Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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Ec
t
−
s
X
i=1
p
i
c
i
= const , ( 1 4 .5 )
d
dt
−Ec
t
+
s
X
i=1
p
i
c
i
!
= 0 ,
c
t
c
i
c
t
c
i
t → t
′
= t + εh(q, t) , q
i
→ q
′
i
= q
i
+ εf
i
(q, t) , i = 1, 2, . . . , s ,
(14.6)
ε
ε
t
2
Z
t
1
L
q,
dq
dt
, t
dt =
t
′
2
Z
t
′
1
L
q
′
,
dq
′
dt
′
, t
′
dt
′
. (14.7)
Eh −
s
X
i=1
p
i
f
i
= const , (14.8)
εh(q, t) = δt, εf
i
(q, t) = δq
i
.
t → t + δt, q
i
→ q
i
+ δq
i
, i = 1, 2, . . . , s (14.9)
Eδt −
s
X
i=1
p
i
δq
i
= const . (14.10)
L
L(r, v) =
1
2
mv
2
− U(r), U(r) =
ar
r
3
, a = const . (14.11)
a
E =
1
2
mv
2
+ U(r )
m[r, v]a
r → r
′
= λr, t → t
′
= λ
2
t , (14.12)
λ
L
r
′
,
dr
′
dt
′
= L
λr,
v
λ
= L(r, v)
1
λ
2
= L(r, v)
dt
dt
′
.
λ λ = 1 + ε
δr = εr, δt = 2εt
2Et − mvr = const . (14.13)
r(t)
r
2
(t) =
2E
m
(t − τ)
2
+ B ,
τ B
U(λr) = U(r)/λ
2
U(r) = β/r
2
t
2
Z
t
1
L
q,
dq
dt
, t
dt =
t
′
2
Z
t
′
1
L
q
′
,
dq
′
dt
′
, t
′
+ ε
dF (q
′
, t
′
)
dt
′
dt
′
,
(15.7a)
Eh −
s
X
i=1
p
i
f
i
− F = const . (15.8a)
U(r, t) = U( r −Vt) V
B
A
x
= A
z
= 0 A
y
= xB
A =
1
2
[B, r]
N
L(r
1
, . . . , r
N
, v
1
, . . . , v
N
, t)
ε
r
a
→ r
a
+ ε, t → t . (15.1 )
δr
a
= ε, δt = 0,
X
a
p
a
!
ε = const ,
ε
N
X
a=1
p
a
= const . (15.2)
ε
n
r
a
→ r
a
+ δr
a
, δr
a
= ε[ n, r
a
], δt = 0 , (15.3)
X
a
p
a
δr
a
= ε
X
a
p
a
[n, r
a
] = const ,
εn
X
a
[r
a
, p
a
] = const .
n
M =
N
X
a=1
[r
a
, p
a
] = const . (15.4)
r
a
→ r
a
, t → t + ε . (15.5)
δr
a
= 0, δt = ε ,
E =
N
X
a=1
p
a
v
a
− L =
3N
X
i=1
p
i
˙q
i
− L = const . (15.6)
O
O
′
R
r
′
r
x
x
′
y
′
y
K(x, y, z)
K
′
(x
′
, y
′
, z
′
)
K
′
(x
′
, y
′
, z
′
)
K(x, y, z)
K
′
R
O
′
= R(t)
K
′
r = R(t) + r
′
R(0) = 0
V =
˙
R K
′
r = r
′
+ Vt
′
, t = t
′
, v = v
′
+ V . (16.1)
K
L(r, v) =
1
2
mv
2
− U(r) , (16.2)
p =
∂L
∂v
= mv , E = pv − L =
1
2
mv
2
+ U(r) . (16.3)
K
′
L
′
1
(r
′
, v
′
, t) = L(r, v) =
1
2
m (v
′
)
2
−mv
′
V+
1
2
m V
2
−U(r
′
+Vt
′
) .
(16.4)
p
′
1
=
∂L
′
1
∂v
′
= m (v
′
+ V) = p ; (16.5a)
E
′
1
= p
′
v
′
− L
′
1
=
1
2
m (v
′
)
2
+ U −
1
2
m V
2
= E − Vp . (16 .5 b)
K
′
K
′
L
′
2
(r
′
, v
′
, t) =
1
2
m (v
′
)
2
− U(r
′
+ Vt
′
) . (16.6)
L
′
2
L
′
1
F (r
′
, t) = − mr
′
V +
1
2
m V
2
t
L
′
1
(r
′
, v
′
, t)
p
′
2
=
∂L
′
2
∂v
′
= mv
′
= p −mV , (16.7a)
E
′
2
=
1
2
m (v
′
)
2
+ U = E − Vp +
1
2
m V
2
. (16.7b)
E
′
1
E
′
2
E E
′
K
E K
′
E
′
(5b)
(7b)