А.Б. Пименов - Задачник по теоретической механике, страница 2

. . , qs)4#+%* $> U (q1, q2, . . . , qs) q0i ((( &# B&∂U (q0 )= 0.∂qiL??M2 $> U (q1, q2, . . . , qs) > + $&*+ L&& ;&M • #* ,* &* &>* 2U (0 &*+ ∂ U (q) *+ $&%*+ $> uij = ∂q ∂q (U )ij = uij i j q**+ B& q0i L??M #* $'*0δ1 = u11 > 0,u u 11 12δ2 = u21 u22 > 0, u11 · · · u1s δs = us1 · · · uss...

> 0.&( D,&' %( * ,&' L? M G&( *+ *+ #•tij ẍj + uij xj = 0.L?)Mj:- %&% * *+ %&>*+& L?)M 0( % $&%+ >xj (t) = d_ Aj eiωtL?FM $ $% Aj % L?FM &( L?)M $&% %&% *+ ,#&+ &2− ω tij + uij Aj = 0,L?EMj&( & &- $& & / $&%(2L)Me_S − ω tij + uij = 0.=&& & L)M #0 s &-2ω(i) &* $&%(/ #* * #* ω(1), ω(2), . . . , ω(s),( $% ω = ω(k) (k = 1, s) L?EM %('% # * +% $/ $% A(k)j " , *+ # L#0 &- *%&>*+ & L?)MM %xj (t) = d_(k)Ck Aj eiω(k) t .L)Mk6$* * Ck , k = 1, s L2s 0*+ M% +%(( 2s *+ xi (0) = x0i ,ẋi (0) = v0i ,i = 1, s.L)M" # * &( /0, , &-( * L?)M %( > & " L?FM , #+% $ %•(ω=0)xj(t) = d_ (Aj (a + bt))L)!M $* a b LAj G $( $%L#* & &>* * L?EMM /0 M• " &*+ #*+ ω(n) &*+ %&>*+ & % /0,, &-( * L?)M -( '%&/ N * ω(n) K *+#*+ & v1, v2, . . .

, vK &>* * L?EML$&%( &, rM & ω $' &% & ω(n)K = s − r,,% s G && &>* * Ls × sM &* $% $ #%* # * B & ,#& L?EM9 K = N /0 &- &(, L?FM $% & #>#*+ & vixj (t) = d_(n)NL)@MCi vji eiω(n) t ;i=19 K < N &- * L?)M , #0 &%' ( % $&%( , $ t $ (N −K) eiω t⎧N −K⎪= d_ a11 + a12 t + . .

. + a1(N −K) teiω t ,⎪⎨ x1L) M···⎪⎪⎩ x = d_ a + a t + . . . + aN −Keiω t .ss1s2s(N −K) t(n)(n)(n)7% $ "&-& '% &/+&&, &( & N &N *+ &- * ,#&+& L?EM L %( '% # * N .& ' N *+ #> $%M K < N %( & $*/0+ #** ' ;% &*+ /0 &- $*( ,% % L)@M4' &-/ %( &*+ %L) M $( &( # %&'0+ &(% B$>* ' $*&* ' $&& #* +&( B&,• ; $*+ $% /0+ &*#* ω(k) ω(n) Lk = nM ((( + &, T LT G &> %& &* B&,(0 + B> tij MAT(k) T A(n) = 0 (k = n)L)?ML T % $% AT &$&M• -+ $>*+ %/0+ # * 0(( $ %#( ,&' L $&%$' - ##0* *((/( >( - & Qi = Qi(t)M ,,ΔL =sL))MQi (t)qi .i=1* ,&' # * $%&'%/ %*+ -+ %& $&#' $* ( $'( &(•(2)L=(2)L0sss11+ ΔL =tij ẋi ẋj −uij xi xj +Qi (t)xi .2 i,j=12 i,j=1i=1L)FM&( *'%*+ # G %&%( *+ %&>*+ &tij ẍj + uij xj = Qi (t),L)EMj&- & $&%( # #>/ #0, &-(%&% * , &-( %&% *#0 %&(t) + x %(t)xi (t) = xLFMii7#0 &- %&% * *+ *+# L)M3 &- %&% * %%&% * L)EM " ,% - * G$&% > * ΩQi (t) = d_ fi eiΩt,LFM &- * L)EM 0( % $&% > ' * Ω%iΩtx(t) = d_ Bk eLFMk $ $% Bi &( +%( ,#&* $ $% LFM L)EM2LF!M−Ω T +U B =FL & % ,% &>* T U #&* B>tij uij #>* B F G B Bi fi MB=2−Ω T +U−1LF@MF.4&* &%* ξiG B ##0* &%* &*+ ,&' # * $&%( # ,&' %&*+ ,&+ >(& &* #* # *•L=i1 ˙2 1 2 2ξ − ω ξ .2 i 2 (i) iLF M2( +'%( &*+ &% #+% & &&* %> L T M $*$%* Ã(k)i Ã(i)T T Ã(k) = δ ik .LF?M• & &>$% Ã⎛(1)Ã1⎜ (1)⎜ Ã2A=⎜⎜ ⎝(1)ÃsA $ #> &&*+⎞(2)(s)Ã1 .

