a) The (covariant) Lagrangian of a particle interacting with an electromagnetic field is given by L (x, u) = -mc₁√√/u³ uz Use the Lagrange equations to derive the equation of motion d ƏL dr Jua dua dT с Here we have used the tensor Faß = Jª A³ - ³ Aª. m Use the Lagrange equations L (№ª A³, A³, x) = == b) The Lagrangian (density) of the electromagnetic field is given by 1 ƏL əxa to derive the inhomogeneous Maxwell equations Ja Faß 1 16п Faß up. = ә ƏL ƏL дха д(дº AB) AB с -Faß Fa = 4π js. Aªua. Faß Aa aja.
a) The (covariant) Lagrangian of a particle interacting with an electromagnetic field is given by L (x, u) = -mc₁√√/u³ uz Use the Lagrange equations to derive the equation of motion d ƏL dr Jua dua dT с Here we have used the tensor Faß = Jª A³ - ³ Aª. m Use the Lagrange equations L (№ª A³, A³, x) = == b) The Lagrangian (density) of the electromagnetic field is given by 1 ƏL əxa to derive the inhomogeneous Maxwell equations Ja Faß 1 16п Faß up. = ә ƏL ƏL дха д(дº AB) AB с -Faß Fa = 4π js. Aªua. Faß Aa aja.
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Transcribed Image Text:a) The (covariant) Lagrangian of a particle interacting with an electromagnetic field is given by
L (x, u) =
= -mc\/ubuz
Use the Lagrange equations
d dL
ƏL
dт дия
to derive the equation of motion
dua
m
dt
Here we have used the tensor Faß = JªA³ – ƏB Aª.
b) The Lagrangian (density) of the electromagnetic field is given by
1
L (ª A® , A® , x)
-Faß Faß
16т
Aª ja-
Use the Lagrange equations
ƏL
ƏL
dxa d(dª A®)
to derive the inhomogeneous Maxwell equations
J Faß
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