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3
(E(t,x),B(t,x)) x∈R
∂ B+∇×E =0, ∇B =0,
t
Z
d
{L(E(t,x)+ε η(t,x),B(t,x)+ε β(t,x))−L(E(t,x),B(t,x))}dxdt =0,
dε
ε=0
(η,β)
L
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(E,B)∈R , E (E,B)
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E −B EB
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h(D,B)= sup ED−L(E,B), ∀D∈R , ∀B∈R
3
E∈R
′
∂ B+∇×(h (D,B))=0, ∇B =0,
t
D
′
∂ D−∇×(h (D,B))=0, ∇D =0,
t
B
∂ (h(D,B))+∇(D×B)=0,
t
∂ (D×B)+∇(Π(D,B))=0,
t
Π
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1 1
2 2 2 2
L(E,B)= (E −B ), h(D,B)= (D +B ),
2 2
q q
2 2 2 2 2 2
L(E,B) =− 1−E +B −(EB) , h(D,B) = 1+D +B +(D×B)
BORN−INFELD system
B,D << 1
B =0
p
2
L(E,0) =− 1−E , ∇×E =0.
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∂ B+∇×(B×v+ ) =0, ∇B =0,
t
h
B
∂ D+∇×(D×v− ) =0, ∇D =0,
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q
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h = 1+D +B +(D×B) , v = .
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augmented adding 4
B⊗B+D⊗D 1
∂ (hv)+∇(hv⊗v− )=∇( ), ∂ h+∇(hv) =0
t t
h h
6
D
∂ B+∇×(B×v+ ) =0, ∇B =0,
t
h
B
∂ D+∇×(D×v− ) =0, ∇D =0,
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DISREGARDING
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10×10
D
∂ B+∇×(B×v+ ) =0, ∇B =0,
t
h
B
∂ D+∇×(D×v− ) =0, ∇D =0,
t
h
B⊗B+D⊗D 1
∂ (hv)+∇(hv⊗v− ) =∇( ), ∂ h+∇(hv) =0,
t t
h h
2 2 2
1+D +B +(hv)
η(h,hv,D,B) = ,
h
classical
(t,x)→ (t,x+Ut), (h,v,D,B)→ (h,v−U,D,B),
3
U∈R
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∂ b+(v∇)b = (b∇)v−τ∇×d, ∂ d+(v∇)d = (d∇)v+τ∇×b,
t t
∂ τ +(v∇)τ = τ∇v, ∂ v+(v∇)v = (b∇)b+(d∇)d+τ∇τ,
t t
1 B D
τ = , b = , d = .
h h h
τ τ <0 τ =0
h∼∞
2 2 2 2
τ >0, τ +v +b +d =1, τv =d×b.
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10×10 linearly degenerate
∞
L
CONVEXHULL BI−MANIFOLD
natural
∇D =∇B = 0
q
2 2 2
h≥ 1+D +B +(hv) +2|D×B−hv|.
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