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r34/xmpl/excalc.tst
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— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 8360) [annotate] [blame] [check-ins using]
%Problem: Calculate the PDE's for the isovector of the heat equation. %-------- % (c.f. B.K. Harrison, f.B. Estabrook, "Geometric Approach...", % J. Math. Phys. 12, 653, 1971); %The heat equation @ psi = @ psi is equivalent to the set of exterior % xx t %equations (with u=@ psi, y=@ psi): % T x pform psi=0,u=0,x=0,y=0,t=0,a=1,da=2,b=2; a:=d psi - u*d t - y*d x; da:=- d u^d t - d y^d x; b:=u*d x^d t - d y^d t; %Now calculate the PDE's for the isovector; tvector v; pform vpsi=0,vt=0,vu=0,vx=0,vy=0; fdomain vpsi=vpsi(psi,t,u,x,y),vt=vt(psi,t,u,x,y),vu=vu(psi,t,u,x,y), vx=vx(psi,t,u,x,y),vy=vy(psi,t,u,x,y); v:=vpsi*@ psi + vt*@ t + vu*@ u + vx*@ x + vy*@ y; factor d; on rat; i1:=v |_ a - l*a; pform o=1; o:=ot*d t + ox*d x + ou*d u + oy*d y; fdomain f=f(psi,t,u,x,y); i11:=v _|d a - l*a + d f; let vx=-@(f,y),vt=-@(f,u),vu=@(f,t)+u*@(f,psi),vy=@(f,x)+y*@(f,psi), vpsi=f-u*@(f,u)-y*@(f,y); factor ^; i2:=v |_ b - xi*b - o^a + zet*da; let ou=0,oy=@(f,u,psi),ox=-u*@(f,u,psi), ot=@(f,x,psi)+u*@(f,y,psi)+y*@(f,psi,psi); i2; let zet=-@(f,u,x)-@(f,u,y)*u-@(f,u,psi)*y; i2; let xi=-@(f,t,u)-u*@(f,u,psi)+@(f,x,y)+u*@(f,y,y)+y*@(f,y,psi)+@(f,psi); i2; let @(f,u,u)=0; i2; % These PDE's have to be solved; clear a,da,b,v,i1,i11,o,i2,xi,t; remfdomain f; clear @(f,u,u); %Problem: %-------- %Calculate the integrability conditions for the system of PDE's: %(c.f. B.F. Schutz, "Geometrical Methods of Mathematical Physics" %Cambridge University Press, 1984, p. 156) % @ z /@ x + a1*z + b1*z = c1 % 1 1 2 % @ z /@ y + a2*z + b2*z = c2 % 1 1 2 % @ z /@ x + f1*z + g1*z = h1 % 2 1 2 % @ z /@ y + f2*z + g2*z = h2 % 2 1 2 ; pform w(k)=1,integ(k)=4,z(k)=0,x=0,y=0,a=1,b=1,c=1,f=1,g=1,h=1, a1=0,a2=0,b1=0,b2=0,c1=0,c2=0,f1=0,f2=0,g1=0,g2=0,h1=0,h2=0; fdomain a1=a1(x,y),a2=a2(x,y),b1=b1(x,y),b2=b2(x,y), c1=c1(x,y),c2=c2(x,y),f1=f1(x,y),f2=f2(x,y), g1=g1(x,y),g2=g2(x,y),h1=h1(x,y),h2=h2(x,y); a:=a1*d x+a2*d y$ b:=b1*d x+b2*d y$ c:=c1*d x+c2*d y$ f:=f1*d x+f2*d y$ g:=g1*d x+g2*d y$ h:=h1*d x+h2*d y$ %The equivalent exterior system:; factor d; w(1) := d z(-1) + z(-1)*a + z(-2)*b - c; w(2) := d z(-2) + z(-1)*f + z(-2)*g - h; indexrange 1,2; factor z; %The integrability conditions:; integ(k) := d w(k) ^ w(1) ^ w(2); clear a,b,c,f,g,h,w(k),integ(k); %Problem: %-------- %Calculate the PDE's for the generators of the d-theta symmetries of %the Lagrangian system of the planar Kepler problem. %c.f. W.Sarlet, F.Cantrijn, Siam Review 23, 467, 1981; %Verify