Solution 2 to problem over

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

Expressions

The solution is given through the following expressions:


r30=0


r31=0


r32=0


r33=0


r34=0


r35=0


r36=0


r37=0


r38=0


r39=0


r312= - r338


r314=0


r315= - r338


r317=0


r318=0


r320=0


r321=r338


r322=0


r323=0


r324=0


r325=0


r327=0


r328= - r342


r329=0


r330=0


r331=0


r332=0


      a11*r342 - b12*kap*r338
r333=-------------------------
                b12


r334=0


r335=0


r336=0


r337=0


r339=0


r340=0


r343=0


r344=0


r345=0


r346=0


r347=0


r348=0


r349=0


r350=0


r351=0


r352=0


      a11*r342 - b12*kap*r338
r353=-------------------------
                b12


r354=0


r355=0


a23=0


a22=a11


a13=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:

 r319, r342, r338, a11, a33, b12

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.

 
{b12}

Relevance for the application:

The system of equations related to the Hamiltonian HAM:

      2                     2                     2
HAM=u1 *a11 + u1*v2*b12 + u2 *a11 - u2*v1*b12 + u3 *a33

has apart from the Hamiltonian and Casimirs the following 3 first integrals: 

        2                       2                          2        2
FI= - u1 *u3*kap + u1*v1*v3 - u2 *u3*kap + u2*v2*v3 - u3*v1  - u3*v2

             3                2                   2
{HAM,FI} = u1 *v1*b12*kap + u1 *u2*v2*b12*kap + u1 *u3*v3*b12*kap

                  2                  2                        2
            - 2*u1 *v1*v2*a11 + u1*u2 *v1*b12*kap + 2*u1*u2*v1 *a11

                        2                                3               2
            - 2*u1*u2*v2 *a11 - 2*u1*u3*v2*v3*a11 - u1*v1 *b12 - u1*v1*v2 *b12

                3                2                     2
            + u2 *v2*b12*kap + u2 *u3*v3*b12*kap + 2*u2 *v1*v2*a11

                                       2               3            2
            + 2*u2*u3*v1*v3*a11 - u2*v1 *v2*b12 - u2*v2 *b12 - u3*v1 *v3*b12

                   2
            - u3*v2 *v3*b12



     2                           2
FI=u1 *u3*a11 + u1*u3*v2*b12 + u2 *u3*a11 - u2*u3*v1*b12

{HAM,FI} = 0



     3
FI=u3

{HAM,FI} = 0





And again in machine readable form:



HAM=u1**2*a11 + u1*v2*b12 + u2**2*a11 - u2*v1*b12 + u3**2*a33$

FI= - u1**2*u3*kap + u1*v1*v3 - u2**2*u3*kap + u2*v2*v3 - u3*v1**2 - u3*v2**2$

FI=u1**2*u3*a11 + u1*u3*v2*b12 + u2**2*u3*a11 - u2*u3*v1*b12$

FI=u3**3$