Сягло И.С. Теоретическая механика
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S
s
q
i
2s
S
S(q
i
, t) = f
o
(t) + f
1
(q
1
) + f
2
(q
2
) + . . . + f
s
(q
s
).
S = f
o
(t)+W (q
i
)
df
o
dt
= −H(q
i
,
∂W
∂q
i
).
df
o
dt
= −E, H(q
i
,
∂W
∂q
i
) = E.
f
o
= −Et
S = −Et + W (q
i
).
S W
1
2m
dW
dq
2
+
mω
2
q
2
2
= E.
S
S = −Et +
Z
v
u
u
u
t
2m(E −
1
2
mω
2
q
2
) dq.
E
∂S
∂E
= −t +
Z
(2mE − m
2
ω
2
q
2
)
−
1
2
m dq = C
q
q =
v
u
u
u
t
2E
mω
2
sin(ω(t + C)).
m
U
x = ξη, y =
1
2
(ξ
2
− η
2
).
L =
m
2
( ˙x
2
+ ˙y
2
) + c(x ˙y − y ˙x) −
k
2
(x
2
+ y
2
)
L = −mc
2
r
1 −
v
2
c
2
H =
p
2
2
+
k
2
x
2
+ λ(
p
2
2
+
k
2
x
2
)
2
.
l m
(x
i
, x
j
), (p
i
, x
j
), (M
i
, x
j
), (M
i
, p
j
), (M
i
, M
j
),
x
i
p
i
M
i
m
m
∆V
∆m ∆V
x
o
, y
o
, z
o
~r
o
~r(t, ~r
o
) x
o
, y
o
, z
o
~r
o
x
o
, y
o
, z
o
~v =
∂~r
∂t
¯
¯
¯
¯
¯
¯
~r
o
=
=
d~r
dt
, ~w =
∂~v
∂t
¯
¯
¯
¯
¯
¯
~r
o
=
=
d~v
dt
.
~r
o
=
~r(t, ~r
o
) ~r
o
(t, ~r)
~r
o
(t, ~r)
t
~r
~r
~r(t, ~r
o
)
ρ(~r, t)
dρ
dt
=
∂ρ(~r(~r
o
, t), t)
∂t
¯
¯
¯
¯
¯
¯
¯
~r
o
=
=
∂ρ
∂t
¯
¯
¯
¯
¯
¯
~r=
+ (~v∇)ρ,
∇ =
~
i
∂
∂x
+
~
j
∂
∂y
+
~
k
∂
∂z
.
~w =
d~v
dt
=
∂~v
∂t
+ (~v∇)~v.
ρ = lim
∆V →0
∆m
∆V
, lim
∆V →0
∆~p
∆V
= ρ~v, lim
∆V →0
∆
~
M
∆V
= ρ[~r ×~v].
m =
Z
V
ρ dV, ~p =
Z
V
ρ~v dV,
~
M =
Z
V
ρ[~r ×~v] dV.
-
-
v∆t
V (t)
dS
~n
~v
dV
S
∆t dS
v∆t
dV
S
= ~v~n∆t dS
∆V =
I
S
dV
S
= ∆t
I
S
~v~n dS.