Манаков Н.Л., Некипелов А.А., Овсянников В.Д. Задачи по теоретической механике
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ϕ
1
(p, q, t) ϕ
2
(p, q, t)
˙ϕ
1
≡ 0 ˙ϕ
2
≡ 0 ϕ
3
= {ϕ
1
, ϕ
2
}
˙ϕ
3
≡ 0
p q
P
α
= P
α
(p, q, t), Q
α
= Q
α
(p, q, t)
P Q
dP
α
dt
= −
∂H
0
∂Q
α
,
dQ
α
dt
=
∂H
0
∂P
α
, α = 1, . . . , s.
H
0
6= H P Q
q
α
, Q
β
; q
α
, P
β
; p
α
, Q
β
; p
α
, P
β
.
F = F(q
α
Q
β
t)
q
α
Q
β
p
α
=
∂F
∂q
α
, P
α
= −
∂F
∂Q
α
, H
0
= H +
∂F
∂t
.
F q Q Q
α
=
Q
α
(p, q, t) s
p
α
=
∂F
∂q
α
, α = 1, . . . , s,
P
α
(p, q, t) P
α
= −
∂F
∂Q
α
Q
α
(p, q, t)
q
α
P
β
Φ = Φ(q, P, t)
Q
α
=
∂Φ
∂P
α
, p
α
=
∂Φ
∂q
α
, H
0
= H +
∂Φ
∂t
.
D ≡
¯
¯
¯
¯
∂(Q
1
, . . . , Q
s
, P
1
, . . . , P
s
)
∂(q
1
, . . . , q
s
, p
1
, . . . , p
s
)
¯
¯
¯
¯
= 1
{ϕ
1
, ϕ
2
} ≡ {ϕ
1
, ϕ
2
}
p,q
= {ϕ
1
, ϕ
2
}
P,Q
.
{P
α
, Q
β
}
p,q
= δ
αβ
, {P
α
, P
β
}
p,q
= {Q
α
, Q
β
}
p,q
= 0.
˙q
α
p
α
L(x, y, z, ˙x, ˙y, ˙z) =
m
2
( ˙x
2
+ ˙y
2
+ ˙z
2
) − mgz,
p
x
=
∂L
∂ ˙x
= m ˙x, p
y
=
∂L
∂ ˙y
= m ˙y, p
z
=
∂L
∂ ˙z
= m ˙z.
H(x, y, z, p
x
, p
y
, p
z
) =
p
2
x
+ p
2
y
+ p
2
z
2m
+ mgz.
˙p
x
= 0, ˙p
y
= 0, ˙p
z
= −mg, ˙x =
p
x
m
, ˙y =
p
y
m
, ˙z =
p
z
m
.
p
x
(t) = p
ox
, p
y
(t) = p
oy
, p
z
(t) = p
oz
− mgt,
x(t) =
p
0x
t
m
+ x
0
, y(t) =
p
0y
t
m
+ y
0
, z(t) = −g
t
2
2
+
p
0z
t
m
+ z
0
.
p
0x
p
0y
p
0z
x
0
y
0
z
0
L(r, ϕ, z, ˙r, ˙ϕ, ˙z) =
m
2
( ˙r
2
+ r
2
˙ϕ
2
+ ˙z
2
) − mgz,
p
r
=
∂L
∂ ˙r
= m ˙r, p
ϕ
=
∂L
∂ ˙ϕ
= mr
2
˙ϕ, p
z
=
∂L
∂ ˙z
= m ˙z.
H(r, ϕ, z, p
r
, p
ϕ
, p
z
) =
1
2m
Ã
p
2
r
+
p
2
ϕ
r
2
+ p
2
z
!
+ mgz.
L(r, ϑ, ϕ, ˙r,
˙
ϑ, ˙ϕ) =
m
2
( ˙r
2
+ r
2
˙
ϑ
2
+ r
2
sin
2
ϑ ˙ϕ
2
) − mgr cos ϑ,
p
r
=
∂L
∂ ˙r
= m ˙r, p
ϑ
=
∂L
∂
˙
ϑ
= mr
2
˙
ϑ, p
ϕ
=
∂L
∂ ˙ϕ
= mr
2
sin
2
ϑ ˙ϕ,
H(r, ϑ, ϕ, p
r
, p
ϑ
, p
ϕ
) =
1
2m
Ã
p
2
r
+
p
2
ϑ
r
2
+
p
2
ϕ
r
2
sin
2
ϑ
!
+ mgr cos ϑ.
R = R(t)
2α
H =
1
2mR
2
(t)
Ã
p
2
ϑ
+
p
2
ϕ
sin
2
ϑ
!
+ mgR(t) cos ϑ;
˙
ϑ =
p
ϑ
mR
2
(t)
,
˙p
ϑ
=
p
2
ϕ
cos ϑ
mR
2
(t) sin
3
ϑ
+ mgR(t) sin ϑ ˙ϕ =
p
ϕ
mR
2
(t) sin
2
ϑ
˙p
ϕ
= 0.
