Яковенко Г.Н. Краткий курс теоретической механики
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{ωωω
ωω
i
} ωωω
ωω
i
O
i
{ωωω
ωω
i
}
a −a
{a
i
} i = 1, N
{a −a}
O
B
a
-a
O
a
O
B
a
-a a
B
a
a
a {−a a}
{a −a}
a b O
c = a + b
a
b
c a b= +
O
c O
O c
{a
i
} i = 1, N
R =
N
X
i=1
a
i
(12.1)
a
i
M
O
(a
i
) a
i
O
M
O
(a
i
) = [r
i
, a
i
], (12.2)
r
i
O a
i
A
O
M
O
OA
r
i
a
i
r
i
O M
O
(a
i
)
B
V
O
= 0 B
V
B
= [ωωω
ωω
, OB] = [ωωω
ωω
, −BO] = [BO, ωωω
ωω
] = M
B
(ωωω
ωω
), (12.3)
M
O
{a
i
} i = 1 , N M
O
(a
i
)
M
O
=
N
X
i=1
M
O
(a
i
) =
N
X
i=1
[r
i
, a
i
]. (12.4)
M
O
R M
O
O A
M
A
= M
O
− [OA, R], (12.5)
¤
M
A
=
N
X
i=1
[ρρρ
ρρ
i
, a
i
] =
N
X
i=1
[r
i
− OA, a
i
] =
N
X
i=1
[r
i
, a
i
] − [OA,
N
X
i=1
a
i
] = M
O
− [OA, R].
¥
M
A
R = 0
{a −a}
a −a
R 6= 0
O A
M
A
= M
O
O
A
R
M
A
M
O
O A
(R, M
A
) = (R, M
O
). (12.6)
(R, M
O
) R
M
O
(R, M
O
)
O A
R
Ïð
R
M
A
A
M
A
B
M
B
O
M
O
Ïð
R
M
O
R
R
M
A
= (R/R, M
A
) = (R/R, M
O
) =
R
M
O
(12.7)
[R, M
B
] = 0
B [R, M
B
] = 0
min
B
M
B
=
1
R
(R, M
O
). (12.8)
R
B
OB ⊥R O M
O
M
B
= M
O
− [OB, R] R [R, M
B
] = 0
OB ⊥R
0 = [R, M
B
] = [R, M
O
] − [R, [OB, R]] = [R, M
O
] − OB(R, R) + R(R, O B) =
= [R, M
O
] − OBR
2
,
B O B = [R, M
O
]/R
2
r =
1
R
2
[R, M
O
] + λR, (12.9)
λ
M
B
B
a
i
a
i
{a
i
} i = 1, N
{a
i
} i = 1, N
M
O
O R 6= 0 R = 0
M
O
O
B a
i
{a
i
} i = 1, N
B {a
i
−a
i
}
a
i
a
i
-a
i
a
i
M
i
B
+{ }ïàðà
~
B
a
1
M
1
B
+{ }ïàðû
M
N
a
N
...
...
a
1
a
N
...
B
M
B
R
a
-a
~
~
a
i
a
i
B {a
i
−a
i
} M
i
a
i
{a
i
} i = 1, N {a
i
}
a
i
B {a
i
−a
i
}
M
i
a
i
B
R =
N
P
i=1
a
i
B
{a −a} {b −b}
m
M
1
M
2
{a −a} {b −b}
M
1
= M
2
= M
¤ ⇓
⇑
⇑
¥
a
b
c
-a
-b
-c
A
B
M
1
M
2
M = M
1
+ M
2
¤ M
1
M
2
A B
{a −a} {b −b} a b
A −a −b B M
1
= [AB, a] M
2
= [AB, b]
A
B
M = M
1
+ M
2
= [AB, a] + [AB, b] = [AB, a + b] = [AB, c]. ¥
B
M
B
=
N
P
i=1
M
i
a
i
B
M
B
M
B
B M
B
{a
i
} i = 1, N
M
O
O
R =
N
X
i=1
a
i
, M
O
=
N
X
i=1
[OO
i
, a
i
], (13.1)
O
i
a
i
R = 0
R 6= 0
r =
1
R
2
[R, M
O
] + λR, (13.2)
λ
B
M
B
=
(R, M
O
)
R
2
R. (13.3)
M
B
{a −a} M
B
{a
i
} {b
i
}
m
{a
i
} {b
i
}
M
O
O
¤ ⇓
M
O
⇑ M
O
{a
i
}
{b
i
} {a
i
}
{b
i
} ¥