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r38/packages/excalc/frames.red
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module frames; % Author: Eberhard Schruefer; global '(basisforml!* basisvectorl!* keepl!* naturalframe2coframe dbaseform2base2form dimex!* indxl!* naturalvector2framevector metricd!* metricu!* coord!* cursym!* detm!* commutator!-of!-framevectors); fluid '(alglist!* indl kord!* subfg!*); % indl needed by Common Lisp. symbolic procedure coframestat; begin scalar framel,metric; flag('(with),'delim); framel := cdr rlis(); remflag('(with),'delim); if cursym!* eq '!*semicol!* then go to a; if scan() eq 'metric then metric := xread t else if cursym!* eq 'signature then metric := rlis() else symerr('coframe,t); a: cofram(framel,metric) end; put('coframe,'stat,'coframestat); %put('cofram,'formfn,'formcofram); symbolic procedure cofram(u,v); begin scalar alglist!*; rmsubs(); u := for each j in u collect if car j eq 'equal then cdr j else list j; putform(caar u,1); basisforml!* := for each j in u collect !*a2k car j; indxl!* := for each j in basisforml!* collect cadr j; dimex!* := length u; basisvectorl!* := nil; if null v then metricd!* := nlist(1,dimex!*) else if car v eq 'signature then if dimex!* neq length cdr v then rerror(excalc,12, "Dimension of coframe and metric are inconsistent.") else metricd!* := for each j in cdr v collect aeval j; if null v or (car v eq 'signature) then <<detm!* := simp car metricd!*; for each j in cdr metricd!* do detm!* := multsq(simp j,detm!*); detm!* := mk!*sq detm!*; metricu!* := metricd!*:= pair(indxl!*,for each j in pair(indxl!*,metricd!*) collect list j)>> else mkmetric v; if flagp('partdf,'noxpnd) then remflag('(partdf),'noxpnd); putform('eps . indxl!*,0); put('eps,'indxsymmetries, list list('lambda,'(indl),list('tot!-asym!-indp, list('evlis,mkquote for j := 1:dimex!* collect list('nth,'indl,j))))); put('eps,'indxsymmetrize, list list('lambda,'(indl),list('asymmetrize!-inds, mkquote(for j := 1: dimex!* collect j),'indl))); flag('(eps),'covariant); setk('eps . for each j in indxl!* collect lowerind j,1); if null cdar u then return; keepl!* := append(for each j in u collect !*a2k car j . cadr j,keepl!*); coframe1 for each j in u collect cadr j end; symbolic procedure coframe1 u; begin scalar osubfg,scoord,v,y,w; osubfg := subfg!*; subfg!* := nil; v := for each j in u collect <<y := partitop j; scoord := pickupcoords(y,scoord); y>>; if null atom car scoord then <<remflag({caar scoord},'covariant); scoord := for each j in scoord collect mvar numr lc partitop j; v := for each j in u collect partitop j>>; if length scoord neq dimex!* then rerror(excalc,3,"badly formed basis"); w := !*pf2matwrtcoords(v,scoord); naturalvector2framevector := v; subfg!* := nil; naturalframe2coframe := pair(scoord, for each j in lnrsolve(w,for each k in basisforml!* collect list !*k2q k) collect mk!*sqpf partitsq!* car j); subfg!* := osubfg; coord!* := scoord; dbaseform2base2form := pair(basisforml!*, for each j in v collect mk!*sqpf repartit exdfpf j) end; symbolic procedure pickupcoords(u,v); %u is a pf, v a list. Picks up vars in exdf and declares them as %zero forms. if null u then v else if null eqcar(ldpf u,'d) then rerror(excalc,4,"badly formed basis") else if null v then <<putform(cadr ldpf u,0); pickupcoords(red u,cadr ldpf u . nil)>> else if ordop(cadr ldpf u,car v) then if cadr ldpf u eq car v then pickupcoords(red u,v) else <<putform(cadr ldpf u,0); pickupcoords(red u,cadr ldpf u . v)>> else pickupcoords(red u,car v . pickupcoords(!*k2pf ldpf u,cdr v)); symbolic procedure !