Яковенко Г.Н. Краткий курс теоретической механики
Подождите немного. Документ загружается.
R
r = CP = R(i
1
cos ϕ + i
2
sin ϕ)
ϕ r i
1
d
OP s = Rϕ
ϕ = s/R
τττ
ττ
=
dr
ds
=
dr
dϕ
dϕ
ds
= −i
1
sin ϕ + i
2
cos ϕ.
K =
dτττ
ττ
ds
=
dτττ
ττ
dϕ
dϕ
ds
= −
1
R
(i
1
cos ϕ + i
2
sin ϕ) = −
1
R
n,
ρ n
ρ = R, n = −(i
1
cos ϕ + i
2
sin ϕ).
V =
˙
r = R ˙ϕ(−i
1
sin ϕ + i
2
cos ϕ) = R ˙ϕτττ
ττ
,
V = R ˙ϕ
W =
˙
V = R ¨ϕ(−i
1
sin ϕ + i
2
cos ϕ) − R ˙ϕ
2
(i
1
cos ϕ + i
2
sin ϕ) = R ¨ϕτττ
ττ
+ R ˙ϕ
2
n,
W
τ
= R ¨ϕ W
n
= R ˙ϕ
2
r
r(q
1
, q
2
, q
3
).
q
1
, q
2
, q
3
q
1
, q
2
, q
3
P
q ,q ,q
1 2 3
0 0 0
r(q ,q ,q )
1 2 3
0 0
H
1 1 2 3
(q ,q ,q )
0 0 0
q
0
1
, q
0
2
, q
0
3
q
0
2
, q
0
3
q
1
r(q
1
, q
0
2
, q
0
3
)
q
1
H
1
(q) = ∂r(q)/∂q
1
q = (q
1
, q
2
, q
3
)
H
i
(q) =
∂r(q)
∂q
i
. (3.1)
H
1
(q)
H
2
(q) H
3
(q) q
H
1
(q) H
2
(q) H
3
(q)
H
i
(q) = |H
i
(q)| (3.2)
(H
i
, H
k
) = 0, i 6= k, (3.3)
r(t)
r(q)
q q(t)
V(q, ˙q) =
3
X
i=1
∂r(q)
∂q
i
˙q
i
=
3
X
i=1
H
i
(q) ˙q
i
(3.4)
V
2
(q, ˙q)) = (V, V) =
3
X
i, k=1
(H
i
(q), H
k
(q)) ˙q
i
˙q
k
. (3.5)
V
2
(q, ˙q)) =
3
X
i=1
H
2
i
(q) ˙q
2
i
. (3.6)
∂V
∂ ˙q
i
=
∂r
∂q
i
(3.7)
q
i
˙q
i
∂V
∂q
i
=
d
dt
∂r
∂q
i
, (3.8)
q
k
(H
i
, W) =
d
dt
∂
∂ ˙q
i
µ
V
2
(q, ˙q)
2
¶
−
∂
∂q
i
µ
V
2
(q, ˙q)
2
¶
(3.9)
W
(H
i
, W) =
µ
∂r
∂q
i
,
dV
dt
¶
=
d
dt
µ
∂r
∂q
i
, V
¶
−
µ
d
dt
∂r
∂q
i
, V
¶
(3.7),(3.8)
=
=
d
dt
µ
∂V
∂ ˙q
i
, V
¶
−
µ
∂V
∂q
i
, V
¶
=
d
dt
∂
∂ ˙q
i
µ
V
2
2
¶
−
∂
∂q
i
µ
V
2
2
¶
.
µ
∂V
∂z
, V
¶
=
1
2
∂
∂z
(V, V) =
∂
∂z
µ
V
2
2
¶
.
mW = F
H
i
T (q, ˙q) = mV
2
(q, ˙q)/2 Q
i
= (F, H
i
) = (F, ∂r/∂q
i
)
d
dt
∂T
∂ ˙q
i
−
∂T
∂q
i
= Q
i
(3.10)
W
1
H
i
(H
i
, W) =
1
H
i
½
d
dt
∂
∂ ˙q
i
µ
V
2
(q, ˙q)
2
¶
−
∂
∂q
i
µ
V
2
(q, ˙q)
2
¶¾
. (3.11)
W
i
W =
3
X
i=1
W
i
e
i
e
i
P
r ϕ z
i j k
r = ir cos ϕ + jr sin ϕ + kz. (3.12)
H
r
=
∂r
∂r
= i cos ϕ + j sin ϕ, H
r
= 1,
H
ϕ
=
∂r
∂ϕ
= −ir sin ϕ + jr cos ϕ, H
ϕ
= r,
H
z
=
∂r
∂z
= k, H
z
= 1.
r(t) ϕ(t) z(t)
V = H
r
˙r + H
ϕ
˙ϕ + H
z
˙z.
