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);
4 2 3 2
CURV := (2*O ^O *(@ FF*GG*B0 - 2*@ GG*FF*B0 + @ GG*GG *U3
4 R R R
3 4 1
- GG*B0*U1 - GG*B0*U5))/GG + O ^O *(@ FF - @ U5)
R R R
4
CURV := 0
1
1 2 3 2
O ^O *(@ GG*GG - B0 )
4 R R 4 2 3
CURV := -------------------------- + (O ^O *( - @ FF*@ GG*GG
2 4 R R
GG
3 3 2
- @ GG*FF*GG + @ GG*GG *U5 - FF*B0
R R R
2 4 4 3
+ 2*GG *B0*U3))/GG + (O ^O
2
*(@ FF*GG*B0 - 2*@ GG*FF*B0 + 2*@ GG*GG *U3 - GG*B0*U5)
R R R
3
)/GG
1 3 3 2
O ^O *(@ GG*GG - B0 )
4 R R 4 2
CURV := -------------------------- + (O ^O *( - @ FF*GG*B0
3 4 R
GG
2 3
+ 2*@ GG*FF*B0 - 2*@ GG*GG *U3 + GG*B0*U5))/GG + (
R R
4 3 3 3 3
O ^O *( - @ FF*@ GG*GG - @ GG*FF*GG + @ GG*GG *U5
R R R R R
2 2 4
- FF*B0 + 2*GG *B0*U3))/GG
1
CURV := 0
4
1 2 3 2
CURV := (2*O ^O *( - @ FF*GG*B0 + 2*@ GG*FF*B0 - @ GG*GG *U3
1 R R R
3
+ GG*B0*U1 + GG*B0*U5))/GG
4 1
+ O ^O *( - @ FF + @ U5)
R R R
1 1 2 3 3 3
CURV := (O ^O *( - @ FF*@ GG*GG - @ GG*FF*GG - @ GG*GG *U1
2 R R R R R
4 2 2 4 1 3
- @ U1*GG - FF*B0 + GG *B0*U3))/GG + (O ^O *(
R
2
- @ FF*GG*B0 + 2*@ GG*FF*B0 + @ GG*GG *U3
R R R
3 3 4 2
+ @ U3*GG + GG*B0*U1))/GG + (O ^O *(
R
4 2 3 3
- @ FF*GG *U1 + @ GG*FF *GG + @ GG*FF*GG *U1
R R R R
3 4 2 2
+ @ GG*FF*GG *U5 + @ U1*FF*GG - FF *B0
R R
2 4 4 2 4
+ 3*FF*GG *B0*U3 + GG *U1*U5 - 2*GG *U3 ))/GG + (
4 3 2 2
O ^O *(@ FF*GG *U3 - 3*@ GG*FF*GG*U3 - @ U3*FF*GG
R R R
2 2
+ FF*B0*U1 + FF*B0*U5 - 2*GG *U1*U3 - GG *U3*U5))/
2
GG
1 1 2 2 3
CURV := (O ^O *(@ FF*GG*B0 - 2*@ GG*FF*B0 - @ GG*GG *U3 - @ U3*GG
3 R R R R
3 1 3 3
- GG*B0*U1))/GG + (O ^O *( - @ FF*@ GG*GG
R R
3 3 4 2
- @ GG*FF*GG - @ GG*GG *U1 - @ U1*GG - FF*B0
R R R R
2 4 4 2 2
+ GG *B0*U3))/GG + (O ^O *( - @ FF*GG *U3
R
2
+ 3*@ GG*FF*GG*U3 + @ U3*FF*GG - FF*B0*U1
R R
2 2 2 4 3
- FF*B0*U5 + 2*GG *U1*U3 + GG *U3*U5))/GG + (O ^O
4 2 3 3
*( - @ FF*GG *U1 + @ GG*FF *GG + @ GG*FF*GG *U1
R R R R
3 4 2 2
+ @ GG*FF*GG *U5 + @ U1*FF*GG - FF *B0
R R
2 4 4 2 4
+ 3*FF*GG *B0*U3 + GG *U1*U5 - 2*GG *U3 ))/GG
2 1 2 3 3 3
CURV := (O ^O *( - @ FF*@ GG*GG - @ GG*FF*GG - @ GG*GG *U1
4 R R R R R
4 2 2 4 1 3
