Сягло И.С. Теоретическая механика
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δ
a δ~r
a
d~r
a
δ~r
a
X
a
~
f
A
a
δ~r
a
+
X
a
~
f
R
a
δ~r
a
= 0.
X
a
~
f
A
a
δ~r
a
= 0.
δ~r
a
~
f
A
a
= 0
δ~r
a
m
a
~w
a
X
a
(
~
f
A
a
− m
a
~w
a
)δ~r
a
= 0.
~
I
a
= −m
a
~w
a
X
a
(
~
f
A
a
+
~
I
a
)δ~r
a
= 0.
δ~r
a
δ~r
a
δq
i
t
~r
a
~r
a
δ~r
a
=
X
i
∂~r
a
∂q
i
δq
i
.
δ~r
a
X
a
(
~
f
A
a
− m
a
~w
a
)
∂~r
a
∂q
i
δq
i
= 0.
δq
i
X
a
(
~
f
A
a
− m
a
~w
a
)
∂~r
a
∂q
i
= 0.
X
a
~
f
a
δ~r
a
= −δU(~r
a
(q
i
), t).
t
δU(q
i
, t) =
X
i
∂U
∂q
i
δq
i
.
f
i
= −
∂U
∂q
i
.
Z
t
2
t
1
(
X
a
(
~
f
a
− m
a
~w
a
)δ~r
a
) dt = 0.
w
x
δx a
Z
t
2
t
1
w
x
δx dt =
Z
t
2
t
1
dv
x
dt
δx dt = v
x
δx
¯
¯
¯
¯
¯
¯
t
2
t
1
−
Z
t
2
t
1
v
x
d(δx)
dt
dt.
t
1
t
2
δx
Z
t
2
t
1
v
x
d(δx)
dt
dt =
Z
t
2
t
1
v
x
δv
x
dt =
Z
t
2
t
1
δ(
1
2
v
2
x
) dt.
Z
t
2
t
1
(
X
a
(
~
f
a
−m
a
~w
a
)δ~r
a
) dt =
Z
t
2
t
1
δ(−U +
X
a
m
a
v
2
a
2
) dt = δ
Z
t
2
t
1
(T −U) dt = 0.
L
L = T − U = L(~v
a
, ~r
a
, t) = L( ˙q
i
, q
i
, t).
S
S =
Z
t
2
t
1
L( ˙q
i
, q
i
, t) dt.
δS = 0.
q
i
(t)
t
1
t
2
q
i
(t)
m
1
, m
2
l
1
, l
2
ϕ
1
, ϕ
2
l
1
l
2
a
1
a
2
m
1
m
2
a
1
a
2
b
b
b
b
b
b
b
b
b
b
b
b
l
1
l
2
ϕ
2
ϕ
1
b
b
b
b
b
b
b
b
b
b
"!
#Ã
l
1
l
2
m
1
m
2
a
1
a
2
m l
C
D A B
C 2ϕ