Em Geometria Diferencial, as fórmulas de Cartan relacionam a derivada de Lie ao longo de um campo vetorial, a derivada exterior e a contração.
Derivada de Lie e Contração
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Seja
X
{\displaystyle X}
(
φ
t
)
t
{\displaystyle (\varphi ^{t})_{t}}
X
{\displaystyle X}
ω
{\displaystyle \omega }
L
X
ω
{\displaystyle {\mathcal {L}}_{X}\omega }
L
X
ω
=
lim
t
→
0
(
φ
t
)
∗
ω
−
ω
t
.
{\displaystyle {\mathcal {L}}_{X}\omega =\lim _{t\to 0}{\frac {(\varphi ^{t})^{\ast }\omega -\omega }{t}}.}
Essa forma diferencial é chamada de derivada de Lie de
ω
{\displaystyle \omega }
X
{\displaystyle X}
L
X
(
ω
∧
θ
)
=
(
L
X
ω
)
∧
θ
+
ω
∧
(
L
X
θ
)
{\displaystyle {\mathcal {L}}_{X}(\omega \wedge \theta )=({\mathcal {L}}_{X}\omega )\wedge \theta +\omega \wedge ({\mathcal {L}}_{X}\theta )}
L
X
(
d
θ
)
=
d
(
L
X
θ
)
.
{\displaystyle {\mathcal {L}}_{X}(d\theta )=d({\mathcal {L}}_{X}\theta ).}
Para
Y
{\displaystyle Y}
L
X
Y
{\displaystyle {\mathcal {L}}_{X}Y}
L
X
Y
|
p
=
lim
t
→
0
φ
∗
−
t
Y
φ
t
(
p
)
−
Y
p
t
.
{\displaystyle {\mathcal {L}}_{X}Y\vert _{p}=\lim _{t\to 0}{\frac {\varphi _{\ast }^{-t}Y_{\varphi ^{t}(p)}-Y_{p}}{t}}.}
Trata-se do colchete de Lie
L
X
Y
=
[
X
,
Y
]
{\displaystyle {\mathcal {L}}_{X}Y=[X,Y]}
[
X
,
Y
]
(
f
)
=
X
(
Y
(
f
)
)
−
Y
(
X
(
f
)
)
{\displaystyle [X,Y](f)=X(Y(f))-Y(X(f))}
Se
ω
∈
Ω
r
(
M
)
{\displaystyle \omega \in \Omega ^{r}(M)}
ω
{\displaystyle \omega }
X
{\displaystyle X}
(
r
−
1
)
{\displaystyle (r-1)}
i
X
ω
(
X
1
,
…
,
X
r
−
1
)
=
ω
(
X
,
X
1
,
…
,
X
r
−
1
)
{\displaystyle i_{X}\omega (X_{1},\ldots ,X_{r-1})=\omega (X,X_{1},\ldots ,X_{r-1})}
X
⌟
ω
{\displaystyle X\mathbin {\lrcorner } \omega }
i
X
(
ω
∧
θ
)
=
(
i
X
ω
)
∧
θ
+
(
−
1
)
r
ω
∧
(
i
X
θ
)
{\displaystyle i_{X}(\omega \wedge \theta )=(i_{X}\omega )\wedge \theta +{(-1)}^{r}\omega \wedge (i_{X}\theta )}
produto exterior .
As fórmulas mágicas de Cartan
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Fórmula de homotopia de Cartan
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A fórmula é
L
X
=
i
X
∘
d
+
d
∘
i
X
{\displaystyle {\mathcal {L}}_{X}=i_{X}\circ d+d\circ i_{X}}
d
f
{\displaystyle df}
ω
{\displaystyle \omega }
θ
{\displaystyle \theta }
ω
∧
θ
{\displaystyle \omega \wedge \theta }
[ 1]
Segunda fórmula mágica
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A segunda fórmula de Cartan é
i
[
X
,
Y
]
=
L
X
∘
i
Y
−
i
Y
∘
L
X
{\displaystyle i_{[X,Y]}={\mathcal {L}}_{X}\circ i_{Y}-i_{Y}\circ {\mathcal {L}}_{X}}
L
X
Y
=
[
X
,
Y
]
{\displaystyle {\mathcal {L}}_{X}Y=[X,Y]}
Definição invariante da derivada exterior
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Seja
ω
∈
Ω
1
(
M
)
{\displaystyle \omega \in \Omega ^{1}(M)}
X
{\displaystyle X}
Y
{\displaystyle Y}
M
{\displaystyle M}
L
X
∘
i
Y
=
d
∘
i
X
∘
i
Y
+
i
X
∘
d
∘
i
Y
{\displaystyle {\mathcal {L}}_{X}\circ i_{Y}=d\circ i_{X}\circ i_{Y}+i_{X}\circ d\circ i_{Y}}
i
Y
∘
L
X
=
i
Y
∘
i
X
∘
d
+
i
Y
∘
d
∘
i
X
{\displaystyle i_{Y}\circ {\mathcal {L}}_{X}=i_{Y}\circ i_{X}\circ d+i_{Y}\circ d\circ i_{X}}
i
[
X
,
Y
]
=
d
∘
i
X
∘
i
Y
+
i
X
∘
d
∘
i
Y
−
i
Y
∘
i
X
∘
d
−
i
Y
∘
d
∘
i
X
{\displaystyle i_{[X,Y]}=d\circ i_{X}\circ i_{Y}+i_{X}\circ d\circ i_{Y}-i_{Y}\circ i_{X}\circ d-i_{Y}\circ d\circ i_{X}}
ω
(
[
X
,
Y
]
)
=
X
(
ω
(
Y
)
)
−
d
ω
(
X
,
Y
)
−
Y
(
ω
(
X
)
)
{\displaystyle \omega ([X,Y])=X(\omega (Y))-d\omega (X,Y)-Y(\omega (X))}
d
ω
(
X
,
Y
)
=
X
(
ω
(
Y
)
)
−
Y
(
ω
(
X
)
)
−
ω
(
[
X
,
Y
]
)
.