*pf2matwrtcoords(u,v); if null u then nil else !*pf2colwrtcoords(car u,v) . !*pf2matwrtcoords(cdr u,v); symbolic procedure !*pf2colwrtcoords(u,v); if null v then nil else if u and (cadr ldpf u eq car v) then lc u . !*pf2colwrtcoords(red u,cdr v) else (nil ./ 1) . !*pf2colwrtcoords(u,cdr v); symbolic procedure coordp u; u memq coord!*; symbolic procedure mkmetric u; begin scalar x,y,z,okord; putform(list(cadr u,nil,nil),0); put(cadr u,'indxsymmetries, '((lambda (indl) (tot!-sym!-indp (evlis '((nth indl 1) (nth indl 2))))))); put(cadr u,'indxsymmetrize, '((lambda (indl) (symmetrize!-inds '(1 2) indl)))); flag(list cadr u,'covariant); okord := kord!*; kord!* := basisforml!*; x := simp!* caddr u; y := indxl!*; metricu!* := t; %to make simpindexvar work; for each j in indxl!* do <<for each k in y do setk(list(cadr u,lowerind j,lowerind k),0); y := cdr y>>; for each j on partitsq(x,'basep) do if ldeg ldpf j = 2 then setk(list(cadr u,lowerind cadr mvar ldpf j, lowerind cadr mvar ldpf j), mk!*sq lc j) else setk(list(cadr u,lowerind cadr mvar ldpf j, lowerind cadr mvar lc ldpf j), mk!*sq multsq(lc j,1 ./ 2)); kord!* := okord; x := for each j in indxl!* collect for each k in indxl!* collect simpindexvar list(cadr u,lowerind j,lowerind k); z := subfg!*; subfg!* := nil; y := lnrsolve(x,generateident length indxl!*); subfg!* := z; metricd!* := mkasmetric x; metricu!* := mkasmetric y; detm!* := mk!*sq detq x end; symbolic procedure mkasmetric u; for each j in pair(indxl!*,u) collect car j . begin scalar w,z; w := indxl!*; for each k in cdr j do <<if numr k then z := (car w . mk!*sq k) . z; w := cdr w>>; return z end; symbolic procedure frame u; begin scalar y; putform(list(car u,nil),-1); flag(list car u,'covariant); basisvectorl!* := for each j in indxl!* collect !*a2k list(car u,lowerind j); if null dbaseform2base2form then return; commutator!-of!-framevectors := for each j in pickupwedges dbaseform2base2form collect list(cadadr j,cadadr cdr j) . mk!*sqpf mkcommutatorfv(j, dbaseform2base2form); y := pair(basisvectorl!*, naturalvector2framevector); naturalvector2framevector := for each j in coord!* collect j . mk!*sqpf mknat2framv(j,y) end; symbolic procedure pickupwedges u; pickupwedges1(u,nil); Symbolic procedure pickupwedges1(u,v); if null u then v else if null cdar u then pickupwedges1(cdr u,v) else if null v then pickupwedges1((caar u . red cdar u) . cdr u, ldpf cdar u . nil) else if ldpf cdar u memq v then pickupwedges1(if red cdar u then (caar u . red cdar u) . cdr u else cdr u,v) else pickupwedges1(if red cdar u then (caar u . red cdar u) . cdr u else cdr u,ldpf cdar u . v); symbolic procedure mkbasevector u; !*a2k list(caar basisvectorl!*,lowerind u); symbolic procedure mkcommutatorfv(u,v); if null v then nil else addpf(mkcommutatorfv1(u,mkbasevector cadaar v,cdar v), mkcommutatorfv(u,cdr v)); symbolic procedure mkcommutatorfv1(u,v,w); if null w then nil else if u eq ldpf w then v .* negsq simp!* lc w .+ nil else if ordop(u,ldpf w) then nil else mkcommutatorfv1(u,v,red w); symbolic procedure mknat2framv(u,v); if null v then nil else addpf(mknat2framv1(u,caar v,cdar v),mknat2framv(u,cdr v)); symbolic procedure mknat2framv1(u,v,w); if null w then nil else if u eq cadr ldpf w then v .* lc w .+ nil else if ordop(u,cadr ldpf w) then nil else mknat2framv1(u,v,red w); symbolic procedure dualframe u; rerror(excalc,5,"Dualframe no longer supported - use frame instead"); symbolic procedure riemannconx u; riemconnection car u; put('riemannconx,'stat,'rlis); smacro procedure mkbasformsq u; mksq(list(caar basisforml!