理论力学全套解疑17

A17-1AmgAAδrδrAπ17-1mgδrmgδrBBmgmg12trrdd=trδrdrδrtδr=0M17-31vMtt+∆tdrδrdrδrdrδr17-3117-3217-320),,,(=tzyxfttδt=0kjirzyxδδδδ++=MxyzM),,(1zzyyxxMδδδ+++0),,,(=+++tzzyyxxfδδδ0),,,(),,,(=+∂∂+∂∂+∂∂+=+++zzfyyfxrftzyxftzzyyxxfδδδδδδ0),,,(=tzyxf0=∂∂+∂∂+∂∂zzfyyfxxfδδδ(1)(1)),,(zyxfkjizfyfxff∂∂+∂∂+∂∂=gradgradf0),,,(=tzyxfM(1)(gradf)⋅δr=0(1)′δr8radfδrMdtt+dt17-32MM′dr=dxi+dyj+dzk0)d,d,d,d(=++++ttzzyyxxf0dddd),,,()d,d,d,d(=+∂∂+∂∂+∂∂+∂∂+=++++ttfzzfyyfxxftzyxfttzzyyxxf=0),,,(tzyxf0dddd=∂∂+∂∂+∂∂+∂∂ttfzzfyyfxxf(2)(2)0dd)grad(=∂∂+⋅ttffr(2)′(2)′dr(1)′(2)′δrdrdrMtf(x,y,z)=00=∂∂tf(2)′0d)grad(=⋅rf(3)(3)(1)′drδrFdrF⋅drFδr),cos(||rFrrFδδδδFW=⋅=n3n3nsniszyxtfiiii,,2,1;,,2,10),,,(===αα3nsk=3nssk3nk=3nskk=3ns17-51xAyAxByB(1)⎪⎭⎪⎬⎫==−+−=+0)()(22222BABABAAylyyxxryx132232=−×=−=nkOAxϕqq=ϕ⎪⎪⎪⎭⎪⎪⎪⎬⎫=−+===0sincossincos222BBAAyrlrxryrxϕϕϕϕ(1)′(1)′(1)(1)′MOlMOlOOxyzMxyz17-52(2)2222lzyx=++xyzsnk−=32113=−×=θϕϕθ==21qq,,,21MM(2)′⎪⎪⎭⎪⎪⎬⎫===θϕθϕθcossinsincossinlzlylxkk=3nsqkkqqkqqqδδδ,,,21kkqqqδδδ,,,21M1M2Oxy17-6CCyxxθCvC17-60)tg(=−CCxyθ(1)(1)(1)0)tg(=−CCxyδθδ(1)′δθδδCCyx(1)′kkqqqδδδ,,,21kkqqqδδδ,,,21kkqqq,,,2117-6M1M20)tg(C=−xyCθδθδδCCyx23ns(k=3ns)kkqqq,,,21niqqqkii,,2,1),,,(21==rrnFFF,,,21⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂×=∂∂⋅=∑∑==jiizjiiyjiixnijiinijqzFqyFqxFqFQ11r(1)qijjqQδQjqjQjpσEHQj(1)ijkjiiniFniqQwiδδδ∑∑∑=====111rFδqjqjδqj0qδqi=12njirδjFniiwδ∑=1jwδ∑jiinijwrFδδ⋅=∑∑=1jjjqQwδδ=∑jjjqwQδδ∑=(2)qjqjwδ∑qjδqjqjQjQjiiziiyiixzVFyVFxVF∂∂−=∂∂−=∂∂−=M),,2,1(),,(nizyxVViii==iri=xii+yij+zik=ri(i=1,2,,qk)Vk),,2,1;,,2,1()()(),(),([kjniqVqzqyqxVVjjijiji====jjiijiijiinijiizjiiyjiixnijqVqzzVqyyVqxxVqzFqyFqxFQ∂∂−=⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂+∂∂∂∂+∂∂∂∂−=⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂=∑∑==11(3)j=1,2,,k1⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂=∂∂⋅=∑∑==jiizjiiyjiixnijiinijqzFqyFqxFqQ11rFj=1,2,,k2jjiqwQδδ∑=j=1,2,,k3jiqVQ∂∂−=j=1,2,,kOAABAO17-812a2bP1P2BFk=2ϕψψϕ==21,qq1(1)ϕψϕϕϕϕψϕϕψϕϕϕϕϕϕϕϕϕϕcos2sin2sin)sin2sin2()coscos2()cos(21212211FaaPaPbaFbaPaPyFxFyPxPyPxPQByBxDyDxCyCx+−−=∂+∂+∂+∂+∂∂=⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂=ψψψψϕψψϕψϕψψψψψψψsin2sin)sin2sin2()coscos2()cos(2212211FbaPbaFbaPaPyFxFyPxPyPxPQByBxDyDxCyCx+−=∂+∂+∂+∂+∂∂=⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂=2δϕδψQϕQϕϕδϕ=0ψδψ17-82ψψδψψδψψδψδψψδψδδψδψψcos2sincos2sincos||)90cos(||212FbbPbFbPFPwQBD+−=⋅+−=++==∑rrDψδψ=0)ϕδϕ17-83δrCδrAδrDδrB|δrA|=|δrD|=|δrB|=2aδϕ17-8217-83ϕϕϕδϕϕδϕϕδϕϕδϕδϕϕδϕδϕδδϕδϕϕcos2sin2sincos2cos2sincos||)90cos(||)90cos(||21121FaaPaPaFaFaPFPPwQBDC+−−=⋅+⋅+−=++++==∑rrrDDQϕQψδψψψδϕϕϕϕψδψϕδϕψδψϕδϕϕδϕψϕδψϕδϕδδδδδδδδδ)cos2sin()cos2sin2sin(cos2cos2)sin)sin2(sin)sin2sin2()coscos2()cos()()()(2212121221111FbbPFaaPaPbFaFbaPaPbaFbaPaPyPxFyPxPyPxPwByBxDyDxCyCxiiniFnii+−++−−=⋅+⋅+−−+−=++++=+++++=⋅=∑∑==rFδψδϕδδψϕQQqQwjjkjFnii+==∑∑==11ψψϕϕϕψϕcos2sincos2sin2sin221FbbPQFaPaPQ+−=+−−=P1P2F117-91SPAxAq=xAxAAδxACδxCxC=Ax51ACxxδδ51=0=−=∑CAFxSxPwδδδ05=⎟⎠⎞⎜⎝⎛−AxSPδAxjjkjiiniFxQqQwAδδδδ==⋅=∑∑∑==11rFxAxQA5SPQAx−=0=AxQPS5SP=217-81ϕψψψϕϕϕψϕcos2sincos2sin2cos221FbbPQFaaPaPQ+−=+−−=ϕψQϕ=0212arctgPPF+2=ϕQψ=02arctgPF2=ψjjqVQ∂∂−=Qj=0),,2,1(0kjqVj==∂∂G2lOAABAO17-92BαODOAODθθk=1θq=θOABG1G2)()]sin()sin(3[)sin()sin(3)]sin()sin(2[)sin(θθαθαθαθαθαθαθαVGlGllGllGGlV=−++−=−−+⋅−=−++++−=Qθ=00dd==∂∂θθVV0)cos()cos(3=−−+θαθαOA⎟⎠⎞⎜⎝⎛=αθctg21arctg3),,2,1(1kjqQjiinij=∂∂⋅=∑=rFnsk=3nskqqq,,,21jiq∂∂⋅rjiiq∂∂⋅rFFiqjqjQjMABF17-101s(q=s)FsτFτFτFvFrFrFrF==⋅=⋅=⋅=⋅=∂∂⋅=),cos(ddddddFvtstssQsQsFsFτFτABFFnk),,2,1(1kjqQjiinij=∂∂⋅=∑=rF17-10117-102ODF1F2OOxy17-102OA=aOB=bθq=θF1F2θQθδθαδθδθδδδ)(212121bFFbFaFwBAF−=⋅−⋅=⋅+⋅=∑rFrF)()()(2121RFFOOOmmmbFaFQ=+=−=θR=F1+F2F1F2θF1F2QθOOO021=−=bFaFQθbFaF21=011=⋅=∑∑==iiniFniiwrFδδ),cos(||11iiiiniiiniFrFrrFδδδ∑∑===⋅)(11iizyiiyiixniiinizFFxFδδδδ++=⋅∑∑==rFjjniiiniqQδδ∑∑===⋅11rF17-11aFPk=1ϕq=ϕ117-11b0)(||1=−=∑=CAFniFPwirrδδδOCδϕEACδrEδrCDAAδrAδϕϕδϕϕϕϕδδ2coscoscos1cos||||LLEA=⎟⎟⎠⎞⎜⎜⎝⎛⋅==rr17-11δrCδϕδRC=||r0cos21=⎟⎟⎠⎞⎜⎜⎝⎛−=∑=δϕϕδFRPLwiFniδϕFRPL−ϕ2cosϕQϕFRPLQ−=ϕϕ2cosQϕ217-11b)()(1CyCxAyAxFniyFxFyPxPwiδδδδδ+++=∑=ϕϕcossin0FFFFPPPyxyx−====ϕδϕϕδδϕϕδδδϕϕϕδδcos)sin(sin)cos(cos)tg(2RRyRRxLLyCCA==−====0cos21=⎟⎟⎠⎞⎜⎜⎝⎛−=∑=δϕϕδFRPLwiFni.FRPLQ−=ϕϕ2cos3⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂=∑=jiizjiiyjiixnijqzFqyFqxFQ1FRPLRFRFLPyFxFyPxPQCyCxAyAx−=∂∂−+∂∂+∂∂=⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂=ϕϕϕϕϕϕϕϕϕϕϕϕϕϕ2cos)sin()cos()cos()sin()tg(Qϕ=00cos2=−FRPLϕPQϕ2cosRPLF=1217-111CQAAδrAAPδϕδϕϕδϕϕδδδδδϕQFRPLRFLPFyPxPwCAyAxFnii=⎟⎟⎠⎞⎜⎜⎝⎛−=⋅−=++=∑=21cos)tg(180cos||)(DrFRPLQ−=ϕϕ2cos