(t)]*h'(t), {t,t1,t2}]=0=> Integrate[D[L,q]*h(t)-h(t)*(d/dt)D[L,q'(t)] dt, {t,t1,t2}]=0=> Integrate[h(t)*(D[L,q]-(d/dt)D[L,q'(t)]) dt, {t,t1,t2}]=0 ÓÉÓÚh(t)ÊÇÈÎÒâСº¯Êý,ÉÏʽ³ÉÁ¢±ØÒªÇó D[L,q]-(d/dt)D[L,q'(t)]=0 ÕâÊǹØÓÚtµÄ³£Î¢·Ö·½³Ì. ÓÉÓÚLÊǸø¶¨µÄº¯Êý,¸Ã·½³Ì¿ÉÇó³öº¯Êýq(t).¼´: ʹ·ºº¯S(q)È¡¼«ÖµµÄº¯ÊýÐÎʽq(t). ÉÏÊö·½³ÌÔÚÎïÀíÖгÆÎª: Lagrange equation. ÉÏÊö˼Ïë¿ÉÓÃÓڷdz£¸´ÔÓµÄÇé¿ö.±ÈÈç˵ÍäÇúÄܾö¶¨µÄĤµÄƽºâÐÎ×´.Ï´Π½éÉÜĤµÄÎÊÌâ. |
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