Манаков Н.Л., Некипелов А.А., Овсянников В.Д. Задачи по теоретической механике
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T U
q
i
˙q
i
˙q
i
L = L(q
i
, ˙q
i
, t)
T =
N
X
k=1
1
2
m
k
˙
r
2
k
, U = U(r
1
, . . . , r
N
)
r
i
˙
r
k
=
s
X
i=1
∂r
k
∂q
i
˙q
i
+
∂r
k
∂t
.
t T ˙q
i
T =
1
2
s
X
i=1
s
X
j=1
a
ij
(q
1
, . . . , q
s
) ˙q
i
˙q
j
, a
ij
=
N
X
k=1
m
k
µ
∂r
k
∂q
i
·
∂r
k
∂q
j
¶
.
F
( )
k
d
dt
∂L
∂ ˙q
i
−
∂L
∂q
i
= Q
i
, i = 1, 2, . . . , s,
Q
i
=
N
X
k=1
µ
F
( )
k
·
∂r
k
∂q
i
¶
δ
Z
Ldt = 0.
E
E =
s
X
i=1
˙q
i
∂L
∂ ˙q
i
− L.
L
∂L
∂t
= 0
dE
dt
= 0
E = const
L q
i
∂L
∂q
i
= 0
d
dt
∂L
∂ ˙q
i
= 0,
∂L
∂ ˙q
i
≡ p
i
= const,
p
i
q
i
s
q
1
q
s
q
i
˙q
i
T U
q
i
˙q
i
C
i
C
i
2s
§§ §§ §§
§ §§ §§
m
1
m
2
l
T =
m
1
˙z
2
1
2
+
m
2
˙z
2
2
2
.
U = −m
1
gz
1
− m
2
gz
2
.
z
1
+ z
2
= l = const
ÁÀ
¿
q
?
z
s
m
2
s
m
1
z = z
1
z
2
= l −z, ˙z
2
= −˙z L = T −U
L =
m
1
+ m
2
2
˙z
2
+ g(m
1
− m
2
)z + gm
2
l.
d
dt
∂L
∂ ˙z
−
∂L
∂z
= 0
(m
1
+ m
2
)¨z + (m
2
− m
1
)g = 0, ¨z =
m
1
− m
2
m
1
+ m
2
g
˙z =
m
1
− m
2
m
1
+ m
2
gt + V
0z
; z =
m
1
− m
2
m
1
+ m
2
g
t
2
2
+ V
0z
t + z
0
.
V
0z
, z
0
m
1
m
2
s(t)
T =
m
1
˙z
2
1
2
+
m
2
˙z
2
2
2
U = −m
1
gz
1
− m
2
gz
2
z
1
+ z
2
+ s(t) = l = const
q z = z
2
z
1
= l − s − z ˙z
1
= −˙s − ˙z
L =
m
1
2
( ˙s + ˙z)
2
+
m
2
˙z
2
2
+ m
1
g(l − s − z) + m
2
gz.
d
dt
∂L
∂ ˙z
−
∂L
∂z
= 0
m
1
(¨z + ¨s) + m
2
¨z − (m
2
− m
1
)g = 0.
˙z = −
m
1
˙s
m
1
+ m
2
+
m
2
− m
1
m
1
+ m
2
gt + C
1
;
z = −
m
1
s
m
1
+ m
2
+
m
2
− m
1
m
1
+ m
2
g
t
2
2
+ C
1
t + C
2
.
s(0) = z = z
0
˙s(0) = ˙z =
˙z = −
m
1
˙s
m
1
+ m
2
+
m
2
− m
1
m
1
+ m
2
gt; z = −
m
1
s
m
1
+ m
2
+
m
2
− m
1
m
1
+ m
2
g
t
2
2
+ z
0
.
m
h
k l
0
x
L =
m ˙x
2
2
−
k
2
³
p
h
2
+ x
2
− l
0
´
2
m
1
m
2
a
π/4
@
@
@
@
@
@
@
t
m
1
t
m
2
ϕ
π/4
L =
a
2
2
(m
1
cos
2
ϕ + m
2
sin
2
ϕ) ˙ϕ
2
−
ag
√
2
(m
1
sin ϕ + m
2
cos ϕ)
ϕ
m
2
2a
a
m
k
a
?
z
f
@
@
@
@
@
@
f
t
m
L =
m ˙z
2
2
−
k
2
³
p
a
2
+ 4z
2
− a
´
2
+ mgz
m
a
l
θ
y
L =
m
2
(l + aθ)
2
˙
θ
2
+ mg[(l + aθ) cos θ − a sin θ]
(l + aθ)
¨
θ + a
˙
θ
2
+ g sin θ = 0.
