Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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E > 0 e > 1
r = r
min
= p/(−1 + e) ϕ = 0
E < 0
C O
a = CA = (1/2) DA b = CB
2b
2a
y
x
A
B
OC
D
P
p = OP
2a
a =
1
2
(OA + DO) =
1
2
(r
min
+ r
max
) =
1
2
p
1 + e
+
p
1 − e
==
p
1 − e
2
=
α
2|E|
. (3.7)
C O
CO =
1
2
(DO − OA) =
1
2
(r
max
− r
min
) = ae . (3.8)
COB OB = a
b = CB = a
p
1 − e
2
. (3.9)
e b
b =
M
p
2m|E|
. (3.10)
r
min
= (1 − e) a, r
max
= (1 + e) a, p = (1 − e
2
) a . (3.11)
x = x
′
− ae
x
′
a
2
+
y
b
2
= 1 ,
x
′
= x + ae = a cos ξ, y = b sin ξ . (3.12)
ξ
˙
ξ
M = m(x ˙y − y ˙x) = ma(cos ξ −e) b
˙
ξ sin ξ −mb sin ξ a
˙
ξ sin ξ =
= mab(1 − e cos ξ)
˙
ξ .
ξ
ξ − e sin ξ =
M
mab
(t − t
0
) , (3.13a)
t
0
t = t
0
a
T
ξ 2π
T = 2π
mab
M
,
ξ − e sin ξ =
2π
T
(t − t
0
) . (3.13b)
T = 2π
r
ma
3
α
. (3.14)
T
2
a
3
= 4π
2
m
α
=
4π
2
Gm
, (3.15)
E
M
n = r/r
˙ϕ
dn
M
˙ϕ = M/(mr
2
)
M × n
dn
dt
=
M
mr
2
× n . (3.16)
m
dv
dt
= −
α
r
2
n ,
dn
dt
= −
M
α
×
dv
dt
,
d
dt
v × M − α
r
r
= 0 .
U(r) = −α/r
A = [v, M] −α
r
r
. ( 3 .17a)
U(r) = α/r
A = [v, M] + α
r
r
. (3.17b)
A
r A
ϕ
rA = rA cos ϕ = r [v, M] − αr =
M
2
m
− αr = α(p − r) ,
p = M
2
/(mα)
r =
p
1 + (A/α) cos ϕ
. (3.18)
A r = r
min
|A| = αe . (3.19)
E M
A
A
M
E M
A
U(r) = −
α
r
+
β
r
2
U (r) = −
α
r
+
˜
M
2
2mr
2
,
˜
M =
p
M
2
+ 2mβ (3.20)
U (r)
ϕ = ±
Z
M
r
2
dr
r
2mE +
2mα
r
−
˜
M
2
r
2
+ ϕ
0
. (3.21)
γ ϕ = ±
Z
˜
M
r
2
dr
r
2mE +
2mα
r
−
˜
M
2
r
2
+ const ,
γ =
˜
M
M
=
r
1 +
2mβ
M
2
, (3.22)
M →
˜
M, ϕ → γ ϕ
r =
˜p
1 + ˜e cos(γϕ)
, (3.23)
˜p =
˜
M
2
mα
, ˜e =
s
1 +
2E
˜
M
2
mα
2
. (3.24)
E < 0 ˜e < 1
β > 0 β < 0
A, B, A
1
A
B
A
1
∆ϕ =
2π
γ
, ( 3 .25)
r = r
min
δϕ = ∠AOA
1
=
2π
γ
− 2π , (3.26)
δϕ < 0
β > 0 δϕ > 0
β < 0
A
A
1
B
B
1
O
а)
A
A
1
B
B
1
O
б)
E < 0
β > 0 β < 0
γ
r = r
min
= ˜p/(1+ ˜e)
r = r
max
= ˜p/(1 − ˜e)
γ γ = n
1
/n
2
n
1, 2
n
1
n
2
ϕ
r
T
ϕ
T
r
∆ϕ
2π 2π/∆ϕ
U(r) = −α/r 2π/∆ϕ = 1
T
ϕ
= T
r
U(r) = kr
2
/2
2π/∆ϕ = 2 T
ϕ
= 2 T
r
E < 0 E > 0
v = 160
U(r) =
1
2
kr
2
(4.1)
xy
M
¨x = −ω
2
x , ¨y = −ω
2
y , ω =
r
k
m
, (4.2)
x(t) = A cos (ωt + α) , y(t) = B cos (ωt + β) , (4.3)
A, B, α β
cos(ωt + β) = cos δ cos(ωt + α) −sin δ si n(ωt + α) , δ = β −α ,
y = x
B
A
cos δ − B sin δ sin(ωt + α)
cos
2
(ωt + α) + sin
2
(ωt + α) =
=
x
2
A
2
+
1
B
2
sin
2
δ
y −
B
A
x cos δ
2
= 1 . (4.4)
xy
xy
x
2
a
2
+
y
2
b
2
= 1 , (4.5)
x(t) = a cos ωt , y(t) = b sin ωt , (4.6)
r
0
= (a, 0, 0)
v
0
= (0, bω, 0)
a b
T = 2π/ω
T
r
= π/ω
M
E
x
E
y
x y
E
x
, E
y
, M A, B, δ
E
x
= kA
2
/2 E
y
= kB
2
/2 M = mABω sin δ
E = E
x
+ E
y
N = m ˙x ˙y + kxy . (4.7)
N