Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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2
◦
y
2 4
◦
x y
xz yz
3
◦ ◦
z
y
x
O O
C
y
x
C C
H
H H
H
y
xB
Cl Cl
Cl
2 2 4 3
L(x,
˙
x, t)
x →
ˆ
S x ,
˙
x →
ˆ
S
˙
x , (24.1)
S
ij
ˆ
S
ˆ
S =
ˆ
S
T
,
ˆ
S
ˆ
S =
ˆ
E . (24.2)
S
x = A cos(ωt + ϕ)
x →
ˆ
Sx
ˆ
Sx
ˆ
Sx
x
ˆ
Sx = λx
ˆ
S
ˆ
S =
ˆ
E
ˆ
S
ˆ
Sx = λ
ˆ
Sx = λ
2
x = x λ = ±1
x = A cos(ωt+ϕ)
ˆ
SA = A
ˆ
SA = −A S
ˆ
S x
x
x ±
ˆ
Sx = (A ±
ˆ
SA) cos(ωt + ϕ)
ˆ
S
ˆ
S = E
P
j
S
ij
S
jk
= δ
ik
120
◦
S
F(t)
ˆ
SF(t) = +F(t) .
x
a
ˆ
Sx
a
= − x
a
ˆ
S F(t),
ˆ
S x
a
= −(F(t), x
a
) .
ˆ
S F(t),
ˆ
S x
a
=
F(t),
ˆ
S
T
ˆ
S x
a
= + (F(t), x
a
)
(F(t), x
a
) = 0
k k k
m m
y
x
A B
AB x
1
x
2
◦
y
L =
1
2
m( ˙x
2
1
+ ˙x
2
2
) −
k
2
x
2
1
+ (x
1
− x
2
)
2
+ x
2
2
(24.3)
x =
x
1
x
2
→
ˆ
S x =
−x
2
−x
1
,
ˆ
S =
0 −1
−1 0
. (24.4)
ˆ
S
ˆ
S x
s
= +x
s
ˆ
S x
a
= −x
a
x
s
=
1
−1
a
s
cos(ω
s
t + ϕ
s
) ,
x
a
=
1
1
a
a
cos(ω
a
t + ϕ
a
) .
x
s
x
1
x
a
x
2
A
B
x
A
= x
B
= b cos(γt + ϕ).
F(t) =
1
1
k b cos(γt + ϕ) .
ˆ
S F(t) =
−F(t) x
a
x
s
γ → ω
a
k k
k k
m
m
m m
1
3
2 4
k k
k k
2m
m
m m
1
3
2 4
N 3N
3N
r
1
, r
2
, . . . r
N
N
X
a=1
m
a
r
a
= 0 , (2 5 .1 )
m
a
a u
a
a
r
a0
r
a
= r
a0
+ u
a
,
˙
r
a
=
˙
u
a
. (25.2)
N
X
a=1
m
a
r
a0
= 0 ,
N
X
a=1
m
a
u
a
= 0 . (25.3)
L(u,
˙
u) =
1
2
N
X
a=1
m
a
˙
u
2
a
−
1
2
N
X
a,b=1
u
a
,
ˆ
k
ab
u
b
, (25.4)
ˆ
k
ab
m
a
¨
u
a
+
N
X
b=1
ˆ
k
ab
u
b
= 0 (25.5)
x
(α)
= (u
1
, . . . , u
N
) , (25.6)
ω
α
ω
0
= 0
V
x
(0)
= (Vt, . . . , Vt) , (25.7)
a
Vt ·
N
X
a=1
m
a
u
a
= 0 ,
V
ε = Ω δt
x
(0)
= ([ ε, r
10
], . . . , [ε, r
n0
]) , (25.8)
a
N
X
a=1
[ε, r
a0
] · m
a
u
a
= 0
ε ·
N
X
a=1
m
a
[r
a0
, u
a
] = 0 .
ε
N
X
a=1
m
a
[r
a0
, u
a
] = 0 . (25.9)
3N −6
z
x y
u
a
3N − 5
2
C
2
H
4
AB CD
N m
k
l
X
n
= n · l
A B
k k k k k
m m m
X
1
X
2
X
N
A B
y
x
l
0
f = k(l − l
0
)
y
x
y
n−1
y
n
(n − 1)l nl
α
x
n
n
y
n
n y
n
n
F
n
= − f sin α = −f
y
n
− y
n−1
l
L =
m
2
N
X
n=1
˙y
2
n
− −
f
2l
N+ 1
X
n=1
(y
n
− y
n−1
)
2
, (26.1)
y
0
y
N+ 1
y
0
≡ 0, y
N
≡ 0 . (26.2)
¨y
n
+ ω
2
0
(2y
n
− y
n−1
− y
n+1
) = 0 , n = 1, 2, . . . , N , (26.3)
ω
0
=
p
f/(ml)
y
n
(t) = Re
e
iωt
f(X
n
)
, X
n
= n · l , ( 2 6 .4 a)
f(X
n
) n
l x
Re
e
iωt
f(X
n
+ l)
(26.4b)
f(X
n
+ l)
f(X
n
) λ
f(X
n
+ l) = λf(X
n
) (26.5a)