Яковенко Г.Н. Краткий курс теоретической механики
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N
P
i=1
[r
i
, F
i
] = 0
˙
K
O
= M
O
− m[V
O
, V
C
] (17.3)
M
O
M
O
=
N
P
i=1
[r
i
, F
i
]
N
P
i=1
m
i
V
i
= mV
C
C
½
1. V
O
= 0,
2. O = C
¾
⇒
˙
K
O
= M
O
. (17.4)
M
z
M
O
z
K
z
=
O
C
OC
e
1
e
2
j
M
O
M
O
ωωω
ωω
= 0
O
e
1
e
2
e
3
ωωω
ωω
e
3
ωωω
ωω
= ωe
3
ω = ˙ϕ
t
e
3
ωωω
ωω
= ωωω
ωω
−ωωω
ωω
= ωωω
ωω
= ωe
3
= ˙ϕe
3
(17.5)
ωωω
ωω
= 0
O
K
O
=
N
P
i=1
[r
i
, m
i
V
i
] =
N
P
i=1
m
i
[r
i
, [ωωω
ωω
, r
i
]] =
=
N
P
i=1
m
i
{ωωω
ωω
(r
i
, r
i
) − r
i
(ωωω
ωω
, r
i
)} = ωωω
ωω
N
P
i=1
m
i
r
2
i
= I
O
ωωω
ωω
,
(17.6)
ωωω
ωω
⊥r
i
I
0
=
N
P
i=1
m
i
r
2
i
ωωω
ωω
= 0
i
= W
O
O
=
N
X
i=1
[r
i
, −m
i
i
] = −[
N
X
i=1
m
i
r
i
, W
O
] = −m[OC, W
O
]. (17.7)
ωωω
ωω
= 0
O
=
N
X
i=1
[r
i
, −m
i
i
] = −2
N
X
i=1
m
i
[r
i
, [ωωω
ωω
, ]] = 0. (17.8)
˙
K
O
= M
O
+ M
O
+ M
O
I
O
˙ω˙ω˙ω
˙ω˙ω
= M
O
− m[OC, W
O
]. (17.9)
V
O
O
W
O
½
1. W
O
= 0,
2. O = C
¾
⇒ I
O
˙ω˙ω˙ω
˙ω˙ω
= M
O
. (17.10)
I
O
¨ϕ = M
O
e
1
e
2
m¨x = R
x
, m¨y = R
y
, I
O
¨ϕ = M
O
, (17.11)
x y R
x
R
y
O
T =
1
2
N
X
i=1
m
i
V
2
i
=
1
2
N
X
i=1
m
i
(V
i
, V
i
). (18.1)
t
m
i
˙
V
i
= F
i
dT
dt
=
N
X
i=1
(m
i
˙
V
i
, V
i
) =
N
X
i=1
(F
i
, V
i
) = N. (18.2)
N
N
N =
N
X
i=1
(F
i
, V
i
), (18.3)
V
i
F
i
δA
δA =
N
X
i=1
(F
i
, dr
i
), (18.4)
dr
i
F
i
dt V
i
dt = dr
i
dT = δA. (18.5)
t ∈ [t
1
, t
2
] r
i
(t)
A
12
r
i
(t)
t ∈ [t
1
, t
2
]
A
12
=
Z
t
2
t
1
δA =
Z
t
2
t
1
N
X
i=1
(F
i
, dr
i
) =
Z
t
2
t
1
N
X
i=1
³
F
i
¡
t, r(t),
˙
r(t)
¢
, V
i
(t)
´
dt. (18.6)
T
2
− T
1
= A
12
. (18.7)
A
12
= 0 T
2
= T
1
V
i
= V
i
+ V
i
T =
1
2
N
X
i=1
m
i
(V
i
, V
i
) = T + T +
N
X
i=1
m
i
(V
i
, V
i
). (18.8)
ωωω
ωω
= 0
O C
O = C
V
i
= V
C
T =
1
2
N
X
i=1
m
i
(V
i
, V
i
) =
1
2
mV
2
C
. (18.9)
V
i
= V
C
V
C
= 0
N
X
i=1
m
i
(V
i
, V
i
) = (V
C
,
N
X
i=1
m
i
V
i
) = (V
C
, mV
C
) = 0. (18.10)
T =
1
2
mV
2
C
+ T , (18.11)
T
V
C
V m R
r
R
r
C
V
ω = V/R
ωωω
ωω
= 0
V
i
= ωr = V r/R
T =
1
2
mV
2
+
1
2
m
r
2
R
2
V
2
=
1
2
m
R
2
+ r
2
R
2
V
2
.
