The classical Lagrangian for the electromagnetic field is
L=−14μ0FμνFμν−JμAμ.
Is there also a Hamiltonian? If so, how to derive it? I know how to write down the Hamiltonian from the Lagrangian where derivatives are taken only with respect to time, but I can't see the obvious way to generalize this.
Answer
Yes. There is a standard way to generalize to field theory.
Let a theory of n≥1 fields ϕi with a Lagrangian density L=L(ϕi,∂μϕi) be given. Here we use that standard abuse of notation in which ϕi denotes the vector whose components are the fields; ϕi=(ϕ1,…,ϕn).
To obtain the corresponding hamiltonian density, one first defines the following canonical momentum corresponding to the field ϕi: πi(x)=∂L∂˙ϕi(ϕi(x),∂μϕi(x)),˙ϕi:=∂tϕi
In your case, the fields are Aμ with corresponding momenta πμ.
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