limitthirteen.mws

The special case v(theta)=-1 , with constraints,

Notation diff(v(thtea),theta$n)=v[n]

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restart:

the following substitution is made.

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P(t,theta) = P[infinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t);

>

grtw();

>	qload(gowdy);

>	grcalc(WeylSq);

>	gralter(_,13,6,7);

>

grmap(_,subs,P(t,theta) = P[infinity](theta)-v(theta)*ln(t)+1/4*exp(P[infinity](theta))^2*diff(Q[infinity](theta),`$`(theta,2))^2*t^2*ln(t)^2+V[1](theta)*t^2+(exp(2*P[infinity](theta))*psi[Q](theta)*diff(Q[infinity](theta),`$`(theta,2))-1/4*diff(Q[infinity](theta),`$`(theta,2))^2-1/4*diff(v(theta),`$`(theta,2)))*t^2*ln(t),Q(t,theta) = Q[infinity](theta)+psi[Q](theta)*t^(2*v(theta))+1/2*diff(Q[infinity](theta),`$`(theta,2))*t^(2*v(theta))*ln(t),`x`);

Keep all derivatives of Q[infinity](theta)

>	grmap(_,subs,diff(Q[infinity](theta),theta$5)=Q[5],`x`);

>	grmap(_,subs,diff(Q[infinity](theta),theta$4)=0,`x`);

>	grmap(_,subs,diff(Q[infinity](theta),theta$3)=0,`x`);

>	grmap(_,subs,diff(Q[infinity](theta),theta)=Q[1],`x`);

>	grmap(_,subs,diff(Q[infinity](theta),theta$2)=0,`x`);

Keep all derivatives of v(theta)

>	grmap(_,subs,diff(v(theta),theta$4)=v[4],`x`);

>	grmap(_,subs,diff(v(theta),theta$3)=v[3],`x`);

>	grmap(_,subs,diff(v(theta),theta$2)=v[2],`x`);

>	grmap(_,subs,diff(v(theta),theta$1)=0,`x`):

Keep all derivatives of psi[Q](theta)

>	grmap(_,subs,diff(psi[Q](theta),`$`(theta,3))=psi[Q3],`x`);

>	grmap(_,subs,diff(psi[Q](theta),`$`(theta,2))=0,`x`);

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>	grmap(_,subs,diff(psi[Q](theta),`$`(theta,4))=0,`x`);

Apply constraints

>	grmap(_,subs,diff(psi[Q](theta),theta)=0,`x`):

>	grmap(_,subs,psi[Q](theta)=0,`x`):

>	grmap(_,subs,v(theta)=-1,`x`):

>	gralter(_,6,7);

>	denom(grcomponent(WeylSq,[]));

>	factor(subs(t=0,numer(grcomponent(WeylSq,[]))));

>	core:=simplify(limit(factor(grcomponent(WeylSq,[])/(t*log(t)^2*exp(gamma(t,theta)))),t=0));

>	Ww:=t*log(t)^2*exp(gamma(t,theta))*core;

>	subs(v[2]=diff(v(theta),theta$2),Ww);

>

latex(%);

3\,t \left( \ln  \left( t \right)  \right) ^{2}{e^{\gamma \left( t,

\theta \right) }} \left( {\frac {d^{2}}{d{\theta}^{2}}}v \left( \theta

 \right)  \right) ^{2}

>