ΘΕΩΡΗΤΙΚΗ ΜΗΧΑΝΙΚΗ ΙΙ - astro.auth.grvarvogli/intro_analyt_dynamics.pdf · I0=IK...
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Transcript of ΘΕΩΡΗΤΙΚΗ ΜΗΧΑΝΙΚΗ ΙΙ - astro.auth.grvarvogli/intro_analyt_dynamics.pdf · I0=IK...
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, . : 2008-09
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. k , n=3N-k .., 3- (, n=2N-k . 2- ). n=6N-k . 3- (3 3 .. ) n=3N-k . 2- (2 1 ..).
, qi, ' , . , , , . , ( ) , . , .. ( ), r - , , . r=l, , x2+y2=l2.
.. qj. , qj.:
r i=ri q j ,t (1)
, , . t. T . ,
T=12 mi r i
2=T q j , q j , t (2)
T=12 m i vi , O
2 v i ,O imir i , K 12 I i i2 (3)
vi0 i mi, i , Ii ( ) i riK Ki i. (.. ). , ( Steiner)
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I 0=I Km 2
, ( ) ( ). .
, 2 , q j . qj, V=V(qj,t) . , Lagrange :
L q j , q j ,t =TV q j , t (4)
Hamilton, (q1,t1) B(q2,t2) - ,
J =L q j , q j , t dt=0(5)
( ) (, ) . , Euler, () Lagrange:
ddt L q j L q j=0 j=1,. .. , n (6)
p j=L q j (7)
. , ( L),
Lqk
=0
(8)
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ddt
pk0=0 pk= (9)
() . Lagrange , Lagrange, , Jacobi :
L t
=0 J=j=1n
q j L q j L= (10)
(. r i=r iq j ), Jacobi , . J=E=T+V.
: Hamilton, Lagrange, L q j , q j ,t L ' q j , q j , :
L=L ' f t L=L ' ddt
g q j , t (11)
(., Lagrange). , Lagrange , Jacobi. . L . , Jacobi , L' ( L).
Legendre, Hamilton:
H=j=1n
p j q jL=H q j , p j , t (12) Hamilton:
q j= H p j
, p j=H q j
(13)
Lagrange. , H , . (7) qj . (12). , Jacobi, . , H
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( ). (qj,pj) () .
, (Lagrange Hamilton) ( ) . , , , . , . qj,0 Lagrange ( Hamilton), ( ), :
q j= q j=0 (14)
qj, ,
H p j
=0=H q j
(15)
, , V=V(qj),
L q j
=H q j
=0=V q j
(16)
, 1 .., , . qj=qj0 . 2 .., V(q1,q2). n>3, . : () , () ri=ri(qj,t). , , V=V(qj) , n>2. , .
, , . j qj qj0,
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j=q jq j ,0 j= q j j=q j (17)
, Lagrange j , j Lagrange j , j , j. . j , j () Taylor, 1 j , j , j (.. 1, 20 ) . , 1 ..,
0 = 0 (18)
0 qj0. ,
t =D cos t 00 (19)
t =C 1 et C2 e
t 00 (20)
, . , . (A02 .., , j, , ( ri=ri(qj), . ).
: 2 .., , (. ), . , .
