Коткин Г.Л., Сербо В.Г., Черных А.И. Лекции по аналитической механике
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f(q
1
, p
1
)
df
dt
=
∂f
∂q
1
˙q
1
+
∂f
∂p
1
˙p
1
˙q
1
=
∂H
∂p
1
=
∂H
∂f
∂f
∂p
1
, ˙p
1
= −
∂H
∂q
1
= −
∂H
∂f
∂f
∂q
1
,
df/dt = 0
f(q
1
, p
1
) = const . (33.22)
H = H (f( q
1
, p
1
, . . . , q
k
, p
k
), q
k+1
, p
k+1
, . . . , q
s
, p
s
) , (3 3.23)
f(q
1
, p
1
, . . . , q
k
, p
k
)
z
H =
p
2
r
2m
+
1
2mr
2
f(θ, p
θ
, ϕ, p
ϕ
) , f = p
2
θ
+
p
2
ϕ
sin
2
θ
+ 2ma cos θ .
f
f = M
2
+ 2ma cos θ . (33.24)
E p
ϕ
L(x, ˙x) =
m ˙x
2
2
−
mω
2
x
2
2
− αx
3
+ βx ˙x
2
.
H(p, r) =
p
2
2m
− pa ,
a
H(p, x) =
p
2
2m
+
mω
2
0
x
2
2
+ λ
p
2
2m
+
mω
2
0
x
2
2
2
.
U(r) =
ar
r
3
,
a
α a
Λ(q, p, ˙q, t) =
s
X
i=1
p
i
˙q
i
− H(p, q, t) , ( 3 4 .1 )
q
i
p
i
δq
i
δp
i
Λ(q, p, ˙q, t)
˙p t
1
A(q
(1)
1
, . . . , q
(1)
s
) t
2
B( q
(2)
1
, . . . , q
(2)
s
)
A B
δq
i
(t
1
) = δq
i
(t
2
) = 0 , i = 1, 2, . . . , s . (34.2)
q
i
(t), p
i
(t)
q
i
(t), p
i
(t)
Σ =
t
2
Z
t
1
Λ(q, p, ˙q, t) dt (34.3)
δΣ = 0
δΣ =
t
2
Z
t
1
s
X
i=1
∂Λ
∂q
i
−
d
dt
∂Λ
∂ ˙q
i
δq
i
+
∂Λ
∂p
i
δp
i
dt +
s
X
i=1
∂Λ
∂ ˙q
i
δq
i
|
t
2
t
1
,
(34.4)
∂Λ
∂q
i
−
d
dt
∂Λ
∂ ˙q
i
= 0 ,
∂Λ
∂p
i
= 0 , i = 1, 2, . . . , s . (34.5)
∂Λ
∂q
i
= −
∂H
∂q
i
,
∂Λ
∂ ˙q
i
= p
i
,
∂Λ
∂p
i
= ˙q
i
−
∂H
∂p
i
. (34.6)
δΣ = 0
δS = 0
p
i
(t) p
i
= ∂L/∂ ˙q
i
Λ
t
1
t
2
Σ S
δΣ = 0
δq
i
δp
i
“ p
i
(t)
δΣ = 0
δS = 0 δp
i
(t)
H(p, q, t)
f(q, p, t) f
˙q
i
, ˙p
i
df(q, p, t)
dt
=
∂f
∂t
+
s
X
i=1
∂H
∂p
i
∂f
∂q
i
−
∂H
∂q
i
∂f
∂p
i
. (35.1)
f = f(q, p, t) g = g(q, p, t)
{f, g}
{f, g} ≡
s
X
i=1
∂f
∂p
i
∂g
∂q
i
−
∂f
∂q
i
∂g
∂p
i
. (35.2)
{f, c} = 0, c = const ,
{f, g} = −{g, f},
{f
1
+ f
2
, g} = {f
1
, g} + {f
2
, g},
{f
1
f
2
, g} = f
1
{f
2
, g} + {f
1
, g}f
2
,
∂
∂t
{f, g} =
∂f
∂t
, g
+
f,
∂g
∂t
,
{p
i
, f} =
∂f
∂q
i
, {q
i
, f} = −
∂f
∂p
i
,
{p
i
, p
j
} = {q
i
, q
j
} = 0, {p
i
, q
j
} = δ
ij
.
