iVB.html

Invariant Euler-Lagrange and Helmholtz Operators

> with(Vessiot):

(a package, designed by Ian Anderson et. al. that contains commands for all the basic operations of calculus on jet spaces. Available from http://www.math.usu.edu/~fg_mp/Pages/SymbolicsPage/Symbolics.html)

> read("iVBstructure"):

(a procedure that computes structure equations for an invariant coframe on a jet bundle. Available from Irina Kogan: iakogan@ncsu.edu)

> read("iVarCalc"):

(a package that implements variational calculus relative to invariant frames. Available from Irina Kogan: iakogan@ncsu.edu)

SE(2) action on plane curves

Input:

Invariant frame:

Invariant coordinates and bases:

Euler-Lagrange operator

SA(2) action on plane curves

Input:

Invariant frame:

Invariant coordinates and bases:

Euler-Lagrange operator

(= )

mfr1 > lambda:=evalV(sigma1);

mfr1 > EL:=mfEL(lambda);rewrite11(%);

EL=Iparts(dV( ))

mfr1 > EL,mu:=Iparts(dV(lambda));

EL=dV( )-dH( )

mfr1 > evalV(dV(lambda)-dH(mu));

mfr1 > lambda:=evalV(u[4]* sigma1);

mfr1 > EL:=mfEL(lambda);rewrite11(%);

EL=Iparts(dV( ))

mfr1 > EL,mu:=Iparts(dV(lambda));

EL=dV(lambda)-dH(mu)

mfr1 > evalV(dV(lambda)-dH(mu));

mfr1 > lambda:=evalV(( 2*u[6]-28/3*u[4]^2 )* sigma1);

mfr1 > EL:=mfEL(lambda);rewrite11(%);

EL=Iparts(dV( ))

mfr1 > EL,mu:=Iparts(dV(lambda));

EL=dV( )-dH( )

mfr1 > evalV(dV(lambda)-dH(mu));

mfr1 > lambda:=evalV(u[4]^2* sigma1);

mfr1 > EL:=mfEL(lambda);rewrite11(%);

EL=Iparts(dV( ))

mfr1 > EL,mu:=Iparts(dV(lambda));

EL=dV( )-dH( )

mfr1 > evalV(dV(lambda)-dH(mu));

Helmholtz operator

applied to a variational source form

dfr> lambda:=evalV(u[4]*sigma1);rewrite11(%);

dfr> EL:=mfEL(lambda);rewrite11(%);

mfr1 > h:=mfHelm(EL);

Helm(EL)=I(dV(EL))

mfr1 > h,nu:=Iparts(dV(EL));

Helm(EL)=dV(EL)-dH( )

mfr1 > h:=simplify(evalV(dV(EL)-dH(nu)));

applied to a non-variational source form

dfr> tau:=evalV((u[4]^3)*sigma1&wedge Theta1);

dfr> h:=mfHelm(tau);

Helm(EL)=I(dV(EL))

dfr> h,nu:=Iparts(dV(tau));

Helm=dV( )-dH( ):

dfr> h:=evalV(dV(tau)-dH(nu));

SE(3) action on surfaces in 3D

Input:

variables:

mf3 > coord_frame([x,y],[u],fr3);

infinitesimal generators:

fr1> vectE3:=map(evalV,[vect(x),vect(y),vect(u[0,0]),-y*vect(x)+x*vect(y), -u[0,0]*vect(y)+y*vect(u[0,0]),-u[0,0]*vect(x)+x*vect(u[0,0])]);

cross-section:

> sectE3:=[x=0,y=0,u[0,0]=0,u[1,0]=0,u[0,1]=0,u[1,1]=0];

Invariant frame:

computing structure equations

fr2> str:=iVB(vectE3,sectE3,7,mfrE3,false):

defining invariant frame:

fr2> iVBframe_init(str[1],7):

Invariant coordinates and bases:

Coordinates

> iJetVariables(mfrE3);

dfr> iJetCoframe(mfr1);

mfr1> iJetFrame(mfr1);

Euler-Lagrange operator

mfr1 > lambda:=(evalV(sigma1 &w sigma2));

mfr1 > EL:=mfEL(lambda);

mfr1 > EL,mu:=Iparts(dV(lambda));

mfr1 > evalV(dV(lambda)-dH(mu));

Helmholtz operator

dfr> h:=mfHelm(EL);

dfr> h,nu:=Iparts(dV(EL));

dfr> h:=evalV(dV(EL)-dH(nu));

mean curvature: ( )

mfr1 > lambda:=(evalV(1/2*(u[2,0]+u[0,2])*sigma1 &w sigma2));

mfr1 > EL:=mfEL(lambda);

mfr1 > EL,mu:=Iparts(dV(lambda));

mfr1 > evalV(dV(lambda)-dH(mu));

Helmholtz operator

dfr> h:=mfHelm(EL);

dfr> h,nu:=Iparts(dV(EL));

dfr> h:=evalV(dV(EL)-dH(nu));

Gaussian curvature:

mfr1 > lambda:=(evalV((u[2,0]*u[0,2])*sigma1 &w sigma2));

mfr1 > EL:=mfEL(lambda);

Willmore Lagrangian:

mfr1 > lambda:=evalV((1/2*(u[2,0]+u[0,2])^2)*sigma1 &w sigma2);

mfr1 > EL:=mfEL(lambda);

mfr1 > EL,mu:=Iparts(dV(lambda));

mfr1 > evalV(dV(lambda)-dH(mu));

SA(3) action on surfaces in 3D

Input:

variables:

mf3 > coord_frame([x,y],[u],fr3);

infinitesimal generators:

fr1> vectA3:=map(evalV,[y*vect(x),u[0,0]*vect(x),x*vect(x)-y*vect(y), x*vect(x)-u[0,0]*vect(u[0,0]),x*vect(y),u[0,0]*vect(y),x*vect(u[0,0]), y*vect(u[0,0]),vect(x),vect(y),vect(u[0,0])]);