. . Ã1(2)(s) ⎟Ã2 . . . Ã2 ⎟ ⎟⎟,⎠(2)(s)Ãs . . . Ãs(j)Aij = Ãi .LF)M! 4&* &%* ξi +%(( $&#& ##0*+&% x → ξ xi =Aij ξjLFFMj & %x = Aξ.LFEM&> A %&( / , LF?MAT T A = I,LEM $( #& LFEM $ L &%Mξ = AT T x.LEM@! 4 #* * * ,&'L = ẋ2 +5 25ẏ + 3 ẋẏ − x2 − 4y 2 − 6xy.22@@ 4 #* * * ,&'57L = ẋ2 + ẏ 2 + ẋẏ − 2x2 − y 2 + xy.22@ 4 *+ # # m % ,%/$> , % $&#* y = ax2 La > 0M +%(0( & $ %&% $ (' g = −gey $&*+ (+ x(0) = 0 ẋ(0) = v0@? 4 *+ *+ # * (0 %+-& 3m 2m %+ %*+ $&' '(k '%(@) 4 *+ # %+ # m '%((*+ %&, %&, $&' '/ k *+ ,%/ ,&/ $>@F 4 *+ *+ # &+ # m M m (*+ %( %* $&' '/ k '%( *+ ,%/ ,&/ $>@E &( m %'( $ (' g = −gey$ & y = a/x + bx La, b > 0M &$' &$ 4 *+ *+ # * & +% $' &( / & v(0) = v0ex 4 *+ # * ,&'16q̇12 + 11q̇22 + 22q̇32 + 12q̇1 q̇2 + 22q̇1 q̇3 + 28q̇2 q̇3 −L=21− 9q12 + 38q22 + 49q32 + 30q1 q2 + 40q1 q3 + 82q2 q3 .2  4 *+ *+ # &+ # m'%( *+ ,% ,& > &% R (*+ %&, %&, &( %* $&' '/k '%( 6#( $*( ,&'L=1q2 q̇12 + q1 q̇222−1+ q1 + q2 .q1 q24 *+ *+ # '*+  ! 6#( $*( ,&'1 21112q̇1 + q̇1 q̇2 + q̇2 − cR q1 q2 −L=− − .2q1 q2 q1 q24 *+ *+ # '*+  @ 4 *+ # * &( $*(,&'L = q̇12 + q̇1 q̇2 +5 2q̇2 − 3 q12 + 2q1 q2 + 3q222$& %*+ *+ (+ q1(0) = q2(0) = 0 q̇1(0) = 0q̇2 (0) = 3 4 *+ #%*+ # (*+ ( $&%*+ *+ (+ α1(0) = α0 α2(0) = 0 α̇1(0) = α̇2(0) == 0 * ( m %* l ' $&'* k & #%, $%( g ? f& m1 m2 &$* $&' '( k 2k 3k , # &( #( % $>* , %$&' &% R ; &$' ,&$ & ,&' *+ #%*+# ; 2m1 = m2 ) 3> m &(% q > 0 %'( $ $&+&, , 2α $& &- $ (' g =−gez # $( B& $ E = E0 ex " &- &$ * &(% Q > 0 4 * $%**+ # # $'( &( F 4 *+ # # m % ,%/$> , % $&#* y = ax2 La > 0M +%(0( & $ %&% $ (' g = −gey$& # % -(( $&( F = ex F0 PZR ω0 t E 4 *+ *'%*+ # * (0 %+ -& 2m m %+ $&' '( 2k k -& 2m % -(( ,&( F = F0 PZR Ωt ; $ (' ? 4 *+ *'%*+ # * (0 %+ -& 2m m %+ $&' '( 2k k &- #( $ x = A PZR Ωt ; $ (' ? 4 *+ *'%*+ # * (0 %+ -& 3m 2m %+ $&' '/ k '%( -& 2m % -(( ,&( F = F0 NQP Ωt ; $ (' ? 4 *+ # %+ # m '%((*+ %&, %&, $&' '/ k *+ ,%/ ,&/ $> ( # $**% ,& * F1 = f1 NQP Ωt $&( G F2 = f2 PZR Ωt?! 2 # m '%( (* &( %*$&' '/ k '%( * ,%,& > &% R 8 A &$( %+ $&'#( $ % αA(t) = a PZR Ωt 4 *+# #?