Solution 1 to problem over

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

Expressions

The solution is given through the following expressions:


r10=0


r11=0


r12=0


     b2*r13
r14=--------
       b3


     b1*r13
r15=--------
       b3


    b2*m3
m2=-------
     b3


    b1*m3
m1=-------
     b3


    b2*n3
n2=-------
     b3


    b1*n3
n1=-------
     b3


c33=0


c23=0


c22=0


c13=0


c12=0


c11=0


b33=0


b32= - b1


b31=b2


b23=b1


b22=0


b21= - b3


b13= - b2


b12=b3


b11=0


      2        2        2        2
a33=b1 *k1 + b2 *k1 + b3 *k1 + b3 *k2


a23=b2*b3*k2


      2        2        2        2
a22=b1 *k1 + b2 *k1 + b2 *k2 + b3 *k1


a13=b1*b3*k2


a12=b1*b2*k2


      2        2        2        2
a11=b1 *k1 + b1 *k2 + b2 *k1 + b3 *k1

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:

 r13, n3, m3, k2, k1, b1, b2, b3

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,b21,b31,b1,b2,b3,b32,b13,r13,r14}

Relevance for the application:

The system of equations related to the Hamiltonian HAM:

       2    2           2           2           3
HAM=(u1 *(b1 *b3*k1 + b1 *b3*k2 + b2 *b3*k1 + b3 *k1) + 2*u1*u2*b1*b2*b3*k2

                     2              2
      + 2*u1*u3*b1*b3 *k2 + u1*v2*b3  - u1*v3*b2*b3 + u1*b1*n3

          2    2           2           2           3                    2
      + u2 *(b1 *b3*k1 + b2 *b3*k1 + b2 *b3*k2 + b3 *k1) + 2*u2*u3*b2*b3 *k2

                2
      - u2*v1*b3  + u2*v3*b1*b3 + u2*b2*n3

          2    2           2           3        3
      + u3 *(b1 *b3*k1 + b2 *b3*k1 + b3 *k1 + b3 *k2) + u3*v1*b2*b3

      - u3*v2*b1*b3 + u3*b3*n3 + v1*b1*m3 + v2*b2*m3 + v3*b3*m3)/b3

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

FI=u1*b1 + u2*b2 + u3*b3

which the program can not factorize further.

{HAM,FI} = 0





And again in machine readable form:



HAM=(u1**2*(b1**2*b3*k1 + b1**2*b3*k2 + b2**2*b3*k1 + b3**3*k1) + 2*u1*u2*b1*b2*
b3*k2 + 2*u1*u3*b1*b3**2*k2 + u1*v2*b3**2 - u1*v3*b2*b3 + u1*b1*n3 + u2**2*(b1**
2*b3*k1 + b2**2*b3*k1 + b2**2*b3*k2 + b3**3*k1) + 2*u2*u3*b2*b3**2*k2 - u2*v1*b3
**2 + u2*v3*b1*b3 + u2*b2*n3 + u3**2*(b1**2*b3*k1 + b2**2*b3*k1 + b3**3*k1 + b3
**3*k2) + u3*v1*b2*b3 - u3*v2*b1*b3 + u3*b3*n3 + v1*b1*m3 + v2*b2*m3 + v3*b3*m3)
/b3$

FI=u1*b1 + u2*b2 + u3*b3$