ϑ = α H =
1
2m
Ã
p
2
r
+
p
2
ϕ
r
2
sin
2
α
!
+ mgr cos α,
˙r =
p
r
m
, ˙p
r
=
p
2
ϕ
mr
3
sin
2
α
− mg cos α, ˙ϕ =
p
ϕ
mr
2
sin
2
α
, ˙p
ϕ
= 0.
m L = −mc
2
p
1 − v
2
/c
2
c v ¿ c
p =
∂L
∂v
=
mv
p
1 − v
2
/c
2
,
H(p) = (p · v) − L(v) =
mc
2
p
1 − v
2
/c
2
= c
p
m
2
c
2
+ p
2
.
v ¿ c
(p
2
/m
2
c
2
) ¿ 1
H(p) = mc
2
+
p
2
2m
,
p mc
2
L(x, ˙x) =
˙x
2
2
−
ω
2
x
2
2
− αx
3
+ βx ˙x
2
,
ω α β
H(p x) =
p
2
2(1 + 2βx)
+
ω
2
x
2
2
+ αx
3
, ˙x =
p
1 + 2βx
,
˙p =
βp
2
(1 + 2βx)
2
− ω
2
x − 3αx
2
.
H(p, r) =
p
2
2m
− (p · a)
L(r,
˙
r) =
m
2
(
˙
r + a)
2
δr
i
= ε i = 1 N
δH = H(r
1
+ ε, . . . , r
N
+ ε) − H(r
1
, . . . , r
N
) =
Ã
ε ·
N
X
i=1
∂H
∂r
i
!
,
ε
N
X
i=1
∂H
∂r
i
= 0,
N
X
i=1
∂H
∂x
i
=
N
X
i=1
∂H
∂y
i
=
N
X
i=1
∂H
∂z
i
= 0.
P =
N
X
i=1
p
i
=
N
X
i=1
[i · p
xi
+ j · p
yi
+ k · p
zi
]
˙
P =
N
X
i=1
[i · ˙p
xi
+ j · ˙p
yi
+ k · ˙p
zi
].
˙
P = −
N
X
i=1
·
i ·
∂H
∂x
i
+ j ·
∂H
∂y
i
+ k ·
∂H
∂z
i
¸
= 0,
P = const
δϕ
δr
i
= [δϕ × r
i
], δp
i
= [δϕ × p
i
], i = 1, . . . , N.
δH =
Ã
δϕ ·
N
X
i=1
½·
∂H
∂r
i
× r
i
¸
+
·
∂H
∂p
i
× p
i
¸¾
!
= 0.
L =
N
X
i=1
[r
i
× p
i
].
˙
L =
N
X
i=1
½·
∂H
∂r
i
× r
i
¸
+
·
∂H
∂p
i
× p
i
¸¾
.
δϕ L = const
ϕ θ ψ
l
J
1
= J
2
6= J
3
T =
1
2
£
(J
1
+ ml
2
)(ω
2
1
+ ω
2
2
) + J
3
ω
2
3
¤
.
§
ω
1
=
˙
θ, ω
2
= ˙ϕ sin θ, ω
3
= ˙ϕ cos θ +
˙
ψ; U = mgl cos θ,
L =
J
1
+ ml
2
2
(
˙
θ
2
+ ˙ϕ
2
sin
2
θ) +
J
3
2
( ˙ϕ cos θ +
˙
ψ)
2
− mgl cos θ.
p
ϕ
=
∂L
∂ ˙ϕ
=
£
(J
1
+ ml
2
) sin
2
θ + J
3
cos
2
θ
¤
˙ϕ + J
3
cos θ
˙
ψ;
p
θ
=
∂L
∂
˙
θ
= (J
1
+ ml
2
)
˙
θ; p
ψ
=
∂L
∂
˙
ψ
= J
3
( ˙ϕ cos θ +
˙
ψ).
H =
1
2(J
1
+ ml
2
)
·
p
2
θ
+
(p
ϕ
− p
ψ
cos θ)
2
sin
2
θ
¸
+
p
2
ψ
2J
3
+ mgl cos θ.
Ω
H =
p
2
2m
− (Ω · [r × p]) + U(r)
U(r) = α/r
R = (m
1
r
1
+ m
2
r
2
)/M
r = r
1
− r
2
M = m
1
+ m
2
L =
M
˙
R
2
2
+
m
˙
r
2
2
−
α
r
H =
P
2
2M
+
p
2
2m
+
α
r
m = m
1
m
2
/M
m
l
J
H =
p
2
θ
2(J + ml
2
)
− mgl cos θ
˙
θ =
p
θ
J + ml
2
, ˙p
θ
= −mgl sin θ
θ ¿ 1 θ = θ
0
cos(ωt + α),
p
θ
= −(J + ml
2
)ωθ
0
sin(ωt + α) ω
2
=
mgl
J + ml
2
f(x p t) = x−
p
m
t
H(x p) =
p
2
2m
df
dt
=
∂f
∂t
+ {H, f} = −
p
m
+
p
m
= 0 f(x, p, t) = const.
