Яковенко Г.Н. Краткий курс теоретической механики
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¥
F = ax
α
y
β
i + bx
k
y
l
j + 0k
a α β b k l
Π(x, y, z)
∂Π
∂x
= −ax
α
y
β
,
∂Π
∂y
= −bx
k
y
l
,
∂Π
∂z
= 0.
∂
2
Π
∂x∂y
= −aβx
α
y
β−1
= −bkx
k−1
y
l
,
x y
aβ = bk
α = k −1 β −1 = l a α β b k l
F = ax
α
y
β
i + bx
k
y
l
j
Π = cx
i
y
j
a α β
b k l c i j
a = −ci α = i − 1 β = j b = −cj k = i l = j − 1
O
m t r
r O
r
O
V
j
j
r
r
V
r
F
r
F = f
r
r
, (20.1)
f
f(t, r, ˙r) t
0
r
0
˙r
0
= V
0
r(t)
˙
K
0
= M
0
= [r, F] = [ r, f
r
r
] = 0 :
K
0
= [r, mV] = const
K
0
r⊥K
0
= const)
O V
0
r ϕ
V = V
r
+ V
ϕ
V
r
= ˙r V
ϕ
= r ˙ϕ
V = V
r
+ V
ϕ
K
0
= [r, mV ] = [r, m(V
r
+ V
ϕ
)] = [r, mV
ϕ
]
K
0
= mr
2
˙ϕ = const, c =
K
0
m
, c = r
2
˙ϕ = const. (20.2)
c = r
2
˙ϕ
O
Dj
r
r+ rD
∆t
∆S =
1
2
(r + ∆r)r sin ∆ϕ =
1
2
r
2
∆ϕ ∆t
∆t → 0
dS
dt
=
1
2
r
2
˙ϕ =
1
2
c =
K
0
2m
= const, (20.3)
dS/dt
r(t)
˙ϕ =
c
r
2
. (20.4)
W = W
r
+ W
ϕ
W
r
= ¨r − r ˙ϕ
2
W
ϕ
= ¨ϕr + 2 ˙ϕ ˙r
O
r
W
j
W
r
mW = F = fr/r
r
m(¨r − r ˙ϕ
2
) = f(t, r, ˙r, ϕ, ˙ϕ). (20.5)
t
0
r
0
˙r
0
ϕ
0
˙ϕ
0
r(t) ϕ(t)
r(ϕ)
t ϕ
˙r =
dr
dt
=
dr
dϕ
dϕ
dt
(20.4)
=
dr
dϕ
c
r
2
= −
d
dϕ
µ
1
r
¶
c = −u
0
c, (20.6)
u =
1
r
, u
0
=
du
dϕ
, (20.7)
¨r =
d ˙r
dt
=
d ˙r
dϕ
dϕ
dt
(20.4)(20.6)
= −u
00
c
2
r
2
. (20.8)
u
00
+ u = −
r
2
mc
2
f(t, r, ˙r, ϕ, ˙ϕ). (20.9)
∂f/∂t = 0
u
00
+ u = −
1
mc
2
u
2
f(1/u, −u
0
c, ϕ, u
2
c) (20.10)
r(ϕ) = 1/u(ϕ)
F = f(r, ϕ)r/r
δA = (F, dr) = f(r, ϕ)
(r, dr)
r
= f(r, ϕ)
rdr
r
= f(r, ϕ)dr,
(r, dr) =
1
2
d(r, r) =
1
2
d(r
2
) = rdr
δA = f(r, ϕ)dr + 0dϕ = −dΠ(r, ϕ) = −
∂Π
∂r
dr −
∂Π
∂ϕ
dϕ.
∂Π
∂r
= −f(r, ϕ),
∂Π
∂ϕ
= 0
∂
2
Π
∂r∂ϕ
= −
∂f(r, ϕ)
∂ϕ
= 0.
f(r) r
O
f(r) Π(r)
f(r) = −
∂Π(r)
∂r
, Π(r) = −
Z
f(r)dr. (21.1)
T + Π =
1
2
mV
2
+ Π(r)
(8.4)
=
1
2
m( ˙r
2
+ ˙ϕ
2
r
2
) + Π(r)
(20.4)
=
=
1
2
m
Ã
˙r
2
+
c
2
r
2
!
+ Π(r) = E = const.
