Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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d
2
x
dt
2
+ ω
2
0
(1 + h cos γt)x + 2λ
dx
dt
= 0 . (31.7)
s =
1
4
p
(hω
0
)
2
− 4ǫ
2
− λ . (31.8)
h > 4λ/ω
0
m¨x = −
dU
dx
+ f(x) cos ωt , (32.1)
ω
U
x(t) = X(t) + ξ(t) X(t)
ξ(t) = ξ
0
cos ωt
ξ
0
m
¨
X +m
¨
ξ = −
dU(X)
dX
+f(X) cos ωt−
d
2
U(X)
dX
2
ξ+
df(X)
dX
ξ cos ωt ,
(32.2)
ξ
0
m
¨
ξ = f(X) cos ωt . (32.3)
X(t)
ξ
0
= −
f(X)
mω
2
. (3 2 .4 )
ξ, f
m
¨
X = −
dU
dX
−
1
2mω
2
df(X)
dX
f(X) . (32.5)
−dU /dX
U = U +
f
2
4mω
2
, (32.6)
U
a
s s
d
dt
∂L
∂ ˙q
i
=
∂L
∂q
i
, i = 1, 2, . . . , s , (33.1)
¨q
i
, ˙q
i
q
i
p
i
=
∂L
∂ ˙q
i
, (33.2)
˙p
i
=
∂L
∂q
i
. (33.3)
q
i
p
i
L = L(q, ˙q, t)
˙q
i
˙p
i
˙q
i
= f
i
(q, p, t), ˙p
i
= g
i
(q, p, t) , i = 1, 2, . . . , s , (33.4)
2s
q
i
, p
i
f
i
g
i
dL(q, ˙q, t) =
s
X
i=1
∂L
∂q
i
dq
i
+
∂L
∂ ˙q
i
d ˙q
i
+
∂L
∂t
dt
dL(q, ˙q, t) =
s
X
i=1
( ˙p
i
dq
i
+ p
i
d ˙q
i
) +
∂L
∂t
dt . (33.5)
H(p, q, t) =
s
X
i=1
p
i
˙q
i
− L (33.6)
dH(p, q, t) =
s
X
i=1
( ˙q
i
dp
i
− ˙p
i
dq
i
) −
∂L
∂t
dt . (33.7)
E(t)
H
˙q
i
=
∂H(p, q, t)
∂p
i
, ˙p
i
= −
∂H(p, q, t)
∂q
i
, i = 1, 2, . . . , s (33 .8 )
∂H(p, q, t)
∂t
= −
∂L(q, ˙q, t)
∂t
.
2s
2s q
1
, . . . , q
s
p
1
, . . . , p
s
L(q, ˙q) =
1
2
a(q) ˙q
2
+ b(q) ˙q + c(q)
p ˙q − L =
1
2
a(q) ˙q
2
− c(q).
˙q
p =
∂L
∂ ˙q
= a(q) ˙q + b(q) ,
˙q =
p − b(q)
a(q)
.
H(p, q) =
[p − b(q)]
2
2a(q)
− c(q) . (33.9)
L(x, ˙x) =
1
2
m ˙x
2
−
1
2
mω
2
x
2
H(p, x) =
p
2
2m
+
mω
2
x
2
2
.
H(p
r
, p
θ
, p
ϕ
, r, θ, ϕ) =
p
2
r
2m
+
p
2
θ
2mr
2
+
p
2
ϕ
2mr
2
sin
2
θ
+U(r) . (33.10)
H(p, r, t) =
1
2m
p −
e
c
A(r, t)
2
+ eϕ(r, t) . (33.11)
H(p, r, t) =
r
p −
e
c
A(r, t)
2
c
2
+ m
2
c
4
+ eϕ(r, t) . (33.12)
m = 0 e = 0
H(p, r) = c|p|.
n(r)
H(p, r) =
c
n(r)
|p|. (33.12a)
˙
r =
c
n
p
p
,
˙
p = −
cp
n
2
∂n
∂r
, p = |p|.
“
r(t)
˙
r
p
∂H(p, q, t)
∂q
k
= 0 ,
p
k
= const .
dH(p, q, t)
dt
=
s
X
i=1
˙q
i
dp
i
dt
− ˙p
i
dq
i
dt
+
∂H
∂t
=
∂H
∂t
. (33.13)
∂H(p, q, t)
∂t
= 0 ,
E(t) = H(p(t) , q(t)) = const .
z
B = (0 , 0, B)
A = (0, xB, 0) . ( 33.14)
H(p
x
, p
y
, p
z
, x, y, z, t) =
1
2m
p
2
x
+
p
y
−
e
c
Bx
2
+ p
2
z
(33.15)
t, y, z
p
y
p
z
p
y
= m ˙y +
eB
c
x = const , p
z
= m ˙z = cons t .
z
z =
p
z
m
t + z
0
. (33.16)
ω =
eB
mc
, x
0
=
cp
y
eB
=
p
y
mω
, (33.17)
H =
p
2
x
2m
+
mω
2
2
(x − x
0
)
2
+
p
2
z
2m
(33.18)
x x
0
x
0
p
z
x = x
0
+ R cos ω(t −t
0
), p
x
= m ˙x = −mωR sin ω(t − t
0
) ,
(33.19)
R, t
0
˙y =
∂H
∂p
y
= mω
2
(x
0
− x)
∂x
0
∂p
y
= − ωR cos ω(t − t
0
),
y(t)
y = y
0
− R sin ω(t − t
0
) . (33.20)
z
p
z
/m ω
xy ω > 0
R
(x
0
, y
0
)
p
y
mω x
q
1
p
1
f(q
1
, p
1
)
H = H (f( q
1
, p
1
), q
2
, p
2
, . . . , q
s
, p
s
) , (33.21)