Манаков Н.Л., Некипелов А.А., Овсянников В.Д. Задачи по теоретической механике
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U(r)
r = |r|
F = −grad U(r) = −
r
r
dU
dr
r
M = [r × F] = 0,
L = [r × p] = const.
L (r, ϕ)
L =
m
2
( ˙r
2
+ r
2
˙ϕ
2
) − U(r).
ϕ
L =
∂L
∂ ˙ϕ
= mr
2
˙ϕ = const.
m
2
( ˙r
2
+ r
2
˙ϕ
2
) + U(r) = E = const.
˙ϕ
˙ϕ =
L
mr
2
,
˙r,
˙r = ±
r
2
m
[E −U(r)] −
L
2
m
2
r
2
˙r > 0 ˙r < 0
r
E > U
eff
,
U
eff
(r) = U(r) +
L
2
2mr
2
r
min
r
max
r
min
r
max
r
min
6
r 6 r
max
ϕ
ϕ(r) =
r
Z
r
0
Ldr
r
2
r
2m[E −U(r)] −
L
2
r
2
.
r = r
min
r = r
max
∆ϕ
0
= 2
r
max
Z
r
min
Ldr
r
2
r
2m[E −U(r)] −
L
2
r
2
.
∆ϕ
0
= 2π
n
k
n k
U(r) = −
A
r
n
, (A > 0) n < 2
U =
L
2
2mr
2
L 6= 0
E
n = 2 L <
√
2mA E
L >
√
2mA E > 0
r
min
=
s
µ
L
2
2m
−
A
¶
/E.
n > 2
E
n
= max{U
eff
(r)} =
(n − 2)A
2
µ
L
2
mnA
¶
n
n − 2
.
E > E
n
0 < E < E
n
r
n
r
0
n
-
r
6
U
eff
(r)
E
n
E
0
r
n
r
0
n
E < 0
U(r)
χ
dσ(χ)
dN(χ)
(χ, χ + dχ), n
dσ =
dN(χ)
n
.
ρ
ρ + dρ
dσ = 2πρdρ,
ρ
(χ, χ+dχ)
6
?
ρ
½
½
½
½
½
½
½
½
½
½
½
½
-
>
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©
2πρdρ
C
C
CO
C
C
CW
r
min
ϕ
0
½
½
½
½
½
½
½
½
½
½
½
χ
O
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
dχ
@
@
@
@
@
@
@
@
@
@
@
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
dΩ = 2π sin χdχ,
ρ =
ρ(χ) dΩ dσ
ρ(χ)
χ = |π − 2ϕ
0
|,
ϕ
0
=
∞
Z
r
min
Ldr
r
2
r
2m [E −U(r)] −
L
2
r
2
=
∞
Z
r
min
ρdr
r
2
r
1 −
U(r)
E
−
ρ
2
r
2
(r = r
min
) r
min
ρ(χ)
dσ
dΩ
=
ρ(χ)
sin χ
¯
¯
¯
¯
dρ
dχ
¯
¯
¯
¯
.
U(r) =
½
0, r 6 a
∞, r > a.
r = r
max
= a
r = r
min
= L/p p =
√
2mE
r r
min
a ( ˙r > 0)
x = L/r
ϕ(r) =
r
Z
r
min
(L/r
2
)dr
p
2mE −L
2
/r
2
= −
L/r
Z
L/r
min
dx
p
p
2
− x
2
= arccos
r
min
r
.
r
r = r
min
/ cos ϕ, a > r > r
min
.
∆ϕ
0
= 2 arccos
r
min
a
= 2 arccos
L
ap
.
∆ϕ
0
= 2π/n n
a r
min
.
U(r) =
½
−U
0
, 0 6 r 6 a
0, r > a
L
∆ϕ
0
ρ
˙r =
r
2
m
(E + U
0
) −
L
2
m
2
r
2
,
˙r =
r
2
m
E −
L
2
m
2
r
2
.
L 6= 0 E 6 0
p
0
=
p
2m(E + U
0
)
L r
min
p
0
= L/r
min
E > −U
0
+
L
2
2mr
2
min
−U
0
6 E 6 0
dr
r
p
2
0
m
2
−
L
2
m
2
r
2
= dt →
1
v
0
r
Z
r
min
dr
0
q
1 − r
2
min
/r
0
2
= t − t
0
→
q
r
2
− r
2
min
= v
0
(t − t
0
)
⇒ r(t) =
q
r
2
min
+ v
2
0
(t − t
0
)
2
,
v
0
=
p
0
m
=
r
2
m
(E + U
0
) v
0
=
const t
0
r = r
min
ϕ(r) =
r
Z
r
min
(L/r
2
)dr
p
2m(E + U
0
) − L
2
/r
2
=
r
Z
r
min
(r
min
/r
2
)dr
p
1 − r
2
min
/r
2
=
= arccos
r
min
r
.
r
r(ϕ) =
r
min
cos ϕ
.
r = r
max
=
∆ϕ
0
= 2 arccos(r
min
/a).