that time translation is a d-theta symmetry and calculate the %corresponding integral; pform t=0,q(k)=0,v(k)=0,lam(k)=0,tau=0,xi(k)=0,et(k)=0,theta=1,f=0, l=0,glq(k)=0,glv(k)=0,glt=0; tvector gam,y; indexrange 1,2; fdomain tau=tau(t,q(k),v(k)),xi=xi(t,q(k),v(k)),f=f(t,q(k),v(k)); l:=1/2*(v(1)**2+v(2)**2)+m/r$ %The Lagrangian; pform r=0; fdomain r=r(q(k)); let @(r,q 1)=q(1)/r,@(r,q 2)=q(2)/r,q(1)**2+q(2)**2=r**2; lam(k):=-m*q(k)/r; %The force; gam:=@ t + v(k)*@(q(k)) + lam(k)*@(v(k))$ et(k) := gam _| d xi(k) - v(k)*gam _| d tau$ y :=tau*@ t + xi(k)*@(q(k)) + et(k)*@(v(k))$ %Symmetry generator; theta := l*d t + @(l,v(k))*(d q(k) - v(k)*d t)$ factor @; s := y |_ theta - d f$ glq(k):=@(q k) _|s; glv(k):=@(v k) _|s; glt:=@(t) _|s; %Translation in time must generate a symmetry; xi(k) := 0; tau := 1; glq k; glv k; glt; %The corresponding integral is of course the energy; integ := - y _| theta; clear l,lam k,gam,et k,y,theta,s,glq k,glv k,glt,t,q k,v k,tau,xi k; remfdomain r,f; %Problem: %-------- %Calculate the "gradient" and "Laplacian" of a function and the "curl" %and "divergence" of a one-form in elliptic coordinates; coframe e u=sqrt(cosh(v)**2-sin(u)**2)*d u, e v=sqrt(cosh(v)**2-sin(u)**2)*d v, e ph=cos u*sinh v*d ph; pform f=0; fdomain f=f(u,v,ph); factor e,^; on rat,gcd; order cosh v, sin u; %The gradient:; d f; factor @; %The Laplacian:; # d # d f; %Another way of calculating the Laplacian: -#vardf(1/2*d f^#d f,f); remfac @; %Now calculate the "curl" and the "divergence" of a one-form; pform w=1,a(k)=0; fdomain a=a(u,v,ph); w:=a(-k)*e k; %The curl:; x := # d w; factor @; %The divergence; y := # d # w; remfac @; clear x,y,w,u,v,ph,e k,a k; remfdomain a,f; %Problem: %-------- %Calculate in a spherical coordinate system the Navier Stokes equations; coframe e r=d r,e th=r*d th,e ph=r*sin th*d ph; frame x; fdomain v=v(t,r,th,ph),p=p(r,th,ph); pform v(k)=0,p=0,w=1; %We first calculate the convective derivative; w := v(-k)*e(k)$ factor e; on rat; cdv := @(w,t) + (v(k)*x(-k)) |_ w - 1/2*d(v(k)*v(-k)); %next we calculate the viscous terms; visc := nu*(d#d# w - #d#d w) + nus*d#d# w; %finally we add the pressure term and print the components of the %whole equation; pform nasteq=1,nast(k)=0; nasteq := cdv - visc + 1/rho*d p$ factor @; nast(-k) := x(-k) _| nasteq; remfac @,e; clear v k,x k,nast k,cdv,visc,p,w,nasteq; remfdomain p,v; %Problem: %-------- %Calculate from the Lagrangian of a vibrating rod the equation of % motion and show that the invariance under time translation leads % to a conserved current; pform y=0,x=0,t=0,q=0,j=0,lagr=2; fdomain y=y(x,t),q=q(x),j=j(x); factor ^; lagr:=1/2*(rho*q*@(y,t)**2-e*j*@(y,x,x)**2)*d