Champs électromagnétique : formalisme lagrangien.

A= (A0, A1, A2, A3)
x= (x0, x1, x2, x3)
Ai=Ai(x)
A= (φ, −A)φ
Ax= (t, x)
E=−∂tA− ∇φ
H=∇ × A
A x
A E H x
Fik =∂Ak
∂xi
−∂Ai
∂xk
Fik =−Fki
F
EiHk
i, j, k
F
∂Fij
∂xk
+∂Fjk
∂xi
+∂Fki
∂xj
= 0
i6=j6=l4
∇ × E=−∂tH
∇.H= 0
ds
ds2=dx2
0−(dx2 1+dx2 2+dx2 3) =
3
X
i=0
ix2
i
0= 1 i≥1=−1
Lchamp =−
3
X
i,j=0
ijF2
ij
A
Fij
L=E2−H2
3
X
j=0
ij
∂Fij
∂xj
= 0
∇ × H=∂tE;∇.E= 0

(x0(u), x1(u), x2(u), x3(u))
u
x0
t(t, x1(t), x2(t), x3(t))
v=
(1,˙x1(t),˙x2(t),˙ x3(t))
q
j=qv
j= (q, j)
J=ˆV jdV
j
(ρ, j)
S=ˆV LchampdV +ˆV LinteractiondV
Linteraction =−ijiAi
j
∇.E=ρ
∇ × H=j+∂tE
S=−mˆC ds
m
S=−mˆC ds −qˆC iAidxi
xixi