*,u),1); symbolic procedure riemconnection u; %calculates the riemannian connection and stores it in u; begin putform(list(u,nil,nil),1); flag(list u,'covariant); put(u,'indxsymmetries, '((lambda (indl) (tot!-asym!-indp (evlis '((nth indl 1) (nth indl 2))))))); put(u,'indxsymmetrize, '((lambda (indl) (asymmetrize!-inds '(1 2) indl)))); for each j in indxl!* do for each k in indxl!* do if (j neq k) and indordp(j,k) then setk(list(u,lowerind j,lowerind k),0); riemconpart1 u; riemconpart2 u; riemconpart3 u end; symbolic procedure riemconpart1 u; begin scalar covbaseform,indx1,indx2,indx3,varl,w,z; for each l in dbaseform2base2form do <<covbaseform := partitindexvar list(caar l, lowerind cadar l); for each j on cdr l do <<varl := cdr ldpf j; indx1 := cadar varl; indx2 := cadadr varl; for each y on covbaseform do <<w := list(u,lowerind indx1,lowerind indx2); z := multsq(-1 ./ 2,!*pf2sq multpfsq(lt y .+ nil, simp!* lc j)); setk(w,mk!*sq addsq(z,mksq(w,1))); indx3 := cadr ldpf y; z := multsq(-1 ./ 2,multsq(lc y,simp!* lc j)); if indx1 neq indx3 then if indordp(indx1,indx3) then <<w := list(u,lowerind indx1,lowerind indx3); setk(w,mk!*sq addsq(multsq(z,mkbasformsq indx2), mksq(w,1)))>> else <<w := list(u,lowerind indx3,lowerind indx1); setk(w,mk!*sq addsq(multsq(negsq z, mkbasformsq indx2),mksq(w,1)))>>; if indx2 neq indx3 then if indordp(indx2,indx3) then <<w := list(u,lowerind indx2,lowerind indx3); setk(w,mk!*sq addsq(multsq(negsq z, mkbasformsq indx1),mksq(w,1)))>> else <<w := list(u,lowerind indx3,lowerind indx2); setk(w,mk!*sq addsq(multsq(z, mkbasformsq indx1),mksq(w,1)))>> >>>>>> end; symbolic procedure riemconpart2 u; begin scalar dgkl,indx1,indx2,varl,w,z; if null(dgkl := mkmetricconx2 metricd!*) then return; for each j in dgkl do for each y on cdr j do <<varl := ldpf y; indx1 := cadar varl; indx2 := cadadr varl; w := list(u,lowerind indx1,lowerind indx2); z := multsq(-1 ./ 2,multsq(!*k2q car j,lc y)); setk(w,mk!*sq addsq(z,mksq(w,1)))>> end; symbolic procedure mkmetricconx2 u; if null u then nil else (if x then (ldpf mkupf list(caar basisforml!*,caar u) . x) . mkmetricconx2 cdr u else mkmetricconx2 cdr u) where x = mkmetricconx21 cdar u; symbolic procedure mkmetricconx21 u; if null u then nil else addpf(wedgepf2(exdf0 simp!* cdar u, !*k2pf list ldpf mkupf list(caar basisforml!*,caar u)), mkmetricconx21 cdr u); symbolic procedure riemconpart3 u; begin scalar dg,dgk,dgkl,w,x,z; if null (dg := mkmetricconx3 metricd!*) then return; remprop(u,'indxsymmetries); remprop(u,'indxsymmetrize); for each j in indxl!* do <<if dg and (dgk := atsoc(j,dg)) then dgk := cdr dgk else dgk := nil; for each k in indxl!* do if indordp(j,k) then <<w := list(u,lowerind j,lowerind k); x := if j eq k then nil ./ 1 else mksq(w,1); if dgk and (dgkl := atsoc(k,dgk)) then dgkl := cdr dgkl else dgkl := nil ./ 1; z := multsq(1 ./ 2,dgkl); setk(w,mk!*sq addsq(z,x)); w := list(u,lowerind k,lowerind j); setk(w,mk!*sq addsq(z,negsq x))>>>> end; symbolic procedure mkmetricconx3 u; if null u then nil else ((if x then (caar u . x) . mkmetricconx3 cdr u else mkmetricconx3 cdr u) where x = mkmetricconx31 cdar u); symbolic procedure mkmetricconx31 u; if null u then nil else ((if x then (caar u . x) . mkmetricconx31 cdr u else mkmetricconx31 cdr u) where x = !*pf2sq exdf0 simp!* cdar u); symbolic procedure basep u; if domainp u then nil else or(if sfp mvar u then basep mvar u else eqcar(mvar u,caar basisforml!*), basep lc u,basep red u); symbolic procedure wedgefp u; if domainp u then nil else or(if sfp mvar u then wedgefp mvar u else eqcar(mvar u,'wedge), wedgefp lc u,wedgefp red u); endmodule; end;