Solution 3 to problem over

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

Expressions

The solution is given through the following expressions:


r310=0


r311=0


r312=0


r313=0


r314=0


r315=0


                  1
      a23*r351 + ---*a33*r327
                  2
r317=-------------------------
                a23


                  1
      a23*r352 + ---*a33*r328
                  2
r318=-------------------------
                a23


r319=0


r320=0


r321=0


r322=0


r323=0


r324=0


r325=0


r329=0


                  1
      a23*r316 - ---*a33*r326
                  2
r330=-------------------------
                a23


r331=r351


r332=r352


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


                  1
      a23*r316 - ---*a33*r326
                  2
r350=-------------------------
                a23


r353=0


r354=0


r355=0


c22=0


c11=0


b33=0


b32=0


b31=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, r31, r32, r33, r34, r35, r36, r37, r38, r39, r316, 
r326, r327, r328, r351, r352, 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.

 
{a23,a33}

Relevance for the application:

The system of equations related to the Hamiltonian HAM:

                    2
HAM=2*u2*u3*a23 + u3 *a33

has apart from the Hamiltonian and Casimirs the following 16 first integrals: 

     2        2        2
FI=u1 *v1 + u2 *v1 + u3 *v1

                                        2     2     2
  = a product of the elements of: {v1,u1  + u2  + u3 }

{HAM,FI} = 0



     2        2        2
FI=u1 *v2 + u2 *v2 + u3 *v2

                                        2     2     2
  = a product of the elements of: {v2,u1  + u2  + u3 }

{HAM,FI} = 0



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

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

   v1,

   u3,

   2*u2*a23 + u3*a33}

{HAM,FI} = 0



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

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

   v2,

   u3,

   2*u2*a23 + u3*a33}

{HAM,FI} = 0



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

                                    1          2         2
  = a product of the elements of: {---,v3, - u1 *a33 - u2 *a33 + 2*u2*u3*a23}
                                    2

{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



     3
FI=v1

  = a product of the elements of: {v1,v1,v1}

{HAM,FI} = 0



     2
FI=v1 *v2

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

{HAM,FI} = 0



     2
FI=v1 *v3

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

{HAM,FI} = 0



        2
FI=v1*v2

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

{HAM,FI} = 0



FI=v1*v2*v3

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

{HAM,FI} = 0



        2
FI=v1*v3

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

{HAM,FI} = 0



     3
FI=v2

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

{HAM,FI} = 0



     2
FI=v2 *v3

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

{HAM,FI} = 0



        2
FI=v2*v3

  = a product of the elements of: {v3,v3,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 + u3**2*a33$

FI=u1**2*v1 + u2**2*v1 + u3**2*v1$

FI=u1**2*v2 + u2**2*v2 + u3**2*v2$

FI=u2*u3*v1*a23 + 1/2*u3**2*v1*a33$

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

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

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

FI=v1**3$

FI=v1**2*v2$

FI=v1**2*v3$

FI=v1*v2**2$

FI=v1*v2*v3$

FI=v1*v3**2$

FI=v2**3$

FI=v2**2*v3$

FI=v2*v3**2$

FI=v3**3$