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