Solution 8 to problem over

Expressions | Parameters | Inequalities | Relevance | Back to problem over

Expressions

The solution is given through the following expressions:


r10=r11


r12=0


r13=0


r14=0


r15=0


r20=r22


r21=2*r22


r23=0


r24=0


      1
     ---*n3*r327
      2
r26=-------------
         a23


      1
     ---*n3*r327
      2
r27=-------------
         a23


r28=0


r29=0


r210=0


r211=0


r212=0


r214=0


r215=0


r216=0


r217=0


r218=0


r219=0


     2
r30=---*r37
     3


r31=0


r32=0


     2
r33=---*r37
     3


r34=0


r35=0


r36=0


r38=r37


r39=0


       1
      ---*b33*r327
       2
r310=--------------
          a23


      b33*r327
r311=----------
        a23


       1
      ---*b33*r327
       2
r312=--------------
          a23


r313=0


r314=0


r315=0


       1
      ---*a33*r327
       2
r316=--------------
          a23


       1
      ---*a33*r327
       2
r317=--------------
          a23


r318=0


r319=0


r320=0


r321=0


r322=0


r323=0


r324=0


r325=0


r326=r327


r328=0


r329=0


r330=0


r331=0


r332=0


r333=0


r334=0


r335=0


r336=0


r337=0


r338=0


r339=0


r340=0


r341=0


r342=0


r343=0


r344=0


r345=0


r346=0


r347=0


r348=0


r349=0


r350=0


r351=0


r352=0


r353=0


r354=0


r355=0


m2=m3


m1=0


c22=0


c11=0


b31=0


b32=b33

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:

 b33, r00, r11, r22, r327, r37, n3, a33, m3, 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.

 
{r37,a23}

Relevance for the application:

The system of equations related to the Hamiltonian HAM:

                    2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v2*b33 + u3*v3*b33 + u3*n3 + v2*m3 + v3*m3

has apart from the Hamiltonian and Casimirs the following 5 first integrals: 

     2        2       2    3    2    3
FI=v1 *v2 + v1 *v3 + ---*v2  + ---*v3
                      3         3

                                    1              2       2                 2
  = a product of the elements of: {---,v2 + v3,3*v1  + 2*v2  - 2*v2*v3 + 2*v3 }
                                    3

{HAM,FI} = 0



                                  1    2           1    2
FI=u2*u3*v2*a23 + u2*u3*v3*a23 + ---*u3 *v2*a33 + ---*u3 *v3*a33
                                  2                2

       1       2                       1              1       2
    + ---*u3*v2 *b33 + u3*v2*v3*b33 + ---*u3*v2*n3 + ---*u3*v3 *b33
       2                               2              2

       1
    + ---*u3*v3*n3
       2

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

   v2 + v3,

   u3,

   2*u2*a23 + u3*a33 + v2*b33 + v3*b33 + n3}

{HAM,FI} = 0



     2               2
FI=v2  + 2*v2*v3 + v3

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

{HAM,FI} = 0



FI=v2 + 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 + u3**2*a33 + u3*v2*b33 + u3*v3*b33 + u3*n3 + v2*m3 + v3*m3$

FI=v1**2*v2 + v1**2*v3 + 2/3*v2**3 + 2/3*v3**3$

FI=u2*u3*v2*a23 + u2*u3*v3*a23 + 1/2*u3**2*v2*a33 + 1/2*u3**2*v3*a33 + 1/2*u3*v2
**2*b33 + u3*v2*v3*b33 + 1/2*u3*v2*n3 + 1/2*u3*v3**2*b33 + 1/2*u3*v3*n3$

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

FI=v2 + v3$

FI=1$