How to simplify the equation of motion for the electromagnetic field ?

[Meng-Sen Ma]

Dear all,

 We can easily derive the field equation of the electromagnetic field in xAct. However, the field equation is expressed according to the gauge field A_a. How can we simplify it to obtain the conventional compact form expressed by the field strength F_{a,b}?  Certainly, Maxwell equations are simple. For some nonlinear EM theories, the field equation may be very complicated. So we need a general method. 

[pedram karimi]

hi

you can consider F[-a,-b] and A[-a] as two independent variables and then by using chain law you can define some operator which can be written in covariant form as below:

EOM = ( \frac{\delta{ F_{a,b} } }{ \delta{ A_{c} } } ) \frac{ \delta{ L( F,A ) } }{ \delta{F_{a,b} } } +  \frac{ \delta{ L( F,A ) } }{ \delta{A_{c} } }.

then,

EOM = ( \delta^{a}_{c} \nabla^{b} - \delta^{b}_{c} \nabla^{a} ) \frac{ \delta{ L( F,A ) } }{ \delta{F_{a,b} } }  +  \frac{ \delta{ L( F,A ) } }{ \delta{A_{c} } }.

now this expression gives the EOM in the form of F and A.

I modify your notebook and find EOM for Maxwell Lagrangian and Proca lagrangian.

Notes: I consider manifold with no boundary, in the case of the manifold with boundary one must consider the boundary terms.

best,

Pedram

[Andrew Coates]

I think if you change the rule slightly it should work:

instead of

Rule$A1 = 

 MakeRule[{cd[-a]@A[-b], F[-a, -b] + cd[-b]@A[-a]}, MetricOn -> All]

try

Rule$A1 = 

 MakeRule[{cd[-a]@A[-b], 1/2(F[-a, -b] + cd[-b]@A[-a]+cd[-a]A[-b])}, MetricOn -> All]

then run a simplification. I think this should work.

[Meng-Sen Ma]

Thanks, Coates. Your method indeed works for the EM field. However, for more complicated nonlinear EM field, it does not work.

在 2017年7月8日星期六 UTC+8上午1:17:23，Andrew Coates写道：

[Leo Stein]

Dear Meng-Sen,

If you are sure that only antisymmetric pairs of ∇A will appear, then the following strategy might work. Replace only half of the ∇A—specifically, the half where the two indices are in some canonical order, say, lexicographic. Here is an example of such a rule:

LexARule = {cd[a_]@A[b_] /; OrderedQ[{a, b}] :> F[a, b] + cd[b]@A[a]}

However, I'm not sure if this will work when there are various dummy indices floating around.

Alternatively, you could try using xTerior. In this case, your Lagrangian will only have exterior derivatives appearing, which means that your equations of motion will also only have exterior derivatives. Then it is easy to replace the exterior derivative of A with F.

Best

Leo 

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[Ali Övgün]

Hi Meng-Sen Ma,

I think now, it is working. You can check from attached file.

Dr. Ali Övgün, Instituto de Física,  Pontificia Universidad Católica de Valparaíso, CHILE http://aovgun.weebly.com

[Ali Övgün]

This is the file.