- @ U1*GG - FF*B0 + GG *B0*U3))/GG + (O ^O *(
R
2
- @ FF*GG*B0 + 2*@ GG*FF*B0 + @ GG*GG *U3
R R R
3 3 4 2
+ @ U3*GG + GG*B0*U1))/GG + (O ^O *(
R
4 2 3 3
- @ FF*GG *U1 + @ GG*FF *GG + @ GG*FF*GG *U1
R R R R
3 4 2 2
+ @ GG*FF*GG *U5 + @ U1*FF*GG - FF *B0
R R
2 4 4 2 4
+ 3*FF*GG *B0*U3 + GG *U1*U5 - 2*GG *U3 ))/GG + (
4 3 2 2
O ^O *(@ FF*GG *U3 - 3*@ GG*FF*GG*U3 - @ U3*FF*GG
R R R
2 2
+ FF*B0*U1 + FF*B0*U5 - 2*GG *U1*U3 - GG *U3*U5))/
2
GG
1 2 3 2
O ^O *(@ GG*GG - B0 )
2 R R 4 2 3
CURV := -------------------------- + (O ^O *( - @ FF*@ GG*GG
1 4 R R
GG
3 3 2
- @ GG*FF*GG + @ GG*GG *U5 - FF*B0
R R R
2 4 4 3
+ 2*GG *B0*U3))/GG + (O ^O
2
*(@ FF*GG*B0 - 2*@ GG*FF*B0 + 2*@ GG*GG *U3 - GG*B0*U5)
R R R
3
)/GG
2
CURV := 0
2
2 2 3 2 2 3 2
CURV := (O ^O *( - 2*@ GG *FF*GG - 2*@ GG*GG *U1 + 6*FF*B0
3 R R
2 2 4
- 6*GG *B0*U3 + GG ))/GG
4 1 3
2*O ^O *(@ FF*GG*B0 - 2*@ GG*FF*B0 - @ U3*GG )
R R R
+ ------------------------------------------------
3
GG
3 1 2 2 3
CURV := (O ^O *(@ FF*GG*B0 - 2*@ GG*FF*B0 - @ GG*GG *U3 - @ U3*GG
4 R R R R
3 1 3 3
- GG*B0*U1))/GG + (O ^O *( - @ FF*@ GG*GG
R R
3 3 4 2
- @ GG*FF*GG - @ GG*GG *U1 - @ U1*GG - FF*B0
R R R R
2 4 4 2 2
+ GG *B0*U3))/GG + (O ^O *( - @ FF*GG *U3
R
2
+ 3*@ GG*FF*GG*U3 + @ U3*FF*GG - FF*B0*U1
R R
2 2 2 4 3
- FF*B0*U5 + 2*GG *U1*U3 + GG *U3*U5))/GG + (O ^O
4 2 3 3
*( - @ FF*GG *U1 + @ GG*FF *GG + @ GG*FF*GG *U1
R R R R
3 4 2 2
+ @ GG*FF*GG *U5 + @ U1*FF*GG - FF *B0
R R
2 4 4 2 4
+ 3*FF*GG *B0*U3 + GG *U1*U5 - 2*GG *U3 ))/GG
1 3 3 2
O ^O *(@ GG*GG - B0 )
3 R R 4 2
CURV := -------------------------- + (O ^O *( - @ FF*GG*B0
1 4 R
GG
2 3
+ 2*@ GG*FF*B0 - 2*@ GG*GG *U3 + GG*B0*U5))/GG + (
R R
4 3 3 3 3
O ^O *( - @ FF*@ GG*GG - @ GG*FF*GG + @ GG*GG *U5
R R R R R
2 2 4
- FF*B0 + 2*GG *B0*U3))/GG
3 2 3 2 2 3 2
CURV := (O ^O *(2*@ GG *FF*GG + 2*@ GG*GG *U1 - 6*FF*B0
2 R R
2 2 4
+ 6*GG *B0*U3 - GG ))/GG
4 1 3
2*O ^O *( - @ FF*GG*B0 + 2*@ GG*FF*B0 + @ U3*GG )
R R R
+ ---------------------------------------------------
3
GG
3
CURV := 0
3
showtime;
Time: 68391 ms
end;
4: 4:
Quitting
Sat Jun 29 14:11:05 PDT 1991