Сягло И.С. Теоретическая механика
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OZ
~
M
XOY
L =
m
2
( ˙r
2
+ r
2
˙ϕ
2
) − U(r).
ϕ
p
ϕ
= mr
2
˙ϕ = M = .
0Z
M
0Z
˙ϕ > 0
ϕ
dS ≈
1
2
r(t)r(t + dt )dϕ ≈
1
2
r
2
dϕ.
dS
dt
=
1
2
r
2
˙ϕ =
M
2m
.
E =
m
2
( ˙r
2
+ r
2
˙ϕ
2
) + U(r).
˙ϕ
>
:
i
dϕ
~r(t + dt)
~r(t)
U
ef
(r)
r
E
r
1
r
2
E =
m
˙
r
2
2
+
M
2
2mr
2
+ U(r) =
m
˙
r
2
2
+ U
ef
(r),
U
ef
(r)
U
ef
(r) =
M
2
2mr
2
+ U(r) .
U
ef
(r)
˙r =
dr
dt
= ±
v
u
u
u
t
2
m
(E − U
ef
(r)).
r(t)
t = ±
Z
dr
r
2
m
(E − U
ef
(r))
+ C.
E ≥ U
ef
(r)
E
M
U
ef
(r)
E =
E ≥ U
ef
(r)
r
1
≤ r ≤ r
2
r
1
r
2
˙r ˙ϕ
dϕ
dr
=
˙ϕ
˙r
= ±
M
r
2
r
2m(E − U
ef
(r))
.
ϕ = ±
Z
M dr
r
2
r
2m(E − U
ef
(r))
+ C.
U(r)
C
ϕ ˙r
˙r ˙r = 0
C = 0 ϕ
r
ϕ
r = r r = r
i
r = r ,
ϕ = 0
ϕ < 0
ϕ > 0
∆ϕ
r
r
ª
K
∆ϕ r = r r = r
∆ϕ =
Z
r
r
M dr
r
2
r
2m(E − U
ef
(r))
.
∆ϕ 2π
n∆ϕ = 2π m,
m, n
r = r r = r
U(r) = ±
α
r
.
~
F = −∇U = ±
α
r
2
~e
r
,
~e
r
α
C
OX
ϕ =
Z
M dr
r
2
r
2m(E −
α
r
−
M
2
2mr
2
)
.
x = M/r
r =
p
1 + e cos ϕ
,
p e
p =
M
2
mα
, e =
v
u
u
u
t
1 +
2EM
2
mα
2
.
e e > 1
E =
mv
2
2
−
α
r
.
r → ∞
E = mv
2
∞
/2 E = 0
e
E e < 1
U(r) = −α /r
m
o
α = γmm
o
a =
1
2
(r + r ) =
p
1 − e
2
=
α
2|E|
, b = a
√
1 − e
2
=
M
r
2m|E|
.