Solution 2 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


r13=0


r14=0


r15=0


        2
     a11 *c22*r214 - a11*a22*c22*r214 - a11*a33*c22*r214 + a22*a33*c22*r214
r20=------------------------------------------------------------------------
                         2                          2
                      a11 *a33 - 2*a11*a22*a33 + a22 *a33


r21=0


        2
     a11 *c22*r214 - a11*a22*c22*r214 - a11*a33*c22*r214 + a22*a33*c22*r214
r22=------------------------------------------------------------------------
                         2                          2
                      a11 *a33 - 2*a11*a22*a33 + a22 *a33


r23=0


r24=0


r26=0


r27=0


r28=0


     a11*r214 - a33*r214
r29=---------------------
          a11 - a22


r210=0


r212=0


r213=0


r215=0


r216=0


r217=0


r218=0


r219=0


m3=0


m2=0


m1=0


n3=0


n2=0


n1=0


        2                  2                            2
     a11 *a22*c22 - a11*a22 *c22 - a11*a22*a33*c22 + a22 *a33*c22
c33=--------------------------------------------------------------
                    2                          2
                 a11 *a33 - 2*a11*a22*a33 + a22 *a33


c23=0


c13=0


c12=0


c11=0


b33=0


b32=0


b31=0


b23=0


b22=0


b21=0


b13=0


b12=0


b11=0


a23=0


a13=0


a12=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:

 r214, c22, a33, a11, a22

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.

 
{c22,a11 - a33,a22 - a33,a11 - a22,a33,a22,a11,r214}

Relevance for the application:

The new Hamiltonian in form of a list of vanishing expressions: 

{a12,
a13,
a23,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
b33,
c11,
c12,
c13,
c23,
 - a11**2*a22*c22 + a11**2*a33*c33 + a11*a22**2*c22 + a11*a22*a33*c22 - 2*a11*
a22*a33*c33 - a22**2*a33*c22 + a22**2*a33*c33,
n1,
n2,
n3,
m1,
m2,
m3}$

The system of equations related to the Hamiltonian HAM:

      2         2         2         2         2  a11*a22*c22 - a22*a33*c22
HAM=u1 *a11 + u2 *a22 + u3 *a33 + v2 *c22 + v3 *---------------------------
                                                     a11*a33 - a22*a33

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

     2                         2               2      2
FI=u2 *(a11*a33 - a22*a33) + u3 *(a11*a33 - a33 ) + v2 *(a11*c22 - a33*c22)

        2
    + v3 *(a11*c22 - a33*c22)

which the program can not factorize further.

{HAM,FI} = 0





And again in machine readable form:



HAM=u1**2*a11 + u2**2*a22 + u3**2*a33 + v2**2*c22 + v3**2*(a11**2*a22*c22 - a11*
a22**2*c22 - a11*a22*a33*c22 + a22**2*a33*c22)/(a11**2*a33 - 2*a11*a22*a33 + a22
**2*a33)$

FI=u2**2*(a11*a33 - a22*a33) + u3**2*(a11*a33 - a33**2) + v2**2*(a11*c22 - a33*
c22) + v3**2*(a11*c22 - a33*c22)$