@ f& m M &$* $&' '( k , # &( #( % $&( ,&$ 4 -& M % -(( $&%( F1(t) = f1 cos Ωt -& m G F2(t) = f2 sin Ω t& ,&' * *+ *'%*+# ; M = 3m/2? f& m M &$* $&' '( k k/2 , # &( #( % $&( ,&$ &( &- *'%* #( $ x(t) = X0 cos Ω0t & ,&' *+ # ; M = 4m?? 2 -& 2m 3m/2 & $&'* '(k k 2k #&/ #/ #&'/ & f& , %,( ,& $&D( &- *'%* #( , X1 (t) = A1 sin Ω1 t $&( G , X2 (t) = A2 cos Ω2 t4 *+ *'%*+ # *?) 2 %* # m '%( * $>,/ $ $&# y = ax2 La > 0M &$' & $ (' g = −g ey (* $&''/ k 4 *+ *+ # &*&%* &( % %&& $&'* &/ :& # $&#&?F 2 # m '%( * ,% ,&> &% R (* $&' '/ k $%&'* %/ -+ $&%+ $&*+ $ > F1 = f1 PZR Ωt F2 = f2 NQP Ωt 4 # #?E 2 -& m '%* % $&'* '( k * &' % l #&/ #/ ; &$' & $ +%( %&% $ (' g "&+ -& % $>* '%,( ,& $& $**% ,& * F (t) = F0 sin ω0t &,&' *+ *'%*+ #) 6 -& 2m $&&$ % $&''( 2k k/2 % L $%- %&,-& m/2 &* -& ' %,( % ,& $&( & G & $&( &- *'%* #( , X(t) = A cos(ωt+ϕ0 ) 4 *'%*+ # -&) 2 ,& m 2m (* $&' '( kk 2k , %,( % $&( ,& $D( &- #( x (t) = A cos ω0 t 4 *+ *'%*+ #*) 2 -& 2m m & $&'* '( 3k/2 k/2 2k #&/ #/ #&'/ &f& , %,( & $& %&%$ (' "&+(( &- *'%* #(, X(t) = A cos Ω4 t 4 *+ *'%*+# -&)! 4 &* &%* # *#&' & & ,&' &%)@ 4 &* &%* # *#&' & & ,&' &%) 4 &* &%* # *#&' & & ,&' &%)? 4 &* &%* # *#&' & & ,&' &%)) 4 &* &%* * $* ,&'L=m1 11ẋ21 + ẋ1 ẋ2 + ẋ22 − k x21 − k x22 − k x1 x2 .222)F 4 &* &%* # *#&' & & ,&' &%)E 4 *+ # &* &%* * ,&' Lγ = NQRPSM 11222 22 2m1 ẋ + m2 ẏ − ω1 x + ω2 y + γ xy.L=22F 4 *+ # &* &%* * ,&' Lβ = NQRPSM1k 2222m1 ẋ + m2 ẏ + β ẋ ẏ −x +y .L=22F ; $*( ,&'L=1 2 2q q̇ + q22 q̇222 1 1−1211+ 2 + 2q1 q2 .2q1 q24 *+ *+ # '*+ 4 &* &%* & ,&' &%F 6#( $*( ,&'mẋ21kx21 kx222+ mẋ2 −−−L=2224 *+ # * x1 (0) = 0,ẋ1 (0) = v1 ,x2 (0) = a,ẋ2 (0) = v2 .3kx1 x284 &* &%*F! 6#( $*( ,&'m 22L=ẋ + ẏ + mΩ xẏ − y ẋ + mg l2 − x2 − y 2 .2;( Ω2 << g/l *+ *+ # $&*+ (+x(0) = 0,ẋ(0) = V,y(0) = 0,ẏ(0) = 0.4 &* &%*F@ & ,&' * %+ (*+ (#&' & $&#' *+ # ;&- #( & $ %&% $( $ (' g 4 *+ # &* &%*F ; $*( ,&'22L = a ẋ + ẏ + β ẋẏ + b l2 − x2 − y 2 .4 *+ # '*+ &* &%* La, b, β, l = NQRPSMF? 2 # m 4m %* $&' '( k 4k '* ,&/ $> ,/ & &' &% R " $&#' *+ #$& ,&' %'( 4 &*&%*F) 2 # m 2m %* $&' '( k2k 2k '* ,%/ ,&/ $> ,/ %$&' &% R & ,&' $&#'*+ # *+ #%*+ # 4&* &%*FF 2 # m 4m '* $&(/ ,%/ $>/ $% , α ,& +%(( &%&% $ (' g :( % & '% &$( $>* & h 4 *+ #* &* #FE 2 -& m '%* $%-* *+ $&'+'( k , %,( % & %&% $( $ (' & ,&' #** # * *+ # &,&' & % &* &%*/ $ •8& &> * N *+ Jij =Nmα r 2α δij− xα i xα j ,LEMα=1,% rα xα i G &%.