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Sat Jun 29 14:08:44 PDT 1991 REDUCE 3.4, 15-Jul-91 ... 1: 1: 2: 2: *** ^ redefined 3: 3: %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; A := - d T*U - d X*Y + d PSI da:=- d u^d t - d y^d x; DA := d T^d U + d X^d Y b:=u*d x^d t - d y^d t; B := - d T^d X*U + d T^d Y %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; V := @ *VT + @ *VU + @ *VX + @ *VY + @ *VPSI T U X Y PSI factor d; on rat; i1:=v |_ a - l*a; I1 := d T*(@ VPSI - @ VT*U - @ VX*Y + L*U - VU) T T T + d U*(@ VPSI - @ VT*U - @ VX*Y) U U U + d X*(@ VPSI - @ VT*U - @ VX*Y + L*Y - VY) X X X + d Y*(@ VPSI - @ VT*U - @ VX*Y) Y Y Y + d PSI*(@ VPSI - @ VT*U - @ VX*Y - L) PSI PSI PSI pform o=1; o:=ot*d t + ox*d x + ou*d u + oy*d y; O := d T*OT + d U*OU + d X*OX + d Y*OY fdomain f=f(psi,t,u,x,y); i11:=v _|d a - l*a + d f; I11 := d T*(L*U - VU) + d U*VT + d X*(L*Y - VY) + d Y*VX - d PSI*L 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; I2 := d T^d U*(@ F + @ F*U + @ F*Y - U*OU + ZET) + d T^d X*( U X U Y U PSI @ F*U - @ F + @ F + @ F*U + @ F*Y - @ F*U - U*OX T U T X X X Y X PSI PSI + U*XI + Y*OT) + d T^d Y *( - @ F + @ F + @ F*U + @ F*Y + @ F - U*OY - XI) T U X Y Y Y Y PSI PSI + d T^d PSI*(@ F + @ F*U + @ F*Y - OT) X PSI Y PSI PSI PSI + d U^d X*(@ F*U + Y*OU) - d U^d Y*@ F - d U^d PSI*OU U U U U + d X^d Y*( - @ F - @ F*U - Y*OY + ZET) U X U Y - d X^d PSI*(@ F*U + OX) + d Y^d PSI*(@ F - OY) U PSI U PSI 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; d T^d U*(@ F + @ F*U + @ F*Y + ZET) + d T^d X*(@ F*U - @ F U X U Y U PSI T U T 2 + @ F*U + @ F + @ F*U + 2*@ F*Y + @ F*U*Y U PSI X X X Y X PSI Y PSI 2 + @ F*Y - @ F*U + U*XI) + d T^d Y PSI PSI PSI *( - @ F - @ F*U + @ F + @ F*U + @ F*Y + @ F - XI) T U U PSI X Y Y Y Y PSI PSI + d U^d X*@ F*U - d U^d Y*@ F U U U U + d X^d Y*( - @ F - @ F*U - @ F*Y + ZET) U X U Y U PSI let zet=-@(f,u,x)-@(f,u,y)*u-@(f,u,psi)*y; i2; 2 d T^d X*(@ F*U - @ F + @ F*U + @ F + @ F*U + 2*@ F*Y T U T U PSI X X X Y X PSI 2 + @ F*U*Y + @ F*Y - @ F*U + U*XI) + d T^d Y Y PSI PSI PSI PSI *( - @ F - @ F*U + @ F + @ F*U + @ F*Y + @ F - XI) T U U PSI X Y Y Y Y PSI PSI + d U^d X*@ F*U - d U^d Y*@ F U U U U - (2*d X^d Y)*(@ F + @ F*U + @ F*Y) U X U Y U PSI let xi=-@(f,t,u)-u*@(f,u,psi)+@(f,x,y)+u*@(f,y,y)+y*@(f,y,psi)+@(f,psi); i2; 2 d T^d X*( - @ F + @ F + 2*@ F*U + 2*@ F*Y + @ F*U T X X X Y X PSI Y Y 2 + 2*@ F*U*Y + @ F*Y ) + d U^d X*@ F*U Y PSI PSI PSI U U - d U^d Y*@ F - (2*d X^d Y)*(@ F + @ F*U + @ F*Y) U U U X U Y U PSI let @(f,u,u)=0; i2; 2 d T^d X*( - @ F + @ F + 2*@ F*U + 2*@ F*Y + @ F*U T X X X Y X PSI Y Y 2 + 2*@ F*U*Y + @ F*Y ) Y PSI PSI PSI - (2*d X^d Y)*(@ F + @ F*U + @ F*Y) U X U Y U PSI % 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; 1 W := d Z + d X*(Z *A1 + Z *B1 - C1) + d Y*(Z *A2 + Z *B2 - C2) 1 1 2 1 2 