-
y
?x
ÁÀ
¿
@
@
@
@
@t
?
mg
¡
¡
θ
l
d
l
m
R ω
m
x = R cos ωt + l cos ϕ,
y = R sin ωt + l sin ϕ.
-
y
?x
H
H
H
H
R
ωt
b
@
@
@
@
@
lϕ
s
?
mg
˙x = −ωR sin ωt − l ˙ϕ sin ϕ, ˙y = Rω cos ωt + l ˙ϕ cos ϕ.
T =
m
2
( ˙x
2
+ ˙y
2
) =
m
2
£
R
2
ω
2
+ l
2
˙ϕ
2
+ 2Rωl ˙ϕ cos(ϕ − ωt)
¤
,
U = −mgx = −mRg cos ωt − mgl cos ϕ,
L = T − U
L =
m
2
£
R
2
ω
2
+ l
2
˙ϕ
2
+ 2Rωl ˙ϕ cos(ϕ − ωt)
¤
+ mgR cos ωt + mgl cos ϕ.
2Rωl ˙ϕ cos(ϕ − ωt)
2Rωl
d
dt
sin(ϕ − ωt) + 2Rω
2
l cos(ϕ − ωt)
L =
ml
2
˙ϕ
2
2
+ mRlω
2
cos(ϕ − ωt) + mgl cos ϕ.
s(t)
¨
θ +
µ
g
l
+
¨s
l
¶
sin θ = 0
l(t)
¨
θ +
2
l
˙
θ
˙
l +
g
l
sin θ = 0.
k
l
X x
L =
M
2
˙
X
2
+
µ
2
˙x
2
−
k
2
(x − l)
2
M = m
1
+m
2
µ =
m
1
m
2
m
1
+ m
2
, E =
M
2
˙
X
2
+
µ
2
˙x
2
+
k
2
(x−l)
2
= const
P
X
= M
˙
X = const
l m
2
m
1
L =
m
1
+ m
2
2
˙x
2
+ m
2
l ˙ϕ
µ
l
2
˙ϕ + ˙x cos ϕ
¶
+ m
2
gl cos ϕ
P
x
= (m
1
+ m
2
) ˙x + m
2
l ˙ϕ cos ϕ = const,
E =
m
1
+ m
2
2
˙x
2
+ m
2
l ˙ϕ
µ
l
2
˙ϕ + ˙x cos ϕ
¶
− m
2
gl cos ϕ = const.
l
2
m
2
m
1
l
1
L =
m
1
+ m
2
2
l
2
1
˙ϕ
2
1
+
m
2
2
l
2
2
˙ϕ
2
2
+
+m
2
l
1
l
2
˙ϕ
1
˙ϕ
2
cos(ϕ
1
−ϕ
2
)+(m
1
+m
2
)gl
1
cos ϕ
1
+m
2
gl
2
cos ϕ
2
.
-
?
H
H
H
H
l
1
m
1
ϕ
1
s
@
@
@
@
@
l
2
m
2
ϕ
2
s
m
1
= m
2
= m
k l
3l
L =
m
2
( ˙x
2
1
+ ˙x
2
2
) − k(x
2
1
+ x
2
2
− x
1
x
2
);
¨x
1
+ ω
2
(2x
1
− x
2
) = 0, ¨x
2
+ ω
2
(2x
2
− x
1
) = 0,
∨∨∨∨∨∨∨∨∨∨∨∨∨∨∨∨
t
m
t
m
¾ -
3l
x
1
x
2
ω
2
= k/m
a b α a
x
2
a
2
+
y
2
b
2
= 1.
ϕ
x = a cos ϕ, y = b sin ϕ.
