А.Б. Пименов - Задачник по теоретической механике, страница 3
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2•;# $&%(( &{A(p, q, t), B(p, q, t)} =s ∂A ∂Bi=1∂A ∂B−∂pi ∂qi ∂qi ∂pi.L!FM; # 10 . 4•{A, B} = − {B, A} .20 .L!EM{c1 A + c2 B, D} = c1 {A, D} + c2 {B, D} ,{D, c1 A + c2 B} = c1 {D, A} + c2 {D, B}L! ML! M(c1 , c2 = NQRPS)30 .J & D#>K{AB, D} = A {B, D} + B {A, D} .40 .L! M4>{A, {B, C}} + {C, {A, B}} + {B, {C, A}} = 0L! !ML'% H#M• %* # {pi , qj } = δij ,{pi , pj } = 0,{qi , qj } = 0.L! @ML! ML! ?M( $&%( $ & $& > F (p, q, t)+ $&*+∂FdL! )MF (p, q, t) =+ {H, F }dt∂t•L& $&%$' / %$> Qdi = 0M• &( L! M , #* $* B& & # L%$> /Mṗi = {H, pi } ,q̇i = {H, qi } .L! FM8& D.3* εijk G $/ &* &&, &, ε123 = 1.L! EML2( & D.3* &, &, * % &('% G &&* % εijk ≡ εijk $& % &+& % $&& $& &%M( & $& /#*+ %+% % •εijk = −εjik ,εijk = −εikj ,L!Mεijk = −εkji .; $0/ & D.3* ' $ & %$* $* &, $&%( & 9A = B × C, Z.( $ & A ' #* $ Ai = εijk Bj Ck ,L! M,% ( $&%* $& &( b-$ %'%*$&(/0( % $( &3j,k=18 Z.( $ & $ L = r × pLi = εijk xj pk , Z.( $* & $&(' ,, $(H = ∇ × AHi = εijk∂Ak .∂xj"'* ((/( %/0 & D.3*10 .
&% %+ $ & D.3* ' #*$&% % $&%( && !×! , δG L& 6&&M δi δi δi a b c = δaj δbj δcj δk δk δk acbL!M δi δi = bj cj = δbi δcj − δci δbj . δb δc L!!Mεijk εabcL& $&&* % $&, & D.3* #>* G &,M20 . ;& %+ & D.3* $ % % '#* $&% % $&%( × , δG L& 6&&Mεijk εkbc30 .;& %+ & D.3* $ % %εijk εijn = 2δkn .40 .L!@M;& %+ & D.3* $ & %εijk εijk = 6.L! M) "* # VM {pk , F (q)} , #M {qk , F (p)} .F 2( >* m %'0( $&B&, $ * # %( $& & {ẋi, ẋj } .
7 *& & $&(',, $(E "* # {Li, xj } , ,% Li G $* & $ L ' %+ !G! M! "* # {Li, pj } .!! "* # Li, r 2 .!! "* # Li, p 2 .!! "* # {Li, p · r} .!@ "* # {Li, Lj } .!! "* # Li, L2 .!? 4 Fijk % # Fijk ="#(xi pj + f (r · p) δij ) , εkmn pm xn ,,% $ %'%* $&(/0( % $( &f G %( >(!) 4 Fijk % # Fijk ="#pi pj + f (r ) xi xj , xj pn pm ,2,% $ %'%* $&(/0( % $( &f G %( >(!F * %1H = Iij Li Lj + Bi Li ,2,% Li G $* & $ Iij BiG $(* &* $ %'%* $&(/0( %$( & ;( # {Li , Lj } = −εijk Lk ,$ &( %'( %( Li!E ; $*( % $&* A B C # &*+ *{A, B} = C −AB,C{A, C} = −A{B, C} = B.
B * %H=1 4C .44$ &( %'( %( $&*+ A, B, C %'( A(t), B(t), C(t) *@ 2( >* %'0( >& $ U (r) = − αr L%6$&M ,* #D>p2αH=−2m r {H, A} ,% AA=G & D$G:,Gαr1p×L−mrLL G & $ >*M@ $ *( # {Lz , H} , $Lz $ >* %'0( >& $((( ,& %'( >*1 222p z + py + pz + UH=x2 + y 2 + z 2 .2m## " ! 05,& %'( *( >( +$&*+ p, q & +&(/0( *$& B/> *•F (p(t), q(t), t) = NQRPS%( ∀t,L!?M B &dF (p(t), q(t), t) = 0.dtL!)M2* , #* >( F (p(t), q(t), t) #*,& %'( ((( %& *$ %+ L, L! )MM•∂F= 0,∂t{H, F } = 0.1)2)L!FML!EM& +'%( ,& %'( $ % , 9 , ( & •H = H(p, q), / %$> LQdi,& %'(= 0M , (((H(p, q) = NQRPS.