(21.2)
˙r
r(t) ϕ(t)
˙r = h(r, c, E) ˙ϕ = c/r
2
r(ϕ)
dr
dϕ
=
r
2
h(r, c, E)
c
h(r, c, E)
m
f = −γ
Mm
r
2
= −
mµ
r
2
, Π = −γ
Mm
r
= −
mµ
r
, (21.3)
γ M O
µ = γM
f
u
00
+ u =
µ
c
2
(21.4)
u = µ/c
2
u =
µ
c
2
+ A cos(ϕ + β), (21.5)
A β A > 0
β u = 1/r
r(ϕ)
r =
p
1 + e cos(ϕ + β)
, (21.6)
p e
p =
c
2
µ
, e =
Ac
2
µ
> 0. (21.7)
O
e
e = 0 p
0 < e < 1 ϕ
e = 1 ϕ + β = π
e > 1 ϕ + β = ±arccos(1/e)
ϕ + β = ±arccos(1/e)
p e
c = r
2
˙ϕ
E
∗
=
1
m
E =
1
m
(T + Π) =
1
2
µ
˙r
2
+
c
2
r
2
¶
−
µ
r
=
1
2
µ
˙r
2
0
+
c
2
r
2
0
¶
−
µ
r
0
= const (21.8)
A r
0
˙r
0
ϕ
0
˙ϕ
0
u
0
=
1
r
0
=
µ
c
2
+ A cos(ϕ
0
+ β), u
0
0
= −
1
c
˙r
0
= −A sin(ϕ
0
+ β),
A
2
=
µ
1
r
0
−
µ
c
2
¶
2
+
1
c
2
˙r
2
0
=
1
r
2
0
− 2
µ
r
0
c
2
+
µ
2
c
4
+
1
c
2
˙r
2
0
=
=
1
c
2
µ
˙r
2
0
+
c
2
r
2
0
− 2
µ
r
0
+
µ
2
c
2
¶
=
µ
2
c
4
µ
2E
∗
c
2
µ
2
+ 1
¶
,
A =
µ
c
2
s
1 +
2E
∗
c
2
µ
2
.
p =
c
2
µ
, e =
Ac
2
µ
=
s
1 +
2E
∗
c
2
µ
2
. (21.9)
T
¤ T
πab a = p/(1 −
e
2
) b = p/
√
1 − e
2
c/2
T
2
a
3
=
µ
πab
c/2
¶
2
,
a
3
=
4π
2
b
2
ac
2
=
4π
2
p
c
2
=
4π
2
µ
.
¥
m
1
m
2
r
1
r
2
O
r
2
Ñ
r
2
r
Ñ
r
1
r
1
r
m
1
m
2
r
1
(t) r
2
(t)
r
0
1
r
0
2
V
0
1
=
˙
r
0
1
V
0
2
=
˙
r
0
2
ρρρ
ρρ
= r
1
−r
2
C
r
C
O ρρρ
ρρ
1
ρρρ
ρρ
2
C
r
C
=
m
1
r
1
+ m
2
r
2
m
1
+ m
2
, ρρρ
ρρ
1
=
m
2
m
1
+ m
2
ρρρ
ρρ
, ρρρ
ρρ
2
= −
m
1
m
1
+ m
2
ρρρ
ρρ
. (22.1)
C
V
0
C
V
0
C
=
m
1
V
0
1
+ m
2
V
0
2
m
1
+ m
2
, r
C
= r
0
C
+ V
0
C
t =
m
1
r
0
1
+ m
2
r
0
2
m
1
+ m
2
+
m
1
V
0
1
+ m
2
V
0
2
m
1
+ m
2
t. (22.2)
m
1
¨
r
1
= −γ
m
1
m
2
ρ
2
ρρρ
ρρ
ρ
, m
2
¨
r
2
= γ
m
1
m
2
ρ
2
ρρρ
ρρ
ρ
,
ρρρ
ρρ
= r
1
− r
2
¨ρ¨ρ¨ρ
¨ρ¨ρ
= −γ
m
1
+ m
2
ρ
2
ρρρ
ρρ
ρ
ρρρ
ρρ
(t)
ρρρ
ρρ
ρρρ
ρρ
1
¨ρ¨ρ¨ρ
¨ρ¨ρ
1
= −γ
m
3
2
(m
1
+ m
2
)
2
ρ
2
1
ρρρ
ρρ
1
ρ
1
.
m
1
C M = m
3
2
/(m
1
+ m
2
)
2
m
2
ρρρ
ρρ
2
(t) = −m
1
ρρρ
ρρ
1
(t)/m
2
ρρρ
ρρ
1
(t) ρρρ
ρρ
2
(t) ρρρ
ρρ
1
(ϕ) ρρρ
ρρ
2
(ϕ)
r
1
(t) = r
C
(t) + ρρρ
ρρ
1
(t) r
2
(t) = r
C
(t) + ρρρ
ρρ
2
(t)
3m 4m 5m
F G H
GH = 3s HF = 4s F G = 5s
5m
3m 4m
5m
Q = 12mV
C
= 0 C
K
C
= 0 E = T + Π
F
G
H
3m
4m
5m
5s
3s
4s
5m
4m
3m
V
1
V
2
C