- #1
RedX
- 970
- 3
For some reason I can't derive the Hamiltonian from the Lagrangian for the E&M field. Here's what I have (using +--- metric):
[tex]
\begin{equation*}
\begin{split}
\mathcal L=\frac{-1}{4}F_{ \mu \nu}F^{ \mu \nu}
\\
\Pi^\mu=\frac{\delta \mathcal L}{\delta \dot{A_\mu}}=-F^{0 \mu}
\\
\mathcal H=\Pi^\mu \dot{A}_\mu -\mathcal L=-F^{0 \mu}\dot{A}_\mu +\frac{1}{4}F_{ \mu \nu}F^{ \mu \nu}
=-F^{0 \mu}\dot{A}_\mu+\frac{1}{4}(2F_{0i}F^{0i}+F_{ij}F^{ij})
\end{split}
\end{equation*}
[/tex]
But F0i=Ei, and Fij=-Bk, so this is equal to:
[tex]
\mathcal H=-F^{0 \mu}\dot{A}_\mu+\frac{1}{2}(-E_{i}^2+B_{i}^2) [/tex]
The Hamiltonian however should be one half the sum of the squares of the electric and magnetic fields. But I can't figure out what I did wrong. I almost have it, as the first term almost adds to the 2nd term to give that, but not quite.
Also, I'm not quite sure when using the (+---) metric whether the canonical momenta is:
[tex]
\Pi^\mu=\frac{\delta \mathcal L}{\partial^0 A_\mu}
[/tex]
or
[tex]
\Pi^\mu=\frac{\delta \mathcal L}{\partial_0 A_\mu}[/tex]
I don't think it matters in the derivation of the Hamiltonian, but which one do you use in the canonical commutation relations for example?
[tex]
\begin{equation*}
\begin{split}
\mathcal L=\frac{-1}{4}F_{ \mu \nu}F^{ \mu \nu}
\\
\Pi^\mu=\frac{\delta \mathcal L}{\delta \dot{A_\mu}}=-F^{0 \mu}
\\
\mathcal H=\Pi^\mu \dot{A}_\mu -\mathcal L=-F^{0 \mu}\dot{A}_\mu +\frac{1}{4}F_{ \mu \nu}F^{ \mu \nu}
=-F^{0 \mu}\dot{A}_\mu+\frac{1}{4}(2F_{0i}F^{0i}+F_{ij}F^{ij})
\end{split}
\end{equation*}
[/tex]
But F0i=Ei, and Fij=-Bk, so this is equal to:
[tex]
\mathcal H=-F^{0 \mu}\dot{A}_\mu+\frac{1}{2}(-E_{i}^2+B_{i}^2) [/tex]
The Hamiltonian however should be one half the sum of the squares of the electric and magnetic fields. But I can't figure out what I did wrong. I almost have it, as the first term almost adds to the 2nd term to give that, but not quite.
Also, I'm not quite sure when using the (+---) metric whether the canonical momenta is:
[tex]
\Pi^\mu=\frac{\delta \mathcal L}{\partial^0 A_\mu}
[/tex]
or
[tex]
\Pi^\mu=\frac{\delta \mathcal L}{\partial_0 A_\mu}[/tex]
I don't think it matters in the derivation of the Hamiltonian, but which one do you use in the canonical commutation relations for example?