Solution 1 to problem h

Equations

The following unsolved equations remain:

     2      2      2      2
0=a11  + a13  + a23  + b12 *kap

Expressions

The solution is given through the following expressions:


r30=0


r31=0


r32=0


r33=0


r34=0


r35=0


r36=0


r37=0


r38=0


r39=0


r310=r315


r311=0


r312=r315


r313=0


r314=0


r316=0


      2*a13*r315
r317=------------
         b12


       - 2*a23*r315
r318=---------------
           b12


r319= - kap*r315


r320=0


r321=0


r322=0


r323=0


r324=0


r325=0


       - 2*a13*r315
r326=---------------
           b12


r327=0


       - 2*a11*r315
r328=---------------
           b12


r329=0


r330=0


r331=0


r332=0


           2             2             2           2
      2*a11 *r315 + 2*a13 *r315 + 2*a23 *r315 + b12 *kap*r315
r333=---------------------------------------------------------
                                  2
                               b12


r334=0


r335=0


r336=0


r337=0


r338=0


r339=0


r340=0


      2*a23*r315
r341=------------
         b12


      2*a11*r315
r342=------------
         b12


r343=0


r344=0


r345=0


r346=0


r347=0


r348=0


r349=0


r350=0


r351=0


r352=0


           2             2             2           2
      2*a11 *r315 + 2*a13 *r315 + 2*a23 *r315 + b12 *kap*r315
r353=---------------------------------------------------------
                                  2
                               b12


r354=0


r355=0


a33= - a11


a22=a11

Parameters

Apart from the condition that they must not vanish to give a non-trivial solution and a non-singular solution with non-vanishing denominators, the following parameters are free:

 r315, a23, a11, a13, b12

Inequalities

In the following not identically vanishing expressions are shown. Any auxiliary variables g00?? are used to express that at least one of their coefficients must not vanish, e.g. g0019*p4 + g0020*p3 means that either p4 or p3 or both are non-vanishing.

 
{a13,b12,r315}

Relevance for the application:

Modulo the following equation:

     2      2      2      2
0=a11  + a13  + a23  + b12 *kap


the system of equations related to the Hamiltonian HAM:

      2                                   2
HAM=u1 *a11 + 2*u1*u3*a13 + u1*v2*b12 + u2 *a11 + 2*u2*u3*a23 - u2*v1*b12

         2
     - u3 *a11

has apart from the Hamiltonian and Casimirs only the following first integral: 

     2          2        2        2      2
FI=u1 *u3*(2*a11  + 2*a13  + 2*a23  + b12 *kap) + 2*u1*u3*v2*a11*b12

                             2          2        2        2      2
    + 2*u1*u3*v3*a23*b12 + u2 *u3*(2*a11  + 2*a13  + 2*a23  + b12 *kap)

                                                  3    2           2
    - 2*u2*u3*v1*a11*b12 - 2*u2*u3*v3*a13*b12 - u3 *b12 *kap - 2*u3 *v1*a23*b12

          2                   2    2        2    2        2    2
    + 2*u3 *v2*a13*b12 + u3*v1 *b12  + u3*v2 *b12  + u3*v3 *b12

  =2*u3*(((u1*v3 - u3*v1)*a23 - (u2*v3 - u3*v2)*a13 + (u1*v2 - u2*v1)*a11)*b12

                2      2      2     2     2
          + (a13  + a23  + a11 )*(u1  + u2 )

             1                            2          2     2     2     2
          + ---*(((u2 + u3)*(u2 - u3) + u1 )*kap + v2  + v3  + v1 )*b12 )
             2

{HAM,FI} = 0





And again in machine readable form:



HAM=u1**2*a11 + 2*u1*u3*a13 + u1*v2*b12 + u2**2*a11 + 2*u2*u3*a23 - u2*v1*b12 - 
u3**2*a11$

FI=u1**2*u3*(2*a11**2 + 2*a13**2 + 2*a23**2 + b12**2*kap) + 2*u1*u3*v2*a11*b12 +
 2*u1*u3*v3*a23*b12 + u2**2*u3*(2*a11**2 + 2*a13**2 + 2*a23**2 + b12**2*kap) - 2
*u2*u3*v1*a11*b12 - 2*u2*u3*v3*a13*b12 - u3**3*b12**2*kap - 2*u3**2*v1*a23*b12 +
 2*u3**2*v2*a13*b12 + u3*v1**2*b12**2 + u3*v2**2*b12**2 + u3*v3**2*b12**2$