(H
r
, H
ϕ
) = 0 (H
r
, H
z
) = 0 (H
z
, H
ϕ
) = 0
V
2
= ˙r
2
+ r
2
˙ϕ
2
+ ˙z
2
.
m(H
r
, W) = m(¨r−r ˙ϕ
2
), m(H
ϕ
, W) = m(r
2
¨ϕ+2r ˙r ˙ϕ), m(H
z
, W) = m¨z (3 .13)
W
(H
r
, W) = ¨r − r ˙ϕ
2
, (H
ϕ
, W)/r = r ¨ϕ + 2 ˙r ˙ϕ, (H
z
, W) = ¨z.
z
x
y
i
r
r
j
k
z
j
P
i
1
r
B
r
O
e
1
e
2
e
3
O
i
3
i
2
r
i
1
i
2
i
3
O e
1
e
2
e
3
(e
k
, e
l
) = δ
kl
. (4.1)
t
(
˙
e
k
, e
l
) = −(e
k
,
˙
e
l
). (4.2)
r
0
(t) e
1
(t) e
2
(t) e
3
(t)
B
y
k
ρρρ
ρρ
=
3
X
k=1
y
k
e
k
, y
k
= const (4.3)
ρρρ
ρρ
OB e
1
e
2
e
3
r = r
0
+ ρρρ
ρρ
= r
0
+
3
X
k=1
y
k
e
k
, (4.4)
r
0
(t) e
1
(t) e
2
(t) e
3
(t)
y
1
y
2
y
3
B
B t
V =
˙
r
0
+
˙
ρρρ
ρρ
= V
0
+
3
X
k=1
y
k
˙
e
k
. (4.5)
˙
e
1
˙
e
2
˙
e
3
ωωω
ωω
t
ωωω
ωω
k
˙
e
k
= [ωωω
ωω
, e
k
], k = 1, 2, 3. (4.6)
ωωω
ωω
ωωω
ωω
=
1
2
3
X
i=1
[e
i
,
˙
e
i
]. (4.7)
¤
[ωωω
ωω
, e
k
] =
1
2
3
X
i=1
[[e
i
,
˙
e
i
], e
k
] =
1
2
3
X
i=1
{
˙
e
i
(e
i
, e
k
) − e
i
(
˙
e
i
, e
k
)} =
=
1
2
3
X
i=1
{
˙
e
i
δ
ik
+ (e
i
,
˙
e
k
)e
i
} =
1
2
(
˙
e
k
+
˙
e
k
) =
˙
e
k
.
a =
3
X
i=1
a
i
e
i
=
3
X
i=1
(e
i
, a)e
i
ωωω
ωω
1
ωωω
ωω
2
ωωω
ωω
1
ωωω
ωω
2
[ωωω
ωω
1
−ωωω
ωω
2
, e
k
] = 0, k = 1, 2, 3,
ωωω
ωω
1
ωωω
ωω
2
¥
t
ωωω
ωω
ρρρ
ρρ
OB
˙ρ˙ρ˙ρ
˙ρ˙ρ
= [ωωω
ωω
, ρρρ
ρρ
]. (4.8)
¤
˙ρ˙ρ˙ρ
˙ρ˙ρ
=
3
X
k=1
y
k
˙
e
k
=
3
X
k=1
y
k
[ωωω
ωω
, e
k
] = [ωωω
ωω
,
3
X
k=1
y
k
e
k
] = [ωωω
ωω
, ρ]. (4.9)
¥
t
ωωω
ωω
O B
V
B
= V
O
+ [ωωω
ωω
, ρρρ
ρρ
], (4.10)
ρρρ
ρρ
OB.
¤ t
¥
O V
O
= 0
B
V
B
= [ωωω
ωω
, r], (4.11)
r = OB.
ωωω
ωω
O B
ωωω
ωω
V
B
V
O
O ωωω
ωω
ωωω
ωω
e
1
, e
2
, e
3
ωωω
ωω
=
3
X
i=1
ω
i
e
i
,
ω
1
= (
˙
e
2
, e
3
) = −(e
2
,
˙
e
3
), ω
2
= (
˙
e
3
, e
1
) = −(e
3
,
˙
e
1
), (4.12)
ω
3
= (
˙
e
1
, e
2
) = −(e
1
,
˙
e
2
).
¤ ω
1
ω
1
= (eee
ee
1
, ωωω
ωω
) = ([eee
ee
2
, eee
ee
3
], ωωω
ωω
) = ([ωωω
ωω
, eee
ee
2
], eee
ee
3
) = (˙e˙e˙e
˙e˙e
2
, eee
ee
3
) = −(e
2
,
˙
e
3
).
eee
ee
1
= [eee
ee
2
, eee
ee
3
]. ¥
B
O
V
O
r
w
εεε
εε
ωωω
ωω
εεε
εε
= ˙ω˙ω˙ω
˙ω˙ω
. (5.1)
εεε
εε
ωωω
ωω
B
O
W
O
r
w
e
O B
W
B
= W
O
+ [εεε
εε
, ρρρ
ρρ
] + [ωωω
ωω
, [ωωω
ωω
, ρρρ
ρρ
]], (5.2)
ρρρ
ρρ
OB.
¤ t
¥
O W
O
= 0
B
W
B
= [εεε
εε
, r] + [ωωω
ωω
, [ωωω
ωω
, r]] = [εεε
εε
, r] + [ωωω
ωω
, V
B
], (5.3)
r = OB.
W = [εεε
εε
, ρρρ
ρρ
]
W = [ωωω
ωω
, [ωωω
ωω
, ρρρ
ρρ
]]