˙p
i
= {H, p
i
}, ˙q
i
= {H, q
i
}.
p
i
q
i
→ q
0
i
= q
i
+ δq
i
{L, x} = {r, L
x
} = [r × i] {L, y} = {r, L
y
} = [r × j]
{L, z} = {r, L
z
} = [r × k];
{L, p
x
} = {p, L
x
} = [p × i] {L, p
y
} = {p, L
y
} = [p × j]
{L, p
z
} = {p, L
z
} = [p × k];
{L, L
x
} = [L × i] {L, L
y
} = [L × j] {L, L
z
} = [L × k]
r p L i j
k
{p, r
2
} {p
2
, r} {p, (a · r)} {(a · p), r}
{L
i
, p
2
} {L
i
, r
2
} {L, (r · p)} {(a · p), (b · r)}
b
2r 2p a a 0 0 0 (a · b)
r p, ϕ(r, p)
{L, ϕ(r, p)} = 0.
L
L = i · L
x
+ j · L
y
+ k · L
z
{L, ϕ(r, p)} = i{L
x
, ϕ(r, p)} + j{L
y
, ϕ(r, p)} + k { L
y
, ϕ(r, p)}.
{L
x
, ϕ(r, p)} =
∂L
x
∂p
x
·
∂ϕ
∂x
−
∂L
x
∂x
·
∂ϕ
∂p
x
+
∂L
x
∂p
y
·
∂ϕ
∂y
−
∂L
x
∂y
·
∂ϕ
∂p
y
+
+
∂L
x
∂p
z
·
∂ϕ
∂z
−
∂L
x
∂z
·
∂ϕ
∂p
z
.
r p
r
2
p
2
(r · p)
∂ϕ
∂x
= 2x
∂ϕ
∂(r
2
)
+ p
x
∂ϕ
∂(r · p)
,
∂ϕ
∂p
x
= 2p
x
∂ϕ
∂(p
2
)
+ x
∂ϕ
∂(r · p)
L
x
= yp
z
− zp
y
{L
x
, ϕ(r, p)} = −z
·
2y
∂ϕ
∂(r
2
)
+ p
y
∂ϕ
∂(r · p)
¸
− p
z
·
2p
y
∂ϕ
∂(p
2
)
+ y
∂ϕ
∂(r · p)
¸
+y
·
2z
∂ϕ
∂(r
2
)
+ p
z
∂ϕ
∂(r · p)
¸
+ p
y
·
2p
z
∂ϕ
∂(p
2
)
+ z
∂ϕ
∂(r · p)
¸
= 0.
{L
x
, F(r p)} {L
y
, F(r p)}
{L
z
, F(r p)}
F(r p) r p
[i × F] [j × F] [k × F] i j k
q Q; q P; p Q; p P
F = F(q, Q, t), p
α
=
∂F
∂q
α
, P
α
= −
∂F
∂Q
α
;
F = F(q, P, t), p
α
=
∂F
∂q
α
, Q
α
=
∂F
∂P
α
;
F = F(p, Q, t), q
α
= −
∂F
∂p
α
, P
α
= −
∂F
∂Q
α
;
F = F(p, P, t), q
α
= −
∂F
∂p
α
, Q
α
=
∂F
∂P
α
.
H
0
= H +
∂F
∂t
F(q, P) =
X
α
q
α
P
α
F(q, Q) =
X
α
q
α
Q
α
F(q, P) =
X
α
f
α
(q, t)P
α
f
α
p
α
=
∂F
∂q
α
= P
α
, Q
α
=
∂F
∂P
α
= q
α
, H
0
= H
p
α
=
∂F
∂q
α
= Q
α
, P
α
= −
∂F
∂Q
α
= −q
α
, H
0
= H
Q
α
=
∂F
∂P
α
= f
α
(q, t), p
α
=
∂F
∂q
α
=
X
β
∂f
β
∂q
α
P
β
,
H
0
= H +
∂F
∂t
= H +
X
α
∂f
α
∂t
P
α
F(q, P, t) = qP + (aq −bP)t a, b
H =
p
2
2m
Q = q − bt, p = P + at, H
0
=
p
2
2m
+ aq − bP.
H
0
P Q H
0
=
(P + at)
2
2m
+ aQ − bP + abt.
˙
P = − a
˙
Q =
P + at
m
− b.
P = −at + P
0
, Q =
P
0
t
m
− bt + Q
0
.
F =
mω
2
q
2
ctg Q