w(2) := d z(-2) + z(-1)*f + z(-2)*g - h; 2 W := d Z + d X*(Z *F1 + Z *G1 - H1) + d Y*(Z *F2 + Z *G2 - H2) 2 1 2 1 2 indexrange 1,2; factor z; %The integrability conditions:; integ(k) := d w(k) ^ w(1) ^ w(2); 1 INTEG := d Z ^d Z ^d X^d Y*Z *( - @ A1 + @ A2 + B1*F2 - B2*F1) + 1 2 1 Y X d Z ^d Z ^d X^d Y*Z 1 2 2 *( - @ B1 + @ B2 + A1*B2 - A2*B1 + B1*G2 - B2*G1) + Y X d Z ^d Z ^d X^d Y 1 2 *(@ C1 - @ C2 - A1*C2 + A2*C1 - B1*H2 + B2*H1) Y X 2 INTEG := d Z ^d Z ^d X^d Y*Z 1 2 1 *( - @ F1 + @ F2 - A1*F2 + A2*F1 - F1*G2 + F2*G1) Y X + d Z ^d Z ^d X^d Y*Z *( - @ G1 + @ G2 - B1*F2 + B2*F1) + 1 2 2 Y X d Z ^d Z ^d X^d Y 1 2 *(@ H1 - @ H2 + C1*F2 - C2*F1 - G1*H2 + G2*H1) Y X 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; 1 1 - Q *M LAM := --------- R 2 2 - Q *M LAM := --------- 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; 1 1 - @ (XI )*Q *M 1 1 1 1 1 2 V GLQ := 2*@ (XI )*V + @ (XI )*V + ------------------ 1 2 R Q Q 1 2 - @ (XI )*Q *M 2 V 1 2 2 + ------------------ + @ (XI ) + @ (XI )*V - @ F R T 1 1 Q Q 1 2 2 2 @ TAU*( - 3*(V ) *R - (V ) *R + 2*M) 1 Q 1 2 + --------------------------------------- - @ TAU*V *V 2*R 2 Q 1 1 2 1 @ TAU*Q *V *M @ TAU*Q *V *M 1 2 V V 1 + ---------------- + ---------------- - @ TAU*V R R T 2 1 1 2 1 2 2 GLQ := @ (XI )*V + @ (XI )*V + 2*@ (XI )*V 2 1 2 Q Q Q 2 1 2 2 - @ (XI )*Q *M - @ (XI )*Q *M 1 2 V V 2 + ------------------ + ------------------ + @ (XI ) - @ F R R T 2 Q 1 2 2 2 @ TAU*( - (V ) *R - 3*(V ) *R + 2*M) 2 1 2 Q - @ TAU*V *V + --------------------------------------- 1 2*R Q 1 2 2 2 @ TAU*Q *V *M @ TAU*Q *V *M 1 2 V V 2 + ---------------- + ---------------- - @ TAU*V R R T glv(k):=@(v k) _|s; 1 1 1 2 2 GLV := @ (XI )*V + @ (XI )*V - @ F 1 1 1 V V V 1 2 2 2 @ TAU*( - (V ) *R - (V ) *R + 2*M) 1 V + ------------------------------------- 2*R 2 1 1 2 2 GLV := @ (XI )*V + @ (XI )*V - @ F 2 2 2 V V V 1 2 2 2 @ TAU*( - (V ) *R - (V ) *R + 2*M) 2 V + ------------------------------------- 2*R glt:=@(t) _|s; 1 1 1 @ (XI )*Q *V *M 1 1 1 2 1 1 2 V GLT := - @ (XI )*(V ) - @ (XI )*V *V + ------------------ 1 2 R Q Q 1 2 1 @ (XI )*Q *V *M 2 V 2 1 2 2 2 2 + ------------------ - @ (XI )*V *V - @ (XI )*(V ) R 1 2 Q Q 2 1 2 2 2 2 @ (XI )*Q *V *M @ (XI )*Q *V *M 1 2 V V + ------------------ + ------------------ - @ F R R T 1 1 2 2 2 2 1 2 2 2 + @ TAU*V *((V ) + (V ) ) + @ TAU*V *((V ) + (V ) ) 1 2 Q Q 1 1 2 2 2 (@ TAU*Q *M)*((V ) + (V ) ) 1 V - ------------------------------- R 2 1 2 2 2 (@ TAU*Q *M)*((V ) + (V ) ) 2 V - ------------------------------- R 1 2 2 2 @ TAU*((V ) *R + (V ) *R + 2*M) 1 1 2 2 T M*(Q *XI + Q *XI ) + --------------------------------- - --------------------- 2*R 3 R %Translation in time must generate a symmetry; xi(k) := 0; K XI := 0 tau := 1; TAU := 1 glq k; 1 NS := - @ F 1 Q 2 NS := - @ F 2 Q glv k; 1 NS := - @ F 1 V 2 