The notation follows that of introduced in the article on jet bundles. Also, let
Γ
¯
(
π
)
{\displaystyle {\bar {\Gamma }}(\pi )}
π
{\displaystyle \pi \,}
A classical field theory is mathematically described by
A fibre bundle
(
E
,
π
,
M
)
{\displaystyle ({\mathcal {E}},\pi ,{\mathcal {M}})}
M
{\displaystyle {\mathcal {M}}}
n
{\displaystyle n\,}
A Lagrangian form
Λ
:
J
1
π
→
Λ
n
M
{\displaystyle \Lambda :J^{1}\pi \rightarrow \Lambda ^{n}M}
Let
⋆
1
{\displaystyle \star 1\,}
volume form on
M
{\displaystyle M\,}
Λ
=
L
⋆
1
{\displaystyle \Lambda =L\star 1\,}
L
:
J
1
π
→
R
{\displaystyle L:J^{1}\pi \rightarrow \mathbb {R} }
Lagrangian function .
We choose fibred co-ordinates
{
x
i
,
u
α
,
u
i
α
}
{\displaystyle \{x^{i},u^{\alpha },u_{i}^{\alpha }\}\,}
J
1
π
{\displaystyle J^{1}\pi \,}
⋆
1
=
d
x
1
∧
…
∧
d
x
n
{\displaystyle \star 1=dx^{1}\wedge \ldots \wedge dx^{n}}
The action integral
S
(
σ
)
=
∫
σ
(
M
)
(
j
1
σ
)
∗
Λ
{\displaystyle S(\sigma )=\int _{\sigma ({\mathcal {M}})}(j^{1}\sigma )^{*}\Lambda \,}
where
σ
∈
Γ
¯
(
π
)
{\displaystyle \sigma \in {\bar {\Gamma }}(\pi )}
open set
σ
(
M
)
{\displaystyle \sigma ({\mathcal {M}})\,}
j
1
σ
{\displaystyle j^{1}\sigma \,}
jet prolongation .
The variation of a section
σ
∈
Γ
¯
(
π
)
{\displaystyle \sigma \in {\bar {\Gamma }}(\pi )\,}
σ
t
=
η
t
∘
σ
{\displaystyle \sigma _{t}=\eta _{t}\circ \sigma \,}
η
t
{\displaystyle \eta _{t}\,}
π
{\displaystyle \pi \,}
V
{\displaystyle V\,}
E
{\displaystyle {\mathcal {E}}\,}
M
{\displaystyle {\mathcal {M}}\,}
σ
∈
Γ
¯
(
π
)
{\displaystyle \sigma \in {\bar {\Gamma }}(\pi )\,}
stationary with respect to the variations if
d
d
t
|
t
=
0
∫
σ
(
M
)
(
j
1
σ
t
)
∗
Λ
=
0
{\displaystyle \left.{\frac {d}{dt}}\right|_{t=0}\int _{\sigma ({\mathcal {M}})}(j^{1}\sigma _{t})^{*}\Lambda =0\,}
This is equivalent to
∫
M
(
j
1
σ
)
∗
L
V
1
Λ
=
0
{\displaystyle \int _{\mathcal {M}}(j^{1}\sigma )^{*}{\mathcal {L}}_{V^{1}}\Lambda =0\,}
where
V
1
{\displaystyle V^{1}\,}
V
{\displaystyle V\,}
Lie derivative .