f g
df(q, p, t)
dt
=
∂f
∂t
+ {H, f}, (35.3)
˙q
i
= {H, q
i
}, ˙p
i
= {H, p
i
}. (35.4)
A(r, t)
v
i
=
p
i
m
−
e
mc
A
i
, ( 35.4)
{v
i
, v
j
} = −
e
m
2
c
({p
i
, A
j
} + {A
i
, p
j
}) =
= −
e
m
2
c
∂A
j
∂x
i
−
∂A
i
∂x
j
= −
e
m
2
c
3
X
k=1
e
ijk
B
k
,
e
ijk
z
B
{v
x
, v
y
} = −
eB
z
m
2
c
= −
ω
m
, ω =
eB
mc
. ( 35.5)
(x
0
, y
0
)
ω = −(e/mc)B
x
0
= x −
v
y
ω
, y
0
= y +
v
x
ω
. (35.6)
{mv
i
, x
j
} = δ
ij
{x
0
, y
0
} =
1
mω
. (35.7)
f(q, p)
g(q, p) h(q, p)
{f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0 , ( 35.8)
h(q, p) “
q
i
p
i
“ τ
t
“
dq
i
dτ
=
∂h
∂p
i
,
dp
i
dτ
= −
∂h
∂q
i
, ( 35.9)
q
i
p
i
τ
f(q(τ ), p(τ)) g( q(τ), p(τ)) τ
df
dτ
= {h, f},
dg
dτ
= {h, g},
d{f, g}
dτ
= {h, {f, g}} . (35.10)
d{f, g}
dτ
=
df
dτ
, g
+
f,
dg
dτ
. (35.11)
f, g, h
f = f(q, p, t) g = g(q, p, t) h = h(q, p, t)
t
p q
f(q, p, t) g(q, p, t)
h = {f, g}
df
dt
=
∂f
∂t
+ {H, f} = 0,
dg
dt
=
∂g
∂t
+ {H, g} = 0 ,
dh
dt
=
∂h
∂t
+ {H, h} = 0 .
U(r) =
kr
2
/2
M = (0, 0, M) , M = xp
y
− yp
x
x y
E
x
=
p
2
x
2m
+
kx
2
2
, E
y
=
p
2
y
2m
+
ky
2
2
.
N = {E
x
, M} =
p
x
p
y
m
+ kxy ,
{E
y
, M} = −N
{E
x
, N} = −{E
y
, N} = −
k
m
M .
{M
i
, x
j
} {M
i
, p
j
} {M
i
, M
j
}
{ap, br} {aM, br} {aM, bM}
{M, rp} {p, r
n
} {p, (ar)
2
}
x
i
p
i
M
i
a b
{M
z
, ϕ} = 0 ϕ
ϕ = ϕ(r
2
, p
2
, (rp))
{M
z
, f} = [n, f ] n
z f
f = r ϕ
1
+ p ϕ
2
+ [r, p] ϕ
3
ϕ
i
= ϕ
i
(r
2
, p
2
, (rp))
q
i
Q
i
q
i
= q
i
(Q, t) , i = 1, . . . , s
∂(q
1
, . . . , q
s
)
∂(Q
1
, . . . , Q
s
)
6= 0) .
“
q
i
, p
i
“ Q
i
, P
i
q
i
= q
i
(Q, P, t), p
i
= p
i
(Q, P, t) , i = 1, . . . , s , (36.1)
s
X
i=1
p
i
dq
i
=
s
X
i=1
P
i
dQ
i
+ dF (q, Q,
˘
t) , (36.2)
dF (q, Q,
˘
t) F (q, Q, t)
dF (q, Q,
˘
t) =
dF (q, Q, t) −
∂F
∂t
dt
t=
˘
t
. (36.3)
t
F (q, Q, t)
p
i
=
∂F
∂q
i
, P
i
= −
∂F
∂Q
i
. (36.4)