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