NS := - @ F 2 V glt; - @ F T %The corresponding integral is of course the energy; integ := - y _| theta; 1 2 2 2 (V ) *R + (V ) *R - 2*M INTEG := ------------------------- 2*R 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; U V @ F*E @ F*E U V ----------------------------- + ----------------------------- 2 2 2 2 SQRT( - SIN(U) + COSH(V) ) SQRT( - SIN(U) + COSH(V) ) PH @ F*E PH + ---------------- COS(U)*SINH(V) factor @; %The Laplacian:; # d # d f; @ F @ F*SIN(U) U U U -------------------- - ----------------------------- 2 2 2 2 COSH(V) - SIN(U) COS(U)*(COSH(V) - SIN(U) ) @ F @ F*COSH(V) V V V + -------------------- + ------------------------------ 2 2 2 2 COSH(V) - SIN(U) SINH(V)*(COSH(V) - SIN(U) ) @ F PH PH + ------------------ 2 2 COS(U) *SINH(V) %Another way of calculating the Laplacian: -#vardf(1/2*d f^#d f,f); @ F @ F*SIN(U) U U U -------------------- - ----------------------------- 2 2 2 2 COSH(V) - SIN(U) COS(U)*(COSH(V) - SIN(U) ) @ F @ F*COSH(V) V V V + -------------------- + ------------------------------ 2 2 2 2 COSH(V) - SIN(U) SINH(V)*(COSH(V) - SIN(U) ) @ F PH PH + ------------------ 2 2 COS(U) *SINH(V) 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; U V PH W := E *A + E *A + E *A U V PH %The curl:; x := # d w; U 2 2 X := (E *(COSH(V)*A *COS(U) - SQRT( - SIN(U) + COSH(V) )*@ (A ) PH PH V 2 2 + COS(U)*SINH(V)*@ (A )))/(SQRT( - SIN(U) + COSH(V) ) V PH V *COS(U)*SINH(V)) + (E *(SIN(U)*A *SINH(V) PH 2 2 + SQRT( - SIN(U) + COSH(V) )*@ (A ) PH U 2 2 - COS(U)*SINH(V)*@ (A )))/(SQRT( - SIN(U) + COSH(V) ) U PH PH 2 2 *COS(U)*SINH(V)) + (E *( - COSH(V) *@ (A ) + COSH(V) *@ (A ) V U U V 2 2 - COSH(V)*A *SINH(V) + SIN(U) *@ (A ) - SIN(U) *@ (A ) U V U U V 2 2 - SIN(U)*A *COS(U)))/(SQRT( - SIN(U) + COSH(V) ) V 2 2 *(COSH(V) - SIN(U) )) factor @; %The divergence; y := # d # w; @ (A ) @ (A ) U U V V Y := ----------------------------- + ----------------------------- 2 2 2 2 SQRT( - SIN(U) + COSH(V) ) SQRT( - SIN(U) + COSH(V) ) @ (A ) PH PH 3 + ---------------- + (COSH(V) *A *COS(U) COS(U)*SINH(V) V 2 2 - COSH(V) *SIN(U)*A *SINH(V) - COSH(V)*SIN(U) *A *COS(U) U V 2 3 + COSH(V)*A *COS(U)*SINH(V) + SIN(U) *A *SINH(V) V U 2 2 2 - SIN(U)*A *COS(U) *SINH(V))/(SQRT( - SIN(U) + COSH(V) ) U 2 2 *COS(U)*SINH(V)*(COSH(V) - SIN(U) )) 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)); R 2 CDV := (E *(V *SIN(TH)*@ (V )*R - (V ) *SIN(TH) + V *@ (V ) R R R PH PH PH R 2 - (V ) *SIN(TH) + V *SIN(TH)*@ (V ) TH TH TH R PH + SIN(TH)*@ (V )*R))/(SIN(TH)*R) + (E *(V *V *SIN(TH) T R R PH + V *SIN(TH)*@ (V )*R + V *V *COS(TH) + V *@ (V ) R R PH PH TH PH PH PH + V *SIN(TH)*@ (V ) + SIN(TH)*@ (V )*R))/(SIN(TH)*R TH TH PH T PH TH ) + (E *(V *V *SIN(TH) + V *SIN(TH)*@ (V )*R R TH R R TH 2 - (V ) *COS(TH) + V *@ (V ) PH PH PH TH + V *SIN(TH)*@ (V ) + SIN(TH)*@ (V )*R))/( TH