∆ϕ
0
= 2π
n
l
n l
l
n
r
min
@
@
@
@
@
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¡
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a
O
∆ ϕ
0
-
v
0
r
min
= a cos
πn
l
, (l > 2n)
E, −U
0
< E 6 0
L = pρ p =
√
2mE
ρ ρ > a
r(t) =
p
ρ
2
+ v
2
(t − t
0
)
2
,
v = p/m t
0
r = ρ
ρ < a
L = pρ = p
0
r
min
p
0
=
p
2m(E + U
0
)
r
min
˙r > 0
r(t) =
(
p
ρ
2
+ v
2
(t − t
0
)
2
, r > a
p
r
2
min
+ v
2
0
(t − t
1
)
2
, r < a.
t
0
t
1
t
0
= t
1
+
p
a
2
− r
2
min
v
0
−
p
a
2
− ρ
2
v
,
r(t
1
) = r
min
.
ϕ(r) =
r
Z
r
min
ρ
dr
r
2
r
1 −
ρ
2
r
2
−
U(r)
E
.
r = r
min
r = ∞
ϕ
0
= ϕ(∞) = ϕ
1
+ ϕ
2
,
r
min
@
@
@
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@
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¡
a
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ϕ
1
ϕ
2
*
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A
A
A
A
A
A
ρ
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H
H
HH
H
Hj
H
H
H
H
H
H
χ
ϕ
1
=
a
Z
r
min
ρ
r
2
dr
r
1 +
U
0
E
−
ρ
2
r
2
=
π
2
− arcsin
r
min
a
; ϕ
2
=
∞
Z
a
ρ
r
2
dr
r
1 −
ρ
2
r
2
= arcsin
ρ
a
χ = 2ϕ
0
− π = 2ϕ
1
+ 2ϕ
2
− π = 2 arcsin
ρ
a
− 2 arcsin
r
min
a
,
sin
χ
2
=
ρ
a
r
E
E + U
0
Ã
r
E + U
0
E
−
ρ
2
a
2
−
r
1 −
ρ
2
a
2
!
.
χ(ρ = 0) = χ(ρ > a) = 0; χ
max
= χ(ρ = a) = 2 arcsin
r
U
0
E + U
0
.
U(r) =
½
U
0
, 0 6 r 6 a
0, r > a.
−U
0
U
0
ρ < a E > U
0
+
L
2
2ma
2
L
2
= 2mEρ
2
, E > E
cr
E
cr
=
U
0
(1 − ρ
2
/a
2
)
.
r
min
= ρ
r
E
E −U
0
.
E 6 E
cr
, r
min
= a
χ=
π − 2 arcsin
ρ
a
, E 6 E
cr
,
2 arcsin
Ã
ρ
a
r
E
E −U
0
!
−2 arcsin
ρ
a
, E > E
cr
.
r
min
<a
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@
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@
a
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H
H
H
H
H
Hj
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
HH
H
H
Hj
:
r
min
=a
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χ
χ
max
= χ
Ã
ρ = a
r
E −U
0
E
!
= π − 2 arcsin
r
E −U
0
E
.
m
U(r) = −α
ln(r/a)
r
2
,
r = a exp
·
L
2
2mα
+
mα
2L
2
(ϕ − ϕ
0
)
2
¸
.
U(r) = −
α
r
6
.
r
2
=
√
2mα
L
sin 2(ϕ − ϕ
0
)
U(r) = −
α
r
+
β
r
2
.
r =
p
1 + ε cos(ϕ/A)
p =
L
2
mα
+
2β
α
, ε =
r
1 +
2Ep
α
, A =
1
p
1 + 2mβ/L
2
.
ϕ
0
= 2πA
U(r) = −
α
r
A = [v ×
L] −
αr
r
A
˙
A
A
˙
v =
F
m
= −
αr
mr
3
,
˙
L = 0,
˙
r = v,
d
dt
µ
1
r
¶
= −
³
r · v
r
3
´
,
˙
A = 0.
A
(A · r) = ([v × L] · r) − αr =
L
2
m
− αr;
A
Ar cos ϕ =
L
2
m
− αr.
r r =
L
2
/m
α + A cos ϕ
r =
p
1 + ε cos ϕ
, p =
L
2
mα
,
= εα
ε
v
2
= ˙r
2
+ r
2
˙ϕ
2
v
2
=
α
mp
(1 + ε
2
+ 2ε cos ϕ) =
α
mp
³
ε
2
− 1 + 2
p
r
´
.
r
p
r
a
a
r
a
= 2a
µ
T
T
¶
2/3
− r
p