9 %( , ( >( &% qk ∂H= 0,∂qk , & H = H(p1 , . . . , pk−1 , pk , pk+1 , . . . , ps ; q1 , . . . , qk−1 , qk+1 , . . . , qs ; t),L!!M / %$> LQdi = 0M ##0* $ pk /0 > &% qk ((( ,&%'(pk = NQRPS.! 9 &>( , $&* $&('*+ $&*+ &/ >/f (pk , qk ) , & H = H f (pk , qk ), p1 , . . . , pk−1 , pk+1 , . . . , ps ; q1 , . . . , qk−1 , qk+1 , . . .
, qs ; t , %$> LQdi,& %'(= 0M>(f (pk , qk )(((f (pk , qk ) = NQRPS.L8' $&% & ##0 ,% &>( + $& $&('*+$&*+M• 8& % > F (p, q, t) G(p, q, t) ((/(,& %'( # {F (p, q, t), G(p, q, t)}' ((( ,& %'(• 2( +'%( %'( * %&&+#+% ,& %'( $#%* @ 4 %'( %&&+ %( * &($*( ,&' a NQPϕm 22 2ρ̇ + ρ ϕ̇ −L=(a = NQRPS).2ρ2@! 3> >& $$*( ,&'L=U (r) &+ &%+m 2ṙ + r2 θ̇2 + r2 PZR2 θϕ̇2 − U (r).24 %'( >* %&&+@@ 3> %'( $ $&+ &* &% R %&%$ (,( g & , >* ,&* %'( 4 %'( %&&+@ 3> m &(% e %'( $ & &% R $(' g = −gez ' %&%*+ $(*+ B&E = −E0 ez , H = H0 ez $(+ & ,>* ,&* %'( 4 %'( %&&+@? 3> m &(% e %'( $ $&+ $&#%az = x2 + y 2 (a = NQRPS) $(*+ %&%*+ $ ('g = −gez B& , $(+ $&(' &*+ E = −E0ez H = H0ez & ,>* ,&* %'( 4 %'( %&&+@) 3> m &(% e %'( $ & $&+ , && α $( %&% $ ('g = −gez ' B& , $(+ $&('&*+ E = E0ez H = H0ez 7 $% / z , &- +%( &% &$' $$&& z > 0 & , ,&* %'( 4 %'( %&&+@F 3> m &(% e %'( $ $&+ $&#%az = x2 + y 2 (a = NQRPS) $(*+ %&%*+ $ (' , $ $&(' &, H = H0ez & , ,&* %'( 4 %'( %&&+@E 3> m &(% e %'( $ & $&+ , && α $( %&% $ ('g = −gez ' , $ $&(' &,H = H0 ez 7 $% / z , &- +%( &% &$' $$&& z > 0& , >* ,&* %'(4 %'( %&&+ 3> m &(% e %'( $ & &% R $* (' g = −gez %&% $( , $H = H0 ez & , >* ,&*%'( 4 %'( %&&+ ; $*( ,g = −gezH=1 2 1 2 1 2 2pp + ω q2 1 2 2 2 0 24 %'( * ; $*( ,H=p232+ω02 q322+ap212p222+ω02 q22+b2+1 2 2ω q .2 0 1p22ω02 q222+24 %'( * ( La, b = NQRPSM ! ; $*( ,H=2p+2mmω02 q 22+λ2p+2mmω02 q 224 %'( * Ln > 2 λ = NQRPSMp232n.+ω02 q322.
