Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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δq
i
b
t
t+δt
“
q
i
, t → q
′
i
, t
′
q
i
= q
i
(q
′
1
, . . . , q
′
s
, t
′
), t = t(q
′
1
, . . . , q
′
s
, t
′
) , i = 1, 2, . . . , s . (9.6)
t
′
“ t
′
= t − λx
t
′
t
′
=
p
(ct)
2
− r
2
t
′
s + 1
S =
t
2
Z
t
1
Ldt =
t
′
2
Z
t
′
1
L
dt
dt
′
dt
′
. (9.7)
L
′
L
′
= L
dt
dt
′
, (9.8a)
S =
t
′
2
Z
t
′
1
L
′
q
′
,
dq
′
dt
′
, t
′
dt
′
. (9.9)
b
d
dt
′
∂L
′
∂(dq
′
i
/dt
′
)
=
∂L
′
∂q
′
i
, i = 1, 2, . . . , s . (9.10)
L = T −U
L
′
= λL T U
λ = const)
λ
a
L
′
q
′
,
dq
′
dt
′
, t
′
= L
q(q
′
, t
′
),
dq
dt
, t(q
′
, t
′
)
·
dt
dt
′
, (9.8b)
dq
i
dt
=
dq
i
dt
′
/
dt
dt
′
,
dq
i
(q
′
, t
′
)
dt
′
=
∂q
i
∂t
′
+
X
k
∂q
i
∂q
′
k
dq
′
k
dt
′
,
dt(q
′
, t
′
)
dt
′
=
∂t
∂t
′
+
X
k
∂t
∂q
′
k
∂q
′
k
dt
′
.
§
L
i
=
1
2
m
i
v
2
i
−
GMm
i
r
i
, i = 1, 2 ,
L = L
1
+ L
2
M m
1,2
r
1,2
v
1,2
=
˙
r
1,2
G
L
L = −Gm
1
m
2
/|r
1
−r
2
|
L → L
′
= λL λ
e
ϕ(r, t) A(r, t)
E B
E(r, t) = −
∂ϕ
∂r
−
1
c
∂A
∂t
, B(r, t) =
∂
∂r
, A
, (10.1)
c
d
dt
∂L
∂v
=
∂L
∂r
(10.2)
m
˙
v = eE +
e
c
[v, B] , (10.3)
L(r, v, t) =
1
2
mv
2
− eϕ +
e
c
Av . (10.4)
x
1
2
mv
2
eϕ
(e/c)Av
p =
∂L
∂v
= mv +
e
c
A . (10.5)
E B
ϕ → ϕ
′
= ϕ −
1
c
∂f
∂t
, (10.6)
A → A
′
= A +
∂f
∂r
,
f
ϕ
′
, A
′
ϕ, A L L
′
ef/c
L
′
=
1
2
mv
2
− eϕ
′
+
e
c
A
′
v =
§
=
1
2
mv
2
− eϕ +
e
c
Av +
e
c
∂f
∂t
+
∂f
∂r
v
= L +
e
c
df(r, t)
dt
,
F (q, t)
L
′
(q, ˙q, t) = L(q, ˙q, t) +
dF (q, t)
dt
, (10.7)
S S
′
S
′
=
t
2
Z
t
1
L
′
dt =
t
2
Z
t
1
Ldt +
t
2
Z
t
1
dF
dt
dt = S + F (q
(2)
, t
2
) −F (q
(1)
, t
1
) ,
p
′
i
=
∂L
′
∂ ˙q
i
=
∂L
∂ ˙q
i
+
∂
∂ ˙q
1
dF
dt
= p
i
+
∂F
∂q
i
. (10.8)
L(r, v, t) = −mc
2
r
1 −
v
2
c
2
− eϕ +
e
c
Av . (11.1)
d
dt
mv
p
1 − (v
2
/c
2
)
= eE +
e
c
[v, B] . (11.2)
p =
∂L
∂v
=
mv
p
1 − (v
2
/c
2
)
+
e
c
A . (11.3)
v ≪ c
L = −mc
2
+
1
2
mv
2
− eϕ +
e
c
Av , (11.4)
−mc
2
S =
t
2
R
t
1
Ldt
c
p
1 − (v
2
/c
2
)dt =
p
(cdt)
2
− (dr)
2
= ds , (11.5)
s
(cϕ − Av)dt = A
0
dx
0
− Adr = A
µ
dx
µ
(11.6)
A
µ
= (ϕ, A)
A
µ
= ( ϕ, −A) dx
µ
= ( cdt, dr)
S =
B
Z
A
(−mcds −
e
c
A
µ
dx
µ
) (11.7)
A(ct
1
, r
1
) B(ct
2
, r
2
) (t, r) → (t
′
, r
′
)
L
′
dt
′
= Ldt L
′
(r
′
, dr
′
/dt
′
, t
′
)
L(r, dr/dt, t)
m
l(t)
xy x
y
r
r ϕ
r = l(t) . (12.1)
mg
R
r
ϕ(t)
ϕ
l(t)
m
ϕ
˜
U
˜q0
“
“
˜
U(˜q)
˜q = r − l(t)
˜
U(˜q)
|˜q|
ϕ r
r ˜q
˜
L(ϕ, ˜q, ˙ϕ,
˙
˜q, t) =
m
2
h
˙
l(t) +
˙
˜q
i
2
+ [l(t) + ˜q]
2
˙ϕ
2
+
+mg [l(t) + ˜q] cos ϕ −
˜
U(˜q) (12.2)
m
d
dt
h
(l(t) + ˜q)
2
˙ϕ
i
= −mg [l(t) + ˜q] sin ϕ , (12.3)
m
h
¨
l(t) +
¨
˜q
i
= m [l(t) + ˜q] ˙ϕ
2
+ mg cos ϕ −
d
˜
U
d˜q
. (12.4)
˜q
˜q = 0
˙
˜q = 0
m
d
dt
l
2
(t) ˙ϕ
= −mgl(t) s in ϕ , (12.5)
l(t)
ϕ(t)
˜q(t)
“ ˜q(t) = 0
“
˜q = 0
˙
˜q = 0
˜
U
ϕ