Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
Подождите немного. Документ загружается.
dE
dt
=
m¨x +
dU(x)
dx
˙x = 0 .
x(t)
E =
1
2
mv
2
0
+ U(x
0
) ,
dx
dt
= ±
r
2
m
[E − U(x)] ( ˙x ≷ 0) (1.3)
t = ±
r
m
2
Z
x
x
0
dx
p
E − U(x)
+ t
0
. (1.4)
x
1
x
2
x
3
x
U
E
U
m
x
m
x
i
U(x
i
) = E
v
i
= 0
a
i
= −U
′
(x
i
)/m
U
m
T = E −U(x) > 0
E < U
m
x
1
< x < x
2
x > x
3
a
1,2,3
6= 0 a
1,3
> 0 a
2
< 0
x
1,2,3
x
1
< x < x
2
T =
√
2m
Z
x
2
x
1
dx
p
E − U(x)
. (1.5)
x > x
3
E = U
m
x
m
F
x
(x
m
) = −U
′
(x
m
)/m = 0
U
′
(x
m
) = 0 U
′′
(x
m
) 6= 0
x
m
U(x) = U
m
+
1
2
U
′′
(x
m
) (x − x
m
)
2
+ . . . , U
′′
(x
m
) < 0
x
m
x(t) = x
m
− (x
m
− x
0
) e
−λ(t−t
0
)
, λ =
r
−
U
′′
(x
m
)
m
,
x(t) → x
m
t → ∞
x
m
E → U
m
m x > 0
U(x) = V
a
2
x
2
1 −
a
x
2
U(x) = −Ax
4
U(x) =
−kx
2
x dw
x, x + dx
dt
dw = 2 dt/T
dw(x)/dx
U(x) = mω
2
x
2
/2 a
U(r) ≡ U(r)
U(r)
“
F = −
∂U
∂r
= −
dU
dr
r
r
E =
1
2
mv
2
+ U(r) , (2.1)
M = m[r, v ] , (2.2)
dM
dt
= [r, F] = 0 .
M
xy r ϕ
E =
1
2
m ˙r
2
+
1
2
m(r ˙ϕ)
2
+ U(r) , (2.3)
∂
∂r
≡
∂
∂x
,
∂
∂y
,
∂
∂z
.
ϕ
r
˙r
r ˙ϕv
x
M = (0, 0, M) , M = mr
2
˙ϕ .
˙ϕ
E =
1
2
m ˙r
2
+ U (r) ,
U (r) = U(r) +
M
2
2mr
2
. (2.5)
U (r) M
2
/(2mr
2
)
dr
dt
= ±
r
2
m
[E − U (r)] ( ˙r ≷ 0)
dt = ±
r
m
2
dr
p
E − U (r)
, (2.6)
t(r)
t = ±
r
m
2
Z
r
r
0
dr
p
E − U ( r)
+ t
0
.
r(t)
x(t) U(x)
U (r) r
i
U (r
i
) = E
˙r
i
= 0 ¨r
i
= −U
′
(r
i
)/m
r
i
U r
U x E < U
m
r
1
< r < r
2
r > r
3
T
r
=
√
2m
Z
r
2
r
1
dr
p
E − U ( r)
. (2.7)
dt =
mr
2
M
dϕ .
dt
ϕ = ±
M
√
2m
Z
r
r
0
dr
r
2
p
E − U ( r)
+ ϕ
0
. (2.8)
˙ϕ = M/(mr
2
)
r
i
r
i
r
1
r
2
T
ϕ
< T
r
T
ϕ
> 4T
r
T
ϕ
T
ϕ
< T
r
T
ϕ
> 4T
r
dt
r dr = v dt
dϕ “
r
r + dr
dS = r
2
dϕ/2
dS
dt
=
1
2
r
2
˙ϕ , (2.9)
M = 2m
dS
dt
, (2.10)
U(r) = −αr
−n
n ≥ 2
r
U(r) = −
α
r
. (3.1a)
α = Gm m m
m G
α = e
2
U (r) = −
α
r
+
M
2
2mr
2
(3.2)
E > 0
r
1
E < 0
r
min
< r < r
max
E = −mα
2
/(2M
2
)
r = M
2
/(mα)
ϕ = ±
Z
M
r
2
dr
r
2mE +
2mα
r
−
M
2
r
2
+ const . (3.3)
u =
p
r
, p =
M
2
mα
, (3.4)
ϕ = ∓
Z
du
p
e
2
− (u − 1)
2
+ ϕ
0
, e =
r
1 +
2EM
2
mα
2
. (3.5)
ϕ = ±arccos
u − 1
e
+ ϕ
0
,
r =
p
1 + e cos(ϕ − ϕ
0
)
.
ϕ
0
= 0 r = r
min
ϕ = 0
r =
p
1 + e cos ϕ
, (3.6)
e p
r
1
r
m in
r
m a x
r
E
1
E
2
U (r)
U (r) = −
α
r
+
M
2
2mr
2
e > 1 E > 0
e = 1 E = 0
e < 1 E < 0
E = −mα
2
/(2M
2
) e = 0
p
ϕ = π/2
p =
M
2
mα
= r|
ϕ=π/2
.
U(r) =
α
r
, (3.1b)
r =
p
−1 + e cos ϕ
, (3.6b)
e p