& , $* * $%& α &% ' ( L&*+ M >& 4$&&** , LEMJij =2dm r δij − xi xj =2d x ρ(r) r δij − xi xj ,3LE!M,% ρ(r) G #A( $ &$&%( * Ld3x ≡ dV GB #A &%, M• 8&% ' $ #, &*+ &( '% /#* %( & ( * &• " #0 &% ? $ #%* %( %, %( , $'( &#( ? ##0*+&% L& &%* $&%(/ $' >& 0 & %/ &>/ &+ ' ( &%* * &% $%' *&%M• 6( B&,( &%, 3112T = MV +Jij ωi ωj ,22 i,j=1LE@M,% M G &%, V G & %'( >& &%, ωi G $&> & , & &0(&%, * &% ' ( •D,&' &%, 3112L = MV +Jij ωi ωj − U (r, t),22 i,j=1LE M,% U (r, t) G $>( B&,( &%, • "*#&( %'* #& $&( * &%' ( &%* ' %#( %,>& &>⎛⎞0⎜⎟Jij = ⎝ 0 J2 0 ⎠ .0 0 J3J1 0LE?M2,* $* J1, J2, J3 & &> */,* &> &%, 7 * &% & & &> %, */ ,* (&> &%, • 9 * $* & &> Jij &% >& $* Jij &&> %&, &% & >& & a L & a $% >& M, #* ** $ &Jij=Jij2+ M a δij − ai ajLE)ML& f&M• ,* b& θ, ϕ, ψ $(/ % &>/ *&% x, y, z ' ( &%* $ -/ ( $%' * &% x, y, z θ G , > G %( , '% ( z z L*( $& z z 0 ≤ θ ≤ πMTϕ G , $&> G %( , '% / x ON L*( $& x ON 0 ≤ϕ < 2π MTG , #, &0( G %( , '% / x L*( $& ON x 0 ≤ ψ < 2πMD( ON G ( $&( &%*+ $ xy x y ψ6 &( b& $(/ $ $&>& , & $&, &0( &%, * &% x, y, z ' ( ωx = θ̇ cos ψ + ϕ̇ sin θ sin ψ,LEFMωy = −θ̇ sin ψ + ϕ̇ sin θ cos ψ,LEEMωz = ψ̇ + ϕ̇ cos θ.LM•E " &-+ %& & 2a &$'* * *m M 4 $* & &> M x, y, z T #M x , y $%/0+ %,( %& z Lz $&$%(& $ &ME 4 ,* &> ,* * &> *(0 *+ m M &$'*+ &-+$&(, & 2a 2bE 4 ,* &> ,* * &> *(0 *+ m 2m &$'*+ &-+$&(,, &, 2a 4aE! 7$&% * &> %&%, $&#% &0(* h &% a $ $&+ &% >& E@ 4 ,* * &> $ %&%$&* m &% RE 2 m &% R *( # $&*( $ $ 1$ ,&' % 4 ,%'(E? 6>* , &'( m % l ( $ $>, $ $&# y = ax2 La > 0M & &$'/ y %&% $ (' 4 $&% *+ #&'(E) 7%&%* $* $>%& m &% R +%( -&+ ,& $&+ ' &-* $$&* #( 4 $&% B+#EF m &% R ' %,( $&*#& $ ,& $&+ & ,&'* %'( %&&+ * *&> * J1 = J2 = J0 J3 = J EE ;&' m % l -&& &$ &-$&(, , $ &, &$' & 6 >&'( -&& $&&$ ' &' > &, $ ,% ,& $&+ ; +%( %&% $ (' & ,&' * %'( 4 / B&,/ %&%, (0,( #$&*( $ ,& $&+ m , * h &% ( R & ,&' %'( %&&+ %(%&%, &, m &( ' % , >& l ,* * &>J1 = J2 J3 " ##0*+ &% $ ,*b& 7%&%* >%& m &% r $&*( #$&*( $ >%& $&+ &% R %&% $ (' & ,&' >%& 4$&% , *+ #! 