Solution 1 to problem over

Equations

The following unsolved equations remain:

       2      2
0=4*a23  + a33

Expressions

The solution is given through the following expressions:


r11=0


r12=0


r13=0


r14=0


r15=0


r21=0


r22=0


r23=0


r24=0


r26=0


r27=0


r28=0


r29=0


r210=0


r211=0


r212=0


r214=0


r215=0


r216=0


r217=0


r218=0


r219=0


r31=0


r33=0


r34=0


r35=0


r36=0


                1
     a23*r32 - ---*a33*r322
                2
r37=------------------------
              a23


r38=0


r39=0


       1
      ---*a33*r320
       2
r310=--------------
          a23


r311=0


       1
      ---*a33*r322
       2
r312=--------------
          a23


r313=0


r314=0


          1
       - ---*a33*r322
          2
r315=-----------------
            a23


r317=0


r318=0


r319=0


r321=0


r323=0


r324=0


r325= - r322


r326=2*r320


r327=0


r328=0


r329=0


      a23*r316 - a33*r320
r330=---------------------
              a23


r331=0


r332=0


r333=0


r334=0


r335=0


r336=0


r337=0


r338=0


r339=2*r322


r340=0


r341=0


r342=0


r343=0


r344=0


r345=0


r346=0


r347=0


r348=0


r349=0


      a23*r316 - a33*r320
r350=---------------------
              a23


r351=0


r352=0


r353=0


r354=0


r355=0


m3=0


n3=0


c33=0


c23=0


     1
b33=---*a33
     2


b32=0


b23=a23

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:

 r00, r10, r20, r30, r32, r316, r320, r322, a33, a23

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.

 
{b33,a33,r322,a23}

Relevance for the application:

Modulo the following equation:

       2      2
0=4*a23  + a33


the system of equations related to the Hamiltonian HAM:

                                2        1
HAM=2*u2*u3*a23 + u2*v3*a23 + u3 *a33 + ---*u3*v3*a33
                                         2

has apart from the Hamiltonian and Casimirs the following 8 first integrals: 

                         2            2        1       2        1       2
FI=2*u1*v1*v2*a23 - u2*v1 *a23 + u2*v2 *a23 - ---*u3*v1 *a33 + ---*u3*v2 *a33
                                               2                2

       1    2
    - ---*v1 *v3*a33
       2

                                    1
  = a product of the elements of: {---,
                                    2

                           2              2            2            2
   4*u1*v1*v2*a23 - 2*u2*v1 *a23 + 2*u2*v2 *a23 - u3*v1 *a33 + u3*v2 *a33

        2
    - v1 *v3*a33}

{HAM,FI} = 0



        2            2                                2        1       2
FI= - u1 *v3*a33 - u2 *v3*a33 + 2*u2*u3*v3*a23 + u2*v3 *a23 + ---*u3*v3 *a33
                                                               2

                                    1
  = a product of the elements of: {---,
                                    2

   v3,

          2           2
    - 2*u1 *a33 - 2*u2 *a33 + 4*u2*u3*a23 + 2*u2*v3*a23 + u3*v3*a33}

{HAM,FI} = 0



     2        2        2
FI=u1 *v3 + u2 *v3 + u3 *v3

                                        2     2     2
  = a product of the elements of: {v3,u1  + u2  + u3 }

{HAM,FI} = 0



     2        2
FI=v1 *v3 + v2 *v3

                                        2     2
  = a product of the elements of: {v3,v1  + v2 }

{HAM,FI} = 0



     3
FI=v3

  = a product of the elements of: {v3,v3,v3}

{HAM,FI} = 0



     2
FI=v3

  = a product of the elements of: {v3,v3}

{HAM,FI} = 0



FI=v3

which the program can not factorize further.

{HAM,FI} = 0



FI=1

which the program can not factorize further.

{HAM,FI} = 0





And again in machine readable form:



HAM=2*u2*u3*a23 + u2*v3*a23 + u3**2*a33 + 1/2*u3*v3*a33$

FI=2*u1*v1*v2*a23 - u2*v1**2*a23 + u2*v2**2*a23 - 1/2*u3*v1**2*a33 + 1/2*u3*v2**
2*a33 - 1/2*v1**2*v3*a33$

FI= - u1**2*v3*a33 - u2**2*v3*a33 + 2*u2*u3*v3*a23 + u2*v3**2*a23 + 1/2*u3*v3**2
*a33$

FI=u1**2*v3 + u2**2*v3 + u3**2*v3$

FI=v1**2*v3 + v2**2*v3$

FI=v3**3$

FI=v3**2$

FI=v3$

FI=1$