Сягло И.С. Теоретическая механика
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~e
r
=
1
h
r
∂~r
∂r
= cos ϕ
~
i + sin ϕ
~
j, h
r
= 1,
~e
ϕ
=
1
h
ϕ
∂~r
∂ϕ
= −sin ϕ
~
i + cos ϕ
~
j, h
ϕ
= r.
h
r
h
ϕ
~e
r
~e
ϕ
~r r ϕ
d~r =
∂~r
∂r
dr +
∂~r
∂ϕ
dϕ = dr ~e
r
+ rdϕ~e
ϕ
.
dl
r
= dr, dl
ϕ
= rdϕ.
dt
~v = ˙r ~e
r
+ r ˙ϕ ~e
ϕ
,
v
r
=
dl
r
dt
= ˙r, v
ϕ
=
dl
ϕ
dt
= r ˙ϕ,
v
2
= ˙r
2
+ r
2
˙ϕ
2
.
oz
oz
~v = ˙r ~e
r
+ r ˙ϕ ~e
ϕ
+ ˙z
~
k,
v
r
= ˙r, v
ϕ
= r ˙ϕ, v
z
= ˙z,
v
2
= ˙r
2
+ r
2
˙ϕ
2
+ ˙z
2
.
x = r cos ϕ sin θ,
y = r sin ϕ sin θ,
z = r cos θ.
θ ϕ
~e
r
=
1
h
r
∂~r
∂r
= cos ϕ sin θ
~
i + sin ϕ sin θ
~
j + cos θ
~
k,
~e
θ
=
1
h
r
∂~r
∂θ
= cos ϕ cos θ
~
i + sin ϕ cos θ
~
j − sin θ
~
k,
~e
ϕ
=
1
h
ϕ
∂~r
∂ϕ
= −sin ϕ sin θ
~
i + cos ϕ cos θ
~
j,
h
r
= 1, h
θ
= r, h
ϕ
= r sin θ.
d~r =
∂~r
∂r
dr +
∂~r
∂θ
dθ +
∂~r
∂ϕ
dϕ = dr ~e
r
+ rdθ ~e
θ
+ r sin θdϕ ~e
ϕ
,
dl
r
= dr, dl
θ
= rdθ, dl
ϕ
= r sin θdϕ.
~v = ˙r ~e
r
+ r
˙
θ ~e
θ
+ r sin θ ˙ϕ ~e
ϕ
,
v
r
=
dl
r
dt
= ˙r, v
θ
=
dl
θ
dt
= r
˙
θ, v
ϕ
=
dl
ϕ
dt
= r sin θ ˙ϕ,
v
2
= ˙r
2
+ r
2
˙
θ
2
+ r
2
sin
2
θ
˙
ϕ
2
.
q
i
i
1, 2, 3
x = f
1
(q
i
), y = f
2
(q
i
), z = f
3
(q
i
).
~e
i
=
1
h
i
∂~r
∂q
i
.
h
i
f
j
(q
i
)
d~r = h
1
dq
1
~e
1
+ h
2
dq
2
~e
2
+ h
3
dq
3
~e
3
,
dl
1
= h
1
dq
1
, dl
2
= h
2
dq
2
, dl
3
= h
3
dq
3
.
~v = h
1
˙q
1
~e
1
+ h
2
˙q
2
~e
2
+ h
3
˙q
3
~e
3
,
v
1
= h
1
˙q
1
, v
2
= h
2
˙q
2
, v
3
= h
3
˙q
3
,
v
2
= h
2
1
˙q
2
1
+ h
2
2
˙q
2
2
+ h
2
3
˙q
2
3
.