TH TH T TH SIN(TH)*R) %next we calculate the viscous terms; visc := nu*(d#d# w - #d#d w) + nus*d#d# w; R 2 2 VISC := (E *( - 2*V *SIN(TH) *NU - 2*V *SIN(TH) *NUS R R - 2*V *SIN(TH)*COS(TH)*NU - V *SIN(TH)*COS(TH)*NUS TH TH 2 2 2 2 + SIN(TH) *@ (V )*R *NU + SIN(TH) *@ (V )*R *NUS R R R R R R 2 2 + 2*SIN(TH) *@ (V )*R*NU + 2*SIN(TH) *@ (V )*R*NUS R R R R 2 2 + SIN(TH) *@ (V )*NU + SIN(TH) *@ (V )*R*NUS TH TH R R TH TH 2 2 - 2*SIN(TH) *@ (V )*NU - SIN(TH) *@ (V )*NUS TH TH TH TH + SIN(TH)*COS(TH)*@ (V )*NU TH R + SIN(TH)*COS(TH)*@ (V )*R*NUS R TH + SIN(TH)*@ (V )*R*NUS - 2*SIN(TH)*@ (V )*NU R PH PH PH PH 2 2 - SIN(TH)*@ (V )*NUS + @ (V )*NU))/(SIN(TH) *R ) PH PH PH PH R PH 2 2 + (E *( - V *SIN(TH) *NU - V *COS(TH) *NU PH PH 2 2 2 + SIN(TH) *@ (V )*R *NU + 2*SIN(TH) *@ (V )*R*NU R R PH R PH 2 + SIN(TH) *@ (V )*NU TH TH PH + SIN(TH)*COS(TH)*@ (V )*NU TH PH + SIN(TH)*@ (V )*R*NUS + 2*SIN(TH)*@ (V )*NU R PH R PH R + 2*SIN(TH)*@ (V )*NUS + SIN(TH)*@ (V )*NUS PH R PH TH TH + 2*COS(TH)*@ (V )*NU + COS(TH)*@ (V )*NUS PH TH PH TH 2 2 + @ (V )*NU + @ (V )*NUS))/(SIN(TH) *R ) + PH PH PH PH PH PH TH 2 2 (E *( - V *SIN(TH) *NU - V *SIN(TH) *NUS TH TH 2 2 - V *COS(TH) *NU - V *COS(TH) *NUS TH TH 2 2 + SIN(TH) *@ (V )*R*NUS + 2*SIN(TH) *@ (V )*NU R TH R TH R 2 2 2 + 2*SIN(TH) *@ (V )*NUS + SIN(TH) *@ (V )*R *NU TH R R R TH 2 2 + 2*SIN(TH) *@ (V )*R*NU + SIN(TH) *@ (V )*NU R TH TH TH TH 2 + SIN(TH) *@ (V )*NUS TH TH TH + SIN(TH)*COS(TH)*@ (V )*NU TH TH + SIN(TH)*COS(TH)*@ (V )*NUS TH TH + SIN(TH)*@ (V )*NUS - 2*COS(TH)*@ (V )*NU PH TH PH PH PH 2 - COS(TH)*@ (V )*NUS + @ (V )*NU))/(SIN(TH) PH PH PH PH TH 2 *R ) %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; @ (V )*(V *R - 2*NU - 2*NUS) R R R NAST := - @ (V )*(NU + NUS) + ------------------------------ R R R R R - @ (V )*NU @ (V )*V PH PH R PH R PH + @ (V ) + ------------------ + ------------- T R 2 2 SIN(TH)*R SIN(TH) *R - @ (V )*NU TH TH R + ------------------ 2 R @ (V )*(V *SIN(TH)*R - COS(TH)*NU) TH R TH + -------------------------------------- 2 SIN(TH)*R - @ (V )*NUS @ (V )*(2*NU + NUS) R PH PH PH PH + ------------------- + ----------------------- SIN(TH)*R 2 SIN(TH)*R - @ (V )*NUS - @ (V )*COS(TH)*NUS R TH TH R TH + ------------------- + ------------------------ R SIN(TH)*R @ (V )*(2*NU + NUS) @ P TH TH R + ----------------------- + ----- + (2*V *SIN(TH)*NU 2 RHO R R 2 2 + 2*V *SIN(TH)*NUS - (V ) *SIN(TH)*R - (V ) *SIN(TH)*R R PH TH 2 + 2*V *COS(TH)*NU + V *COS(TH)*NUS)/(SIN(TH)*R ) TH TH - @ (V )*NUS (2*@ (V ))*(NU + NUS) R TH R TH R NAST := ------------------ - ------------------------ TH R 2 R - @ (V )*NUS @ (V )*COS(TH)*(2*NU + NUS) PH TH PH PH PH + -------------------- + ------------------------------- 2 2 2 SIN(TH)*R SIN(TH) *R @ (V )*(V *R - 2*NU) R TH R - @ (V )*NU + ----------------------- + @ (V ) R R TH R T TH - @ (V )*NU @ (V )*V PH PH TH PH TH PH + ------------------- + -------------- 2 2 SIN(TH)*R SIN(TH) *R @ (V )*(NU + NUS) TH TH TH - ------------------------ 2 R @ (V )*(V *SIN(TH)*R - COS(TH)*NU - COS(TH)*NUS) TH TH TH + ----------------------------------------------------- 2 SIN(TH)*R @ P TH 2 2 + ------- + (V *V *SIN(TH) *R - (V ) *SIN(TH)*COS(TH)*R R*RHO R TH PH 2 2 2 + V *SIN(TH) *NU + V *SIN(TH) *NUS + V *COS(TH) *NU TH TH TH 2 2 2 + V *COS(TH) *NUS)/(SIN(TH) *R ) TH - @ (V )*NUS (2*@ (V ))*(NU + NUS) R PH R PH R NAST := ------------------ - ------------------------ PH SIN(TH)*R 2 SIN(TH)*R @ (V )*(V *R - 2*NU) R PH R - @ (V )*NU + ----------------------- + @ (V ) R R PH R T PH @ (V )*(NU + NUS) @ (V )*V PH PH PH PH PH PH - ------------------------ + -------------- 2 2 SIN(TH)*R SIN(TH) *R - @ (V )*NU TH TH PH + ------------------- 2 R @ (V )*(V *SIN(TH)*R - COS(TH)*NU) TH PH TH + --------------------------------------- 2 SIN(TH)*R - @ (V )*NUS PH TH TH + -------------------- 2 SIN(TH)*R @ (V )*COS(TH)*( - 2*NU - NUS) @ P PH TH PH + ---------------------------------- + --------------- + ( 2 2 SIN(TH)*R*RHO SIN(TH) *R 2 2 V *(V *SIN(TH) *R + V *SIN(TH)*COS(TH)*R + SIN(TH) *NU PH R TH 2 2 2 + COS(TH) *NU))/(SIN(TH) *R ) 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; 2 2 d T^d X*( - @ Y *Q*RHO + @ Y *E*J) T X X LAGR := -------------------------------------- 2 vardf(lagr,y); d T^d X *(@ J*@ Y*E + 2*@ J*@ Y*E + @ Y*Q*RHO + @ Y*E*J) X X X X X X X X T T X X X X %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); C := d T*E*(@ J*@ Y*@ Y - @ Y*@ Y*J + @ Y*@ Y*J) X T X X T X X X T X X X 2 2 d X*(@ Y *Q*RHO + @ Y *E*J) T X X - ------------------------------- 2 %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; d T^d X*@ Y T *( - @ J*@ Y*E - 2*@ J*@ Y*E - @ Y*Q*RHO - @ Y*E*J) X X X X X X X X T T X X X X %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); PHI PSI CHCK := 0 R PSI CHCK := 0 R THETA CHCK := 0 R PHI CHCK := 0 THETA PSI CHCK := 0 THETA PHI CHCK := 0 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); 1 3 4 2 O *B0 O *COS(THETA) O *(FF*B0 - 2*GG *U3) GAM := ------- + --------------- + ----------------------- 2 3 2 SIN(THETA)*GG 2 GG GG 3 2 2 O *(@ GG*FF + GG*U1) O *(FF*B0 - GG *U3) R GAM := --------------------- - ---------------------- 4 3 2 GG GG 4 GAM := O *(@ FF - U5) 4 1 R 2 O *(@ GG*FF + GG*U1) 3 2 R O *( - FF*B0 + GG *U3) GAM := - ---------------------- + ------------------------ 4 2 GG 2 GG 3 2 O *@ GG O *B0 R GAM := ------- + --------- 1 3 2 GG GG 2 O *@ GG 3 R - O *B0 GAM := --------- + ---------- 1 2 GG 2 GG pform curv(a,b)=2; antisymmetric curv; factor ^; curv(-a,b):=d gam(-a,b) + gam(-c,b)^gam(-a,c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showtime; Time: 68391 ms end; 4: 4: Quitting Sat Jun 29 14:11:05 PDT 1991