@ ; $*( ,p21H= 2+ PZR2 q1 .2p2 + PZR q24 %'( * ; $*( ,mω02 q12 p22mω02 q22p21++.H=2m22m24 %'( * ? 5$( % ,& %'( ( %'(* &( $*( ,&' La = NQRPSM a NOym 24 2L=.ẋ + x ẏ −2x4 ) 3> m &(% e %'( %&% $(, $ H = H0ez " #& &, $>A = −H0 y ex$& , >* ,&* %'(4 %'( F 3> $*( ,&' La = NQRPSMm 22 2222ṙ + r θ̇ + r PZR θϕ̇ − aϕ̇ cos θ.L=24 %'( >* %&&+ E 4 ( % %'( * &( $*(,&' La = NQRPSM a NOym 24 2L=.ẋ + x ẏ −2x4? ; $*( , Lλ = NQRPSMH=p212+ω02 q122+p222+ω02 q222+λp212+ω02 q122sinp222+ω02 q222.4 %'( * ( %? f& m &(% e ' %,( $ &$&+ #& &0 ,& & >y = f (x) &, y %&%*+ $(*+ , $H = H0 ey $ (' g = −g ey & , %'( %&&+? 3> m &(% e %'( $( %&%, $ H = H0 ey $ $&+ #& &0&+ $&#* y = a x3 La > 0M &, y &, >* %'( %&&+?! $&+ , $&& α ' %,(> m &(% e " &- &$* &(% q ; +%( %&%*+ $(*+$ (' g = −gez ' B& , $(+$&('( E = E0ez H = H0ez &, %'( %&&+?@ ; $*( ,&'m 2U (ϕ)2 2222.L=ṙ + r θ̇ + r sin θ ϕ̇ − 22r sin2 θ& , ,&* %'( %'( %&&+ LU (ϕ) G &( %( >(M? 3> m &(% e %'( $ $&+&, , $& &- 2γ %&% $* (' g = −g ez " &- &$ * &(%Q& , >* %'( %&&+?? 3> m &(% e %'( $ $&+ &0(z = f (x2 + y 2 ) %&%*+ $(*+ , H = H0 ez B& E = −E0 ez $(+ & , >* $ , % ,&* %'( %'(>* %&&+?) " &- $&#% z = βρ2 Lβ > 0M +%( $%'** B& &(% Q $&+ $&#%' %,( * &(% e m ; +%( %&%*+ $(*+ $ (' g = −g ez , $H = H0 ez & , %'( &(% %&&+#( * '.•&#&( + $&*+ (p, q) → (P, Q)Pi = Pi (p, q, t),Qi = Qi (p, q, t),L!! M&* (/ % & L&*+ &( &*M */( $&#&(• , J&*+K & J&*+K $&*+ (p, q) , H(p, q, t) J*+K & J*+K $&*+ (P, Q) , K(P, Q, t)cipi dqi − Hdt=Pi dQi − Kdt + dF1 (q, Q, t),L!!Mi,% c = NQRPS• 4#+% % $&#&(L!! M 0 % *&+ $&%(0+> %&(/0+ $&%* &( L&, $&#&( #>M•6 c *( / ,$&#&(• 6 $&#& / c = 1 *(*• 6&& $&#& L!! M ((( ,% ,% ,% *$(/( &⎧ "#⎪= c δij ,Pi (p, q, t), Qj (p, q, t)⎪⎪⎪# p,q⎨ "= 0,Pi (p, q, t), Pj (p, q, t)p,q⎪#"⎪⎪⎪⎩ Qi (p, q, t), Qj (p, q, t)= 0,p,q,% ' % # {, }p,q *(/( $ J&*K $&* (p, q)• :K = cH +∂F∂t &+ , $&#&( % $%/0 #& L $&& $&%(0 > F1 (q, Q, t)M&* ,$&%(0+ > &*+ '()*6$&%(0+>F1 (q, Q, t)F2 (q, P, t)F3 (p, Q, t)F4 (p, P, t)0($&%(0>e_Se_Se_Se_S ∂Q i∂pj ∂P i∂pj ∂Q i∂qj ∂P i∂qj= 0= 0= 0= 0$&#&(&*,$&#&(⎧∂F1⎪⎪=,cp⎪i⎪∂q⎪i⎨∂F1Pi = −,⎪∂Qi⎪⎪⎪⎪⎩ K = cH + ∂F1 .∂t⎧∂F2⎪⎪=,cp⎪i⎪∂q⎪i⎨∂F2Qi =,⎪∂P⎪i⎪⎪∂F2⎪⎩ K = cH +.∂t⎧∂F3⎪⎪=−,cq⎪i⎪∂p⎪i⎨∂F3Pi = −,⎪∂Qi⎪⎪⎪⎪⎩ K = cH + ∂F3 .