V
C
ωωω
ωω
T ωωω
ωω
= 0
ωωω
ωω
= ωωω
ωω
C
h
i
r
i
w
V
i
V
i
= h
i
ω h
i
C
ωωω
ωω
T
T =
1
2
N
X
i=1
m
i
(V
i
)
2
=
1
2
N
X
i=1
m
i
h
2
i
ω
2
=
1
2
I
ω
ω
2
, (18.12)
I
ω
=
N
P
i=1
m
i
h
2
i
ωωω
ωω
C
T =
1
2
mV
2
C
+
1
2
I
ω
ω
2
, (18.13)
V
C
C ω
I
ω
ωωω
ωω
C
m R
C V
0
V
C
C
G
F
P
G C
V
P
= 0 F
δA = (F, dr) = (F, V
P
)dt = 0
T =
1
2
mV
2
C
+
1
2
I
C
ω
2
=
1
2
mV
2
C
+
1
2
1
2
mR
2
µ
V
C
R
¶
2
=
3
4
mV
2
C
,
I
C
= mR
2
/2 ω = V
C
/R
T =
3
4
mV
2
C
=
3
4
mV
2
0
V
C
≡ V
0
r
i
i = 1, N
F
i
(t, r
1
, . . . , r
N
)
F
i
(t, r
1
, . . . , r
N
) F
i
(t, r
1
, . . . , r
N
)
Π(t, r
1
, . . . , r
N
)
F
i
(t, r
1
, . . . , r
N
) = −grad
i
Π(t, r
1
, . . . , r
N
) = −∇
i
Π(t, r
1
, . . . , r
N
) (19.1)
grad
i
Π(t, r
1
, . . . , r
N
) = ∇
i
Π(t, r
1
, . . . , r
N
r
i
F
i
(r
1
, . . . , r
N
)
F
i
(r
1
, . . . , r
N
) F
i
Π t
F
i
(r
1
, . . . , r
N
) = −grad
i
Π(r
1
, . . . , r
N
) = −∇
i
Π(r
1
, . . . , r
N
).
U = −Π
r
i
=
3
X
k=1
x
i k
i
k
, dr
i
=
3
X
k=1
i
k
dx
i k
, F
i
=
3
X
k=1
F
i k
i
k
. (19.2)
F
i
=
3
P
k=1
F
i k
i
k
F
i k
(t, x) = −
∂Π(t, x)
∂x
i k
. (19.3)
δA F
i
δA =
N
P
i=1
(F
i
, dr
i
) =
=
N
X
i=1
3
X
k=1
F
i k
(t, x)dx
i k
= −
N
X
i=1
3
X
k=1
∂Π(t, x)
∂x
i k
dx
i k
= −dΠ(t, x) +
∂Π(t, x)
∂t
dt.
F
i
(x) =
3
P
k=1
F
i k
(x)i
k
δA Π(x)
δA =
N
P
i=1
(F
i
, dr
i
) =
=
N
X
i=1
3
X
k=1
F
i k
(x)dx
i k
= −
N
X
i=1
3
X
k=1
∂Π(x)
∂x
i k
dx
i k
= −dΠ(x).
(19.4)
F
i
(r
1
, . . . , r
N
)
F
i
(r
1
, . . . , r
N
)
m
F
i
(r
1
, . . . , r
N
)
¤ ⇓
⇑
dΠ(x)
N
X
i=1
3
X
k=1
F
i k
(x)dx
i k
= −
N
X
i=1
3
X
k=1
∂Π(x)
∂x
i k
dx
i k
dx
i k
x
i k
¥
E
E = T + Π (19.5)
E
E = T + Π = const. (19.6)
¤
dT = δA
dT = −dΠ r
i
(t) t ∈ [t
1
, t
2
]
T
2
−T
1
= Π
1
−Π
2
E
2
= T
2
+Π
2
= T
1
+Π
1
= E
1