Using Cartan's formula ,
L
X
=
i
X
d
+
d
i
X
{\displaystyle {\mathcal {L}}_{X}=i_{X}d+di_{X}\,}
Stokes' theorem and the compact support of
σ
{\displaystyle \sigma \,}
∫
M
(
j
1
σ
)
∗
i
V
1
d
Λ
=
0
{\displaystyle \int _{\mathcal {M}}(j^{1}\sigma )^{*}i_{V^{1}}d\Lambda =0\,}
Considering a
π
{\displaystyle \pi \,}
E
{\displaystyle {\mathcal {E}}}
V
=
β
α
∂
∂
u
α
{\displaystyle V=\beta ^{\alpha }{\frac {\partial }{\partial u^{\alpha }}}\,}
where
β
α
=
β
α
(
x
,
u
)
{\displaystyle \beta ^{\alpha }=\beta ^{\alpha }(x,u)\,}
contact forms
θ
j
=
d
u
j
−
u
i
j
d
x
i
{\displaystyle \theta ^{j}=du^{j}-u_{i}^{j}dx^{i}\,}
J
1
π
{\displaystyle J^{1}\pi \,}
V
{\displaystyle V\,}
V
1
=
β
α
∂
∂
u
α
+
(
∂
β
α
∂
x
i
+
∂
β
α
∂
u
j
u
i
j
)
∂
∂
u
i
α
{\displaystyle V^{1}=\beta ^{\alpha }{\frac {\partial }{\partial u^{\alpha }}}+\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}u_{i}^{j}\right){\frac {\partial }{\partial u_{i}^{\alpha }}}\,}
where
γ
i
α
=
γ
i
α
(
x
,
u
α
,
u
i
α
)
{\displaystyle \gamma _{i}^{\alpha }=\gamma _{i}^{\alpha }(x,u^{\alpha },u_{i}^{\alpha })\,}
i
V
1
d
Λ
=
[
β
α
∂
L
∂
u
α
+
(
∂
β
α
∂
x
i
+
∂
β
α
∂
u
j
u
i
j
)
∂
L
∂
u
i
α
]
⋆
1
{\displaystyle i_{V^{1}}d\Lambda =\left[\beta ^{\alpha }{\frac {\partial L}{\partial u^{\alpha }}}+\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}u_{i}^{j}\right){\frac {\partial L}{\partial u_{i}^{\alpha }}}\right]\star 1\,}
and hence
(
j
1
σ
)
∗
i
V
1
d
Λ
=
[
(
β
α
∘
σ
)
∂
L
∂
u
α
∘
j
1
σ
+
(
∂
β
α
∂
x
i
∘
σ
+
(
∂
β
α
∂
u
j
∘
σ
)
∂
σ
j
∂
x
i
)
∂
L
∂
u
i
α
∘
j
1
σ
]
⋆
1
{\displaystyle (j^{1}\sigma )^{*}i_{V^{1}}d\Lambda =\left[(\beta ^{\alpha }\circ \sigma ){\frac {\partial L}{\partial u^{\alpha }}}\circ j^{1}\sigma +\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}\circ \sigma +\left({\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}\circ \sigma \right){\frac {\partial \sigma ^{j}}{\partial x^{i}}}\right){\frac {\partial L}{\partial u_{i}^{\alpha }}}\circ j^{1}\sigma \right]\star 1\,}
Integrating by parts and taking into account the compact support of
σ
{\displaystyle \sigma \,}
∫
M
(
j
1
σ
)
∗
i
V
1
d
Λ
{\displaystyle \int _{\mathcal {M}}(j^{1}\sigma )^{*}i_{V^{1}}d\Lambda \,}
=
∫
M
[
∂
L
∂
u
α
∘
j
1
σ
−
∂
∂
x
i
(
∂
L
∂
u
i
α
∘
j
1
σ
)
]
(
β
α
∘
σ
)
⋆
1
{\displaystyle =\int _{\mathcal {M}}\left[{\frac {\partial L}{\partial u^{\alpha }}}\circ j^{1}\sigma -{\frac {\partial }{\partial x^{i}}}\left({\frac {\partial L}{\partial u_{i}^{\alpha }}}\circ j^{1}\sigma \right)\right](\beta ^{\alpha }\circ \sigma )\star 1\,}
=
0
{\displaystyle =0\,}
and since the
β
α
{\displaystyle \beta ^{\alpha }\,}
∂
L
∂
u
α
∘
j
1
σ
−
∂
∂
x
i
(
∂
L
∂
u
i
α
∘
j
1
σ
)
=
0
{\displaystyle {\frac {\partial L}{\partial u^{\alpha }}}\circ j^{1}\sigma -{\frac {\partial }{\partial x^{i}}}\left({\frac {\partial L}{\partial u_{i}^{\alpha }}}\circ j^{1}\sigma \right)=0\,}
These are the Euler-Lagrange Equations
![{\displaystyle i_{V^{1}}d\Lambda =\left[\beta ^{\alpha }{\frac {\partial L}{\partial u^{\alpha }}}+\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}u_{i}^{j}\right){\frac {\partial L}{\partial u_{i}^{\alpha }}}\right]\star 1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87ba7cd06e04b7189eaf530643a81d7373125205)
![{\displaystyle (j^{1}\sigma )^{*}i_{V^{1}}d\Lambda =\left[(\beta ^{\alpha }\circ \sigma ){\frac {\partial L}{\partial u^{\alpha }}}\circ j^{1}\sigma +\left({\frac {\partial \beta ^{\alpha }}{\partial x^{i}}}\circ \sigma +\left({\frac {\partial \beta ^{\alpha }}{\partial u^{j}}}\circ \sigma \right){\frac {\partial \sigma ^{j}}{\partial x^{i}}}\right){\frac {\partial L}{\partial u_{i}^{\alpha }}}\circ j^{1}\sigma \right]\star 1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92f663fa798bc9ce0bf294aff72496766a0b69c9)
\star 1\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e57ff7a47bb2a268eefb2dcaa145a5b657b0ff2)