∂t⎧∂F4⎪⎪=−,cq⎪i⎪∂pi⎪⎨∂F4Qi =,⎪∂P⎪i⎪⎪∂F4⎪⎩ K = cH +.∂t%(K(P, Q, t) = cH p(P, Q, t), q(P, Q, t), t +∂F1 q(P, Q, t), Q, t∂tL!!!ML, %( F2 F3 F4M $( (* %J,K , * $ &-( ,$&#&( L!! M?F "*( $&#&/ $&%(0( >(F =Qi qi .i?E 4 $&%(0/ >/ '%, $&#&((p, q) → (P, Q)Pi = p i ,Qi = qi .) 4 $&%(0/ >/ $&#&( -#&((p, q) → (P, Q)Pi = αpi ,Qi = βqi ,,% α, β = NQRPS.) 4 $&%(0/ >/ $&#&( (p, q) → (P, Q)Pi = αqi ,Qi = βpi ,,% α, β = NQRPS.) 2 $&#& (p, q) → (P, Q) ((( , $&%(0/ >/P = q + e−q + ln p,Q = peq .)! 2 $&#& (p, q) → (P, Q) ((( , $&%(0/ >/⎧⎨ P = q −4 p4 − 1 q 6 ,2⎩Q = pq −1 .)@ 2 $&#& (p, q) → (P, Q) ((( , $&%(0/ >/P = −qp + q 5 ,Q = ln(p − q 4 ).) 2 $&#& (p, q) → (P, Q) ((( , $&%(0/ >/⎧p⎪⎨ P = ln 3 ,4q1⎪⎩ Q = pq.4)? %&, %&*,H=,&1p2+ mω 2 q 22m 2>(& $&#&/ $&%(0>1F = mωq 2 NSXQ.24 (* % $&#&( & J*K ,K 1$ J*K &( + &- 1(&-( $%+ $ &-( J&*+K &)) %&, %&*,,&>(&1p2+ mω 2 q 22m 2$&#&/ (p, q) → (P, Q)⎧i ⎪mωq − ip e−iωt ,⎨ P =√2mω 1⎪⎩ Q= √mωq + ip eiωt .2mωH=2 $&#& ((( 4 ,$&%(0/ >/ & J*K , K 1$ J*K &( + &- 1( &-($%+ $ &-( J&*+K & )F %&, ,pq 3H=2t$&#&/ (p, q) → (P, Q)⎧⎪⎪⎨ P = pq 3 1 + t exp(q −2 ) ,⎪⎪⎩ Q = q −2 + ln tpq 3 .2 $&#& ((( 4 ,$&%(0/ >/ 4 J*K , K )E 2 $&#& (p, q) → (P, Q)Q = (γ p)1/α q (1−α)/α ,P = − (γ p)(α−1)/α q (2α−1)/α((( Lα, γ>/=NQRPSM 4 , $&%(0/F 2 $&#& (p, q) → (P, Q)⎧⎨ Q = −pq,1 γ α⎩ P = lnq pα((( Lα, γ = NQRPSM 4 , $&%(0/>/F 2 $&#& (p, q) → (P, Q)Q = − (b q)α+1 pα+2 ,P = (b q)−α p−α−1((( Lα, b = NQRPSM 4 , $&%(0/>/F 2 $&#& (p, q) → (P, Q)⎧⎨ Q = 2 NOq t,⎩P =pt POq t((( 4 , $&%(0/ >/F! 2 $&#& (p, q) → (P, Q)⎧1/(b−1)⎪3q⎪⎨ Q = ap +,abt⎪3q⎪⎩P =−a((( La, b = NQRPSM 4 , $&%(0/>/F@ 2 $&#& (p, q) → (P, Q)Q = ln qP = pq((( 4 , $&%(0/ >/F 2 $&#& (p, q) → (P, Q)⎧p⎨ Q = lnα q α−1⎩P = pq((( Lα = NQRPSM 4 , $&%(0/>/F? 2 $&#& (p, q) → (P, Q)Q = −γ qP = p + exp(γq − 1)((( Lγ = NQRPSM 4 , $&%(0/>/F) 2 $&#& (p, q) → (P, Q)Q = q + ln p − ep ,P =p((( 4 , $&%(0/ >/FF 2 $&#& (p, q) → (P, Q)Q = −γ p e−q ,P = eq((( Lγ = NQRPSM 4 , $&%(0/>/FE 2 $&#& (p, q) → (P, Q)⎧⎨ Q = − 1 γ p sin 2q,2⎩ P = ln SXq((( Lγ>/=NQRPSM 4 , $&%(0/E 2 $&#& (p, q) → (P, Q)⎧⎨ Q = (γ p)1/2 √1 ,q√⎩P = −q 3/2 γ p((( Lγ = NQRPSM 4 , $&%(0/>/E 2 $&#& (p, q) → (P, Q)⎧⎨ Q = −pq,1 α β⎩ P = lnp q ,α((( Lα, β = NQRPSM 4 , $&%(0/>/E 2 $&#& (p, q) → (P, Q)⎧⎪⎨pQ=−α+1⎪⎩ P = −p,1/α− q,((( Lα = NQRPSM 4 , $&%(0/>/E! 2 $&#& (p, q) → (P, Q)Q = −q − γ p + t cos(q t),P = q,((( Lγ = NQRPSM 4 , $&%(0/>/E@ 2 $&#& (p, q) → (P, Q)Q = −q + ln p,P = −p((( 4 , $&%(0/ >/E 2 $&#& (p, q) → (P, Q)Q = VWNSX (p q), P = λ 1 + (p q)2 ln q((( Lλ = NQRPSM 4 , $&%(0/>/E? 2 $&#& (p, q) → (P, Q)P = ln p19 q 20 ,Q = pq((( 4 , $&%(0/ >/E) 2 $&#& (p, q) → (P, Q)⎧⎨ Q = q −2 + ln 2 p q 3 ,1⎩ P = p q 3 + 2 p q 3 exp 2q((( 4 , $&%(0/ >/EF 2 $&#& (p, q) → (P, Q)(p+q)2P = 2q e(p+q)2+ 1 + 2p e−1 ,Q=p+q((( 4 , $&%(0/ >/EE 2 $&#& (p, q) → (P, Q)Q = −q NSX p,P = 2 ln cos p((( 4 , $&%(0/ >/! 2 $&#& (p, q) → (P, Q)Q = q p,P = ln q 20 p19((( 4 , $&%(0/ >/! 2 $&#& (p, q) → (P, Q)√Q = cR (1 + q cos p) ,P = 2 (1 +√√q cos p) q sin p((( 4 , $&%(0/ >/! & + (+ $&& α β $&#&(p, q) → (P, Q)Q = q α cos βp,P = q α sin βp((( ^ 6 B , $&%(0(>(^!! 4 $&%(0/ >/ $&#&( +$&*+ (p, r) → (P , R)P =p+R = r,e∇α(r, t),c/0, #& $&#&/A → A + ∇α(r, t),ϕ→ϕ−1 ∂α(r, t).c ∂t%( &(' >* B&, $ Lα(r, t) G$&( (&( >(M!@ %&, , H = H(p, q) $&#&/(p, q) → (P, Q)⎧⎨ P = p,⎩ Q = μq + ∂ϕ(p, t) ,∂t,% μ = NQRPS ϕ(p, t)G %( >( 2 $&#& ((( 4 , $&%(0/>/ 4 J*K , K ! 2$&#&(px , py ; x, y) → (Px , Py ; X, Y )+$&*+⎧a ⎪=2pcosx−y,P⎪xx⎪2⎪⎪⎪⎪⎪⎪⎪1 ⎪⎪⎪2px sin x + py ,X=⎪⎨a⎪⎪a ⎪⎪− 2px sin x + py ,Py =⎪⎪2⎪⎪⎪⎪⎪⎪⎪⎪⎩Y =12px cos x + ya((( 4 , $&%(0/ >/!? 2 $&#& (p, q) → (P, Q)⎧⎨ Qi = qi ,⎩ Pi = p i −∂g(q, t),∂qi,% g(q, t) G &( %( >( ##0*+ &% & ((( 4 , $&%(0/ >/!) 2( * ,&'mẋ2+ fxL=2(f = NQRPS)$& (* % $&#&( (p, x) → (P, X)P (t) = p(t + τ ),X(t) = x(t + τ ) , $&%(0/ >/!F 2( %&, ,&, >(& ,&'mq̇ 2 mω 2 q 2L=−22$& (* % $&#&( (p, x) → (P, X)P (t) = p(t + τ ),Q(t) = q(t + τ ) , $&%(0/ >/!E 3> m %'0(( >& $ $*( >%&+ &%+ ,&'m 2ρ̇ + ρ2 ϕ̇2 + ż 2L=222−Uρ +z .;&- $&#&(pρ , pϕ , pz ; ρ, ϕ, z) → (P1 , P2 , P3 ; Q1 , Q2 , Q3 ) $&%(0 >ρF = P1 ρ2 + z 2 + P2 VWNSX + P3 ϕ.z1$ (* % B+ $&#& $& J*K, K # 034.% GH# % $&>$' %( /#, , $%#& & $&#& $&%(0 JK ,K = 0• & GH#•∂F+H∂t∂F, q, t = 0,∂qL!FM,% $&* &, > % $&% > #+% ##0*$* * $&%* $ /0##0* &%pi →∂F.∂qiL!FM * >( F & GH# L! M $&%(0( >( &, , ,$&#&( $&%(0, JK > K = 0• " + $&'(+ #+% &-&( GH# L! M G $* ,&G &&- (0 + $&*+ *+ $&*+ L & s + 1M & $& % + +% %%* #&•F = F (q1 , q2 , .