{\displaystyle \mathrm {d} \omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y]).}
Com um pouco mais de trabalho, chegamos à fórmula
d
ω
(
X
1
,
…
,
X
k
+
1
)
=
∑
1
≤
i
≤
k
+
1
(
−
1
)
i
−
1
X
i
(
ω
(
X
1
,
…
,
X
i
^
,
…
,
X
k
+
1
)
)
+
∑
1
≤
i
<
j
≤
k
+
1
(
−
1
)
i
+
j
ω
(
[
X
i
,
X
j
]
,
X
1
,
…
,
X
i
^
,
…
,
X
j
^
,
…
,
X
k
+
1
)
,
{\displaystyle \mathrm {d} \omega (X_{1},\ldots ,X_{k+1})=\sum _{1\leq i\leq k+1}\left(-1\right)^{i-1}X_{i}{\bigl (}\omega (X_{1},\ldots ,{\widehat {X_{i}}},\ldots ,X_{k+1}){\bigr )}+\sum _{1\leq i<j\leq k+1}\left(-1\right)^{i+j}\omega {\bigl (}[X_{i},X_{j}],X_{1},\ldots ,{\widehat {X_{i}}},\ldots ,{\widehat {X_{j}}},\ldots ,X_{k+1}{\bigr )},}
que pode ser usada para definir a derivada exterior, sem menção a sistemas locais de coordenadas.
Parênteses de Poisson
editar
Considere uma variedade simplética
(
M
,
ω
)
{\displaystyle (M,\omega )}
{
⋅
,
⋅
}
:
C
∞
(
M
)
×
C
∞
(
M
)
→
C
∞
(
M
)
.
{\displaystyle \{\cdot ,\cdot \}\,\colon C^{\infty }(M)\times C^{\infty }(M)\to C^{\infty }(M).}
X
f
{\displaystyle \mathbf {X} _{f}}
X
g
{\displaystyle \mathbf {X} _{g}}
f
,
g
{\displaystyle f,g}
i
X
f
ω
=
−
d
f
,
i
X
g
ω
=
−
d
g
{\displaystyle i_{\mathbf {X} _{f}}\omega =-df,i_{\mathbf {X} _{g}}\omega =-dg}
i
[
X
f
,
X
g
]
ω
=
L
X
f
(
i
X
g
ω
)
−
i
X
g
(
L
X
f
ω
)
=
i
X
f
(
d
(
−
d
g
)
⏟
=
0
)
+
d
(
i
X
f
(
i
X
g
ω
)
)
−
i
X
g
(
i
X
f
d
ω
⏟
=
0
)
−
i
X
g
(
d
(
−
d
f
)
)
=
d
(
i
X
f
i
X
g
ω
)
=
−
d
{
f
,
g
}
.
