Special affine group action on curves
| > | with(Vessiot): |
| f > | coord_frame([x],[u],f): |
| f > | action := transform(f,f,[x=a1*x+a2*u[0]+t1,u[0]=a3*x+a4*u[0]+t2]);
J := linalg[inverse](total_Jacobian(action)): action := pr_transform(action, 4, J); |
![action := [x = a1*x+a2*u[0]+t1, u[0] = a3*x+a4*u[0]+t2]](images/amf_88.gif){kind=small}
![action := [x = a1*x+a2*u[0]+t1, u[0] = a3*x+a4*u[0]+t2, u[1] = (a3+a4*u[1])/(a1+a2*u[1]), u[2] = (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, u[3] = (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[...](images/amf_89.gif)
![action := [x = a1*x+a2*u[0]+t1, u[0] = a3*x+a4*u[0]+t2, u[1] = (a3+a4*u[1])/(a1+a2*u[1]), u[2] = (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, u[3] = (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[...](images/amf_90.gif)
![action := [x = a1*x+a2*u[0]+t1, u[0] = a3*x+a4*u[0]+t2, u[1] = (a3+a4*u[1])/(a1+a2*u[1]), u[2] = (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, u[3] = (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[...](images/amf_91.gif)
![action := [x = a1*x+a2*u[0]+t1, u[0] = a3*x+a4*u[0]+t2, u[1] = (a3+a4*u[1])/(a1+a2*u[1]), u[2] = (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, u[3] = (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[...](images/amf_92.gif){kind=small}
![action := [x = a1*x+a2*u[0]+t1, u[0] = a3*x+a4*u[0]+t2, u[1] = (a3+a4*u[1])/(a1+a2*u[1]), u[2] = (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, u[3] = (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[...](images/amf_93.gif){kind=small}
![action := [x = a1*x+a2*u[0]+t1, u[0] = a3*x+a4*u[0]+t2, u[1] = (a3+a4*u[1])/(a1+a2*u[1]), u[2] = (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, u[3] = (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[...](images/amf_94.gif)
| f > | z := frameJetVariables(f);
lambda := [a1,a2,a3,a4,t1,t2]; |
![z := [x, u[0], u[1], u[2], u[3], u[4]]](images/amf_95.gif){kind=small}
![lambda := [a1, a2, a3, a4, t1, t2]](images/amf_96.gif){kind=small}
| f > | G := [a1*a4-a2*a3-1]; |
![G := [a4*a1-a2*a3-1]](images/amf_97.gif){kind=small}
| f > | g := map2(pullback, action, z); |
![g := [a1*x+a2*u[0]+t1, a3*x+a4*u[0]+t2, (a3+a4*u[1])/(a1+a2*u[1]), (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[1])^5, (-a2*a3+a4*a1)*(15*a2^2*u[2]^3...](images/amf_98.gif)
![g := [a1*x+a2*u[0]+t1, a3*x+a4*u[0]+t2, (a3+a4*u[1])/(a1+a2*u[1]), (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[1])^5, (-a2*a3+a4*a1)*(15*a2^2*u[2]^3...](images/amf_99.gif)
![g := [a1*x+a2*u[0]+t1, a3*x+a4*u[0]+t2, (a3+a4*u[1])/(a1+a2*u[1]), (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[1])^5, (-a2*a3+a4*a1)*(15*a2^2*u[2]^3...](images/amf_100.gif)
![g := [a1*x+a2*u[0]+t1, a3*x+a4*u[0]+t2, (a3+a4*u[1])/(a1+a2*u[1]), (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[1])^5, (-a2*a3+a4*a1)*(15*a2^2*u[2]^3...](images/amf_101.gif){kind=small}
![g := [a1*x+a2*u[0]+t1, a3*x+a4*u[0]+t2, (a3+a4*u[1])/(a1+a2*u[1]), (-a2*a3+a4*a1)*u[2]/(a1+a2*u[1])^3, (-a2*a3+a4*a1)*(-3*u[2]^2*a2+u[3]*a1+u[3]*a2*u[1])/(a1+a2*u[1])^5, (-a2*a3+a4*a1)*(15*a2^2*u[2]^3...](images/amf_102.gif)
| > | amf(G, g, [x,u[0],u[1],u[2]-1,u[3]], z, lambda, r); |
![[r1 = 1/27*(-5*u[3]^2+3*u[2]*u[4])^3/u[2]^8], [-u[3], -u[2]+1, -u[1], -u[0], -x, r1-u[4]^3]](images/amf_103.gif)