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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