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MathGroup Archive 1993

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Re: functional derivatives

  • To: pmartin at landau.ucdavis.edu
  • Subject: Re: functional derivatives
  • From: Steven M. Christensen <stevec at yoda.physics.unc.edu>
  • Date: Thu, 18 Feb 93 23:44:21 EST
  • Cc: mathgroup at yoda.physics.unc.edu

With regard to Dr. Martin's question about functional
derivatives in field theory:

MathTensor can do this in:

Consider the Einstein action form constructed from
the Sqrt of the determinant of the spacetime metric
times the scalar curvature:

In[2]:= Sqrt[Detg] ScalarR

Out[2]= Sqrt[g] R

We assume that the above object is inside a spacetime integral.

In[3]:= Variation[%,Metricg]

MetricgFlag::off: MetricgFlag is turned off by this operation

                                                     pq
                                          Sqrt[g] R g   h
                    pq             p  q                  pq                pq
Out[3]= Sqrt[g] h      - Sqrt[g] h      + ----------------- - Sqrt[g] R   h
                 pq;              p ;q            2                    pq

It is easy to move the covariant derivatives around:

In[4]:=  PIntegrate[%,Metricg]

                   pq
        Sqrt[g] R g   h
                       pq                pq
Out[4]= ----------------- - Sqrt[g] R   h
                2                    pq

Here is a more complicated example:

In[5]:= Sqrt[Detg] RicciR[la,lb] RicciR[ua,ub]

                     ab
Out[5]= Sqrt[g] R   R
                 ab

In[6]:= Variation[%,Metricg]

                   pq  r                                       p qr
        -(Sqrt[g] h      R  )                        Sqrt[g] h      R
                     ;r   pq             p  rq                p ;    qr
Out[6]= --------------------- + Sqrt[g] h      R   - ------------------ - 
                  2                       q;    pr           2
 
                  r  pq                                  p     qr
     Sqrt[g] h      R                          Sqrt[g] h      R
              pq;r                     q  pr            p ;qr
>    ------------------ + Sqrt[g] h      R   - ------------------ + 
             2                     pq;r                2
 
              pq      rs
     Sqrt[g] g   R   R   h
                  rs      pq                qr  p                 pr  q
>    ----------------------- - Sqrt[g] R   R   h   - Sqrt[g] R   R   h
                2                       pq       r            pq       r


where h is the variation of the metric.  One chapter in our
manual does this with a simple example of variation of
a scalar field.  Other example computations using variations
of the metric are also shown.

I know of no other system that can handle this sort of thing since
a serious tensor system is required.

Steve Christensen






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