Solution 2 to problem over

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

Expressions

The solution is given through the following expressions:


      1
     ---*b33*r210
      2
r20=--------------
         a23


     b32*r210
r21=----------
       a23


      1     2
     ---*b32 *r210
      2
r22=---------------
        a23*b33


     b31*r210
r23=----------
       a23


     b31*b32*r210
r24=--------------
       a23*b33


      1     2
     ---*b31 *r210
      2
r25=---------------
        a23*b33


      1
     ---*a33*r210
      2
r26=--------------
         a23


      1
     ---*a33*b32*r210
      2
r27=------------------
         a23*b33


      1
     ---*a33*b31*r210
      2
r28=------------------
         a23*b33


r29=0


      b31*r210
r212=----------
        b33


r213=0


r214=0


r215=0


r216=0


r217=0


r218=0


r219=0


r220=0


c22=0


c11=0


      b32*r210
r211=----------
        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:

 r210, b33, b31, a33, b32, 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}

Relevance for the application:

The system of equations related to the Hamiltonian HAM:

                    2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v1*b31 + u3*v2*b32 + u3*v3*b33

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

                                                             1    2
FI=u2*u3*v1*a23*b31 + u2*u3*v2*a23*b32 + u2*u3*v3*a23*b33 + ---*u3 *v1*a33*b31
                                                             2

       1    2               1    2               1       2    2
    + ---*u3 *v2*a33*b32 + ---*u3 *v3*a33*b33 + ---*u3*v1 *b31
       2                    2                    2

                                             1       2    2
    + u3*v1*v2*b31*b32 + u3*v1*v3*b31*b33 + ---*u3*v2 *b32  + u3*v2*v3*b32*b33
                                             2

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

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

   u3,

   v1*b31 + v2*b32 + v3*b33,

   2*u2*a23 + u3*a33 + v1*b31 + v2*b32 + v3*b33}

{HAM,FI} = 0





And again in machine readable form:



HAM=2*u2*u3*a23 + u3**2*a33 + u3*v1*b31 + u3*v2*b32 + u3*v3*b33$

FI=u2*u3*v1*a23*b31 + u2*u3*v2*a23*b32 + u2*u3*v3*a23*b33 + 1/2*u3**2*v1*a33*b31
 + 1/2*u3**2*v2*a33*b32 + 1/2*u3**2*v3*a33*b33 + 1/2*u3*v1**2*b31**2 + u3*v1*v2*
b31*b32 + u3*v1*v3*b31*b33 + 1/2*u3*v2**2*b32**2 + u3*v2*v3*b32*b33 + 1/2*u3*v3
**2*b33**2$