T
=
m
2
( ˙
x
2
+ ˙
y
2
) =
m
2
(
a
2
sin
2
ϕ
+
b
2
cos
2
ϕ
) ˙
ϕ
2
.
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡
¡ª
x
a
b
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@
@R
y
J
J
J
J
J
J
J^
r
t
?
mg
α
U = −m(gr) = −mg
x
a cos ϕ − mg
y
b sin ϕ.
g
x
= g cos α g
y
= g sin α
U = −mg(a cos ϕ cos α + b sin ϕ sin α) =
= −
mg
2
[(a + b) cos(ϕ − α) + (a − b) cos(ϕ + α)].
L L = T − U =
m
2
(a
2
sin
2
ϕ + b
2
cos
2
ϕ) ˙ϕ
2
+
+
mg
2
[(a + b) cos(ϕ − α) + (a − b) cos(ϕ + α)] .
d
dt
∂L
∂ ˙q
i
−
∂L
∂q
i
= 0
(a
2
sin
2
ϕ + b
2
cos
2
ϕ) ¨ϕ +
a
2
− b
2
2
˙ϕ
2
sin 2ϕ+
+
g
2
[(a + b) sin(ϕ − α) + (a − b) sin(ϕ + α)] = 0.
α
-
x
6
y
α
t
@
@
@
@
¡
¡
¡ª
¡
¡
¡µ
s
y
2
m
x
1
x
2
x s x s
m
x
1
= x y
1
= 0 x
2
= x + s cos α, y
2
= s sin α
T =
M
2
˙x
2
1
+
m
2
( ˙x
2
2
+ ˙y
2
2
) =
M
2
˙x
2
+
m
2
¡
˙x
2
+ 2 ˙x ˙s cos α + ˙s
2
¢
.
U = mgy
2
= mgs sin α
L =
M
2
˙x
2
+
m
2
¡
˙x
2
+ 2 ˙x ˙s cos α + ˙s
2
¢
− mgs sin α.
d
dt
∂L
∂ ˙x
−
∂L
∂x
= 0,
d
dt
∂L
∂ ˙s
−
∂L
∂s
= 0
(M + m)¨x + m¨s cos α = 0,
m¨x cos α + m¨s + mg sin α = 0.
¨x ¨s
¨x =
mg sin α cos α
M + m sin
2
α
, ¨s = −
M + m
M + m sin
2
α
g sin α.
˙x =
mg sin α cos α
M + m sin
2
α
t, ˙s = −
M + m
M + m sin
2
α
gt sin α,
x =
mg sin α cos α
M + m sin
2
α
t
2
2
+ x
0
, s = −
M + m
M + m sin
2
α
g
t
2
2
sin α + s
0
.
t = 0
x(0) = x
0
, s(0) = s
0
, ˙x(0) = 0, ˙s(0) = 0.
x(t), s(t) x
2
y
2
x(t) s(t)
m
x = a(u − sin u), y = −a(1 − cos u), 0 6 u 6 2π
y
t = 0 y = −2a, |v| = v
0
.
6
y
-
x
0
2πa
−2a
s
m
T =
m
2
¡
˙x
2
+ ˙y
2
¢
, U = mgy. u
˙x = a(1 − cos u) ˙u, ˙y = −a ˙u sin u
L = T − U =
ma
2
2
˙u
2
£
(1 − cos u)
2
+ sin
2
u
¤
+ mga(1 − cos u) =
= ma ˙u
2
(1 − cos u) + mga(1 − cos u).
u
L u/2
L = 2ma ˙u
2
sin
2
u
2
+ 2mga sin
2
u
2
= 8ma
2
·
d(cos
u
2
)
dt
¸
2
+ 2mga
³
1 − cos
2
u
2
´
.
s = cos
u
2
4a
L = 8ma
2
˙s
2
+ 2mga(1 − s
2
).
¨s +
g
4a
s = 0
s = A cos(ωt + α),
ω =
p
g/4a A α
˙s(0) = 0 s
0
s = 0 τ = π
p
a/g
s
0
α = π/2, |A| = v
0
/
p
4ag
A t =
0
E = ˙s
∂L
∂ ˙s
− L = 8ma
2
˙s
2
− 2mga(1 − s
2
) =
mv
2
0
2
− 2mga.