Solution 1 to problem over

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

Expressions

The solution is given through the following expressions:


r31=0


r32=r37


r33=0


r34=0


r35=0


r36=0


r38=0


r39=0


       1
      ---*b33*r320
       2
r310=--------------
          a23


r311=0


r312=0


r313=0


r314=0


r315=0


                  1
      a23*r330 + ---*a33*r320
                  2
r316=-------------------------
                a23


r317=0


r318=0


r319=0


r321=0


r322=0


r323=0


r324=0


r325=0


r326=r320


r327=0


r328=0


r329=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=r330


r351=0


r352=0


r353=0


r354=0


r355=0


c23=0


b12=0


b22=0

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:

 r30, r37, r320, r330, b33, a33, c33, 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 - 2*c33,a23,c33}

Relevance for the application:

The system of equations related to the Hamiltonian HAM:

                                  2                     2
HAM=2*u2*u3*a23 + 2*u2*v3*a23 + u3 *a33 + u3*v3*b33 + v3 *c33

has apart from the Hamiltonian and Casimirs the following 4 first integrals: 

     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        1    2           1       2
FI=u2*u3*v3*a23 + u2*v3 *a23 + ---*u3 *v3*a33 + ---*u3*v3 *b33
                                2                2

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

   v3,

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

{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





And again in machine readable form:



HAM=2*u2*u3*a23 + 2*u2*v3*a23 + u3**2*a33 + u3*v3*b33 + v3**2*c33$

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

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

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

FI=v3**3$