Сягло И.С. Теоретическая механика
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yr
±
i
ϕ
F
1
F
2
a
b
r
r
t
m
m
O
a b
S = πab
S =
Z
T
0
M
2m
dt =
M
2m
T = πab.
a b
T = πα
v
u
u
u
t
m
2|E|
3
= 2π
v
u
u
t
m
α
a
3/2
.
α/m
γm
o
U(r) = α /r
r =
p
−1 + e cos ϕ
.
e p
E > 0 e > 1
v
∞
ρ n
dN dS
-
-
-
-
-
-
-
-
-
t
θ
θ + dθ
ρ
dρ
0
dS
n =
dN
dS
.
dN(θ)
θ
θ θ + dθ
dσ
dσ =
dN(θ)
n
.
dσ
ρ ρ+dρ
θ θ + dθ
dσ =
dN(θ)
n
=
dN(ρ)
n
= 2πρdρ.
ρ(θ)
dσ = 2πρ
¯
¯
¯
¯
¯
¯
dρ
dθ
¯
¯
¯
¯
¯
¯
dθ.
ρ θ
dΩ
t
-
y
α
θ
ϕ
o
ϕ = 0
6
ρ
~v
∞
~r
θ θ+dθ
dΩ = 2π sin θdθ, dσ =
ρ
sin θ
¯
¯
¯
¯
¯
¯
dρ
dθ
¯
¯
¯
¯
¯
¯
dΩ.
U(r) = α /r
ϕ = 0
ϕ
o
r
cos ϕ
o
=
1
e
.
θ = π − 2ϕ
o
e ρ
ρ(θ)
E =
mv
2
∞
2
, M = |m ~r ×~v| = mv
∞
r sin α = mv
∞
ρ.
e =
v
u
u
u
t
1 +
m
2
v
4
∞
α
2
ρ
2
.
ϕ
o
θ ρ(θ)
ρ =
α
mv
2
∞
θ
2
.
dσ = π
α
mv
2
∞
2
cos θ/2
sin
3
θ/2
dθ =
α
mv
2
∞
2
dΩ
sin
4
θ/2
.
α
m
U(r) = −α/r − β/r
2
α β
m U(r) =
α/r + β/r
2
m U(r) =
−α/r + β/r
2
∆ϕ
r = r
a
0Z
r = Az
n
, 0 < n < 1
r = b sin z/a
0 < z < πa/2
L =
1
2
a(q) ˙q
2
− U(q).
q = q
o