How can I determine the equations and other conserved quantity of a field theory with Mathematica?
I would like solve the following Lagrangian density:
\[ScriptCapitalL]=-(1/4)Subscript[F,\[Mu]\[Nu]]F^\[Mu]\[Nu]+(Subscript[D,\[Mu]]\[Phi])SuperStar[(D^\[Mu]\[Phi])]-U(\[LeftBracketingBar]\[Phi]\[RightBracketingBar]).
?=-(1/4)Subscript[F,??]F^??+(Subscript[D,?]?)SuperStar[(D^??)]-U(\?\).
Hi Alexsandro,
Mathematica has some nice utilities for Lagrangian mechanics. Below is a link to an earlier posting. One of the replies is mine, and includes an answer which employs those utilities to obtain the equations of motion to a simple system.
Best regards,
David
http://community.wolfram.com/groups/-/m/t/110018?p_p_auth=cd0vYgL0
Well, but is it possible to do this in the covariant form?
Hi Alexsandro,
I am sure it is. I suspect you need to determine the covariant derivative for your field and then use it on the Lagrangian in the usual way. In the classical noncovariant form I used on the oscillator, it would have been almost as easy to do the work in Mathematica with explicit symbolic differentiation as as it was to use the built-in functionality. However, my memory of guage theory is not up to taking this any further. Maybe there is a physicist here with fresher knowledge?
Best,
David