4 $%'* ,&* >%& &% R $'#/ -&+* #& m $&(,*$$&* * 2l $&% $&#& $&$%(& >%& 7$&% $&% *+# #& , ,* &> $&+%(0 & , >& $& >%&& ma2@ 4 *+ # %&%, , &'( m % l >* &, ( $ &$' & $ ,% #& &% R * &'( &(( % %, + ,> 4 * &0( &, &, & $& & (((* m &( ' % , >& l ,* * &> J1 = J2 J3? 4 * &0( % $, &#& $& & ((( * % m ,&% R ,* * &> J1 = J2 = J0 J3 = J ) 4 / & (0,( % $& & ,%' ((( * % m , &% R ,* * &> J1 = J2 = J0 J3 = J F 7%&%* >%& m &% R $& %&%* &' m &$'* &( b >%& $& Y%& ' # $&*($&*( $ ,& $&+ 4 *+# >%& # $'( , &( )! #!#  0• >( , LB & #% '$M $&%(( -H = H(p, q, t) =si=1pi q̇i − L(q̇, q, t),L!Mq̇i =q̇i (p,q,t),% $&%$,( $& #+% *&##0* & q̇i & $&* pi, qi * &pi =∂L∂ q̇i(i = 1, s),$&%(/0+ ##0* $* > ##0*+&% &• : L!M $(/0 $ % ,&' L$& , H *&' $&( $&#&D'%& & , & ( > ' #*0 $& *$ (e_S∂pi∂ q̇j≡ e_S∂ 2L∂ q̇i ∂ q̇j= 0.L!M7#& $&#& D'%& $( $ %, H $& &' L•L(q̇, q, t) =si=1pi q̇i − H(p, q, t),L!!Mpi =pi (q̇,q,t),% $&%$,( $& #+% *&##0* $* pi & $&* q̇i, qi *&q̇i =∂H∂pi(i = 1, s) , & ( > B ' #* 0$& *$ (e_S∂ q̇i∂pj≡ e_S2∂ H∂pi ∂pj2 * , &( •= 0.H(p, q, t)⎧∂H⎪⎨ q̇i =,∂pi∂H⎪⎩ ṗi = −+ Qdi .∂qiL!@M$*(L! M,% Qdi G ##0( %$( L MQdi=Nα=1F dα ·∂r α.∂qiL!?M * , $&%( # ##0/B&,/ * $/ + $&*+ p, q•H(p, q, t) = E(p, q, t).L!)ME $&% & D,&' %/&(  $&% & %/&( D,&' & , * &( $*(,&'m 22 2222L=ṙ + r θ̇ + r PZR θϕ̇ − U (r).2 & ,&',*! & ,&',*&(p2ϕp2ρ+H=+ U (ρ, ϕ).2m 2mρ2H=@ & ,,&' & ,,&'L = −mc2$*(2e1p − A(r, t)2mcL = −mc&($*(*+ eϕ(r, t).&($*(2ṙ 21 − 2 − U (r, t).c*&($*(ṙ 2 e1 − 2 + A(r, t) · ṙ − eϕ(r, t).cc? & ,&',*&($*(2p2ρ1eH0 2p2zp+ρ+ mgz.H=−+ϕ2m 2mρ22c2m) & ,&',*H=&($*(c 2p,n(r),% n(r) G %( >( &% 7#A( $*&F %( * ,c 2H=p,n(r) L!@M &-E & %( * ,H=cn(r)p2 ,% $&( ;n(r)PZR θ = NQRPS,,% θ G , '% & & ṙ & ∇n(r) f& m &(% q $%- % l &- #( & $ %&% $(' g # ,& B& $ E &, -& 1$ &(  3> m &(% q %'( $ $&+ , $, && 2α %&% $ (' g & B& $ E & ,>* m &(% q $> ,/ $$&# y = bx2 Lb > 0M &( &0( &, y $(, &/ ω %&%*+ $(*+ $ (' g , $ H & , #! m &(% q $&(/ $> &(&0( &, & $( , &/ω #&( $(* , α &/ %&%*+ $(*+ $ (' g , $ H &, #@ 2 # m1 m2 &(% q '%( * $> ,/ $ $&# y = ax2 &( &0( &, y $( , &/ ω %&%*+ $(*+$ (' g , $ H (* $&''/ k % l0 %&& ( &, * D,&' , ( %122L = mϕ̇ − mω 1 − NQP ϕ .2& , ( *#&( ##0&%*ϕx = 2 PZR .2? 3> m &(% e %'( %&% $(, $ H = H0ez " #& &, $>Ax = −H0 y,Ay = Az = 0$& , ; &( %'(# 1.