x^d t; vardf(lagr,y); %The Lagrangian does not explicitly depend on time; therefore the %vector field @ t generates a symmetry. The conserved current is pform c=1; factor d; c := noether(lagr,y,@ t); %The exterior derivative of this must be zero or a multiple of the %equation of motion (weak conservation law) to be a conserved current; remfac d; d c; %i.e. it is a multiple of the equation of motion; clear lagr,c; %Problem: %-------- %Show that the metric structure given by Eguchi and Hanson induces a %self-dual curvature. %c.f. T. Eguchi, P.B. Gilkey, A.J. Hanson, "Gravitation, Gauge Theories % and Differential Geometry", Physics Reports 66, 213, 1980; for all x let cos(x)**2=1-sin(x)**2; pform f=0,g=0; fdomain f=f(r), g=g(r); coframe o(r) =f*d r, o(theta) =(r/2)*(sin(psi)*d theta-sin(theta)*cos(psi)*d phi), o(phi) =(r/2)*(-cos(psi)*d theta-sin(theta)*sin(psi)*d phi), o(psi) =(r/2)*g*(d psi+cos(theta)*d phi); frame e; pform gamma1(a,b)=1,curv2(a,b)=2; antisymmetric gamma1,curv2; factor o; gamma1(-a,-b):=-(1/2)*( e(-a) _|(e(-c) _|(d o(-b))) -e(-b) _|(e(-a) _|(d o(-c))) +e(-c) _|(e(-b) _|(d o(-a))) )*o(c)$ curv2(-a,b):=d gamma1(-a,b) + gamma1(-c,b)^gamma1(-a,c)$ factor ^; curv2(a,b):= curv2(a,b)$ let f=1/g; let g=sqrt(1-(a/r)**4); pform chck(k,l)=2; antisymmetric chck; %The following has to be zero for a self-dual curvature; chck(k,l):=1/2*eps(k,l,m,n)*curv2(-m,-n)+curv2(k,l); clear gamma1(a,b),curv2(a,b),f,g,chck(a,b),o(k),e(k); remfdomain f,g; %Problem: %-------- %Calculate for a given coframe and given torsion the Riemannian part and %the torsion induced part of the connection. Calculate the curvature. %For a more elaborate example see E.Schruefer, F.W. Hehl, J.D. McCrea, %"Application of the REDUCE package EXCALC to the Poincare gauge field %theory of gravity", to be submited to GRG Journal; pform ff=0, gg=0; fdomain ff=ff(r), gg=gg(r); coframe o(4)=d u+2*b0*cos(theta)*d phi, o(1)=ff*(d u+2*b0*cos(theta)*d phi)+ d r, o(2)=gg*d theta, o(3)=gg*sin(theta)*d phi with metric g=-o(4)*o(1)-o(4)*o(1)+o(2)*o(2)+o(3)*o(3); frame e; pform tor(a)=2,gwt(a)=2,gam(a,b)=1, u1=0,u3=0,u5=0; antisymmetric gam; fdomain u1=u1(r),u3=u3(r),u5=u5(r); tor(4):=0$ tor(1):=-u5*o(4)^o(1)-2*u3*o(2)^o(3)$ tor(2):=u1*o(4)^o(2)+u3*o(4)^o(3)$ tor(3):=u1*o(4)^o(3)-u3*o(4)^o(2)$ gwt(-a):=d o(-a)-tor(-a)$ %The following is the combined connection; %The Riemannian part could have equally well been calculated by the %RIEMANNCONX statement; gam(-a,-b):=(1/2)*( e(-b) _|(e(-c) _|gwt(-a)) +e(-c) _|(e(-a) _|gwt(-b)) -e(-a) _|(e(-b) _|gwt(-c)) )*o(c); pform curv(a,b)=2; antisymmetric curv; factor ^; curv(-a,b):=d gam(-a,b) + gam(-c,b)^gam(-a,c); showtime; end;