{\displaystyle {\begin{aligned}i_{[\mathbf {X} _{f},\mathbf {X} _{g}]}\omega &={\mathcal {L}}_{\mathbf {X} _{f}}(i_{\mathbf {X} _{g}}\omega )-i_{\mathbf {X} _{g}}({\mathcal {L}}_{\mathbf {X} _{f}}\omega )\\&=i_{\mathbf {X} _{f}}(\underbrace {d(-dg)} _{=0})+d(i_{\mathbf {X} _{f}}(i_{\mathbf {X} _{g}}\omega ))-i_{\mathbf {X} _{g}}(i_{\mathbf {X} _{f}}\underbrace {d\omega } _{=0})-i_{\mathbf {X} _{g}}(d(-df))\\&=d(i_{\mathbf {X} _{f}}i_{\mathbf {X} _{g}}\omega )\\&=-d\{f,g\}.\end{aligned}}}
Portanto,
[
X
f
,
X
g
]
=
X
{
f
,
g
}
{\displaystyle [\mathbf {X} _{f},\mathbf {X} _{g}]=\mathbf {X} _{\{f,g\}}}
X
f
(
X
g
(
h
)
)
−
X
g
(
X
f
(
h
)
)
=
X
{
f
,
g
}
(
h
)
{\displaystyle \mathbf {X} _{f}(\mathbf {X} _{g}(h))-\mathbf {X} _{g}(\mathbf {X} _{f}(h))=\mathbf {X} _{\{f,g\}}(h)}
{
f
,
{
g
,
h
}
}
−
{
g
,
{
f
,
h
}
}
=
{
{
f
,
g
}
,
h
}
{\displaystyle \{f,\{g,h\}\}-\{g,\{f,h\}\}=\{\{f,g\},h\}}
{
f
,
{
g
,
h
}
}
+
{
g
,
{
h
,
f
}
}
+
{
h
,
{
f
,
g
}
}
=
0
{\displaystyle \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0}
identidade de Jacobi , que imediatamente nos dá o
Teorema de Poisson-Jacobi . Se
f
{\displaystyle f}
g
{\displaystyle g}
{
f
,
g
}
{\displaystyle \{f,g\}}
↑ Spivak, Michael D. (2010). Physics for Mathematicians: Mechanics I . Houston, TX: Publish or Perish 
![{\displaystyle {\mathcal {L}}_{X}Y=[X,Y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af17ef64b6fa6093e1fb097a39df26d067c2cb5a)
=X(Y(f))-Y(X(f))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50a880c76221522bb6235aad777a134b52e8d10e)
![{\displaystyle i_{[X,Y]}={\mathcal {L}}_{X}\circ i_{Y}-i_{Y}\circ {\mathcal {L}}_{X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4aa4b222df1518ae2535b10d2047f8a50ff4542)
![{\displaystyle i_{[X,Y]}=d\circ i_{X}\circ i_{Y}+i_{X}\circ d\circ i_{Y}-i_{Y}\circ i_{X}\circ d-i_{Y}\circ d\circ i_{X}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53c783de85382a1e02c30e292be793bcf9b7bb61)
![{\displaystyle \omega ([X,Y])=X(\omega (Y))-d\omega (X,Y)-Y(\omega (X))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f12e085c70751d1c788b85a8e9ae7ff164832aa)
![{\displaystyle \mathrm {d} \omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y]).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32b5204a3be11a311018941f5fe501a72fe76995)
![{\displaystyle \mathrm {d} \omega (X_{1},\ldots ,X_{k+1})=\sum _{1\leq i\leq k+1}\left(-1\right)^{i-1}X_{i}{\bigl (}\omega (X_{1},\ldots ,{\widehat {X_{i}}},\ldots ,X_{k+1}){\bigr )}+\sum _{1\leq i<j\leq k+1}\left(-1\right)^{i+j}\omega {\bigl (}[X_{i},X_{j}],X_{1},\ldots ,{\widehat {X_{i}}},\ldots ,{\widehat {X_{j}}},\ldots ,X_{k+1}{\bigr )},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31e582791b127ee218a94aa13de75c48c68c6bf0)
![{\displaystyle {\begin{aligned}i_{[\mathbf {X} _{f},\mathbf {X} _{g}]}\omega &={\mathcal {L}}_{\mathbf {X} _{f}}(i_{\mathbf {X} _{g}}\omega )-i_{\mathbf {X} _{g}}({\mathcal {L}}_{\mathbf {X} _{f}}\omega )\\&=i_{\mathbf {X} _{f}}(\underbrace {d(-dg)} _{=0})+d(i_{\mathbf {X} _{f}}(i_{\mathbf {X} _{g}}\omega ))-i_{\mathbf {X} _{g}}(i_{\mathbf {X} _{f}}\underbrace {d\omega } _{=0})-i_{\mathbf {X} _{g}}(d(-df))\\&=d(i_{\mathbf {X} _{f}}i_{\mathbf {X} _{g}}\omega )\\&=-d\{f,g\}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c804d0f0fcfc630f5b186c662a89d9fcfb76c922)
![{\displaystyle [\mathbf {X} _{f},\mathbf {X} _{g}]=\mathbf {X} _{\{f,g\}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5aacac33a3161fd3d5f37771be49c6b640d74f)
