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


                       1
     2*a33*c12*r212 + ---*b11*b12*r212
                       4
r20=-----------------------------------
                  a33*b11


r21=0


r22=0


r23=0


      1
     ---*b11*r212
      4
r24=--------------
         a33


r26=0


r27=0


r28=0


r210=0


r213=0


r214=0


r215=0


r216= - r212


r217=0


r218=0


r219=0


r220=0


m3=0


m2=0


m1=0


n3=0


n2=0


n1=0


                         1      3    1         2
      - 2*a33*b12*c12 + ----*b11  - ---*b11*b12
                         16          4
c33=---------------------------------------------
                       a33*b11


c23=0


c22=0


c13=0


c11=0


     1
b33=---*b11
     2


b32=0


b31=0


b23=0


b22=0


b21= - b12


b13=0


a23=0


a22=0


a13=0


a12=0


a11=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:

 r212, b12, b11, c12, a33

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.

 
{b11,r212,a33}

Relevance for the application:

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

{a11,
a12,
a13,
a22,
a23,
b13,
b12 + b21,
b22,
b23,
b31,
b32,
 - 1/2*b11 + b33,
c11,
c13,
c22,
c23,
a33*b11*c33 + 2*a33*b12*c12 - 1/16*b11**3 + 1/4*b11*b12**2,
n1,
n2,
n3,
m1,
m2,
m3}$

The system of equations related to the Hamiltonian HAM:

                                          2        1
HAM=u1*v1*b11 + u1*v2*b12 - u2*v1*b12 + u3 *a33 + ---*u3*v3*b11 + 2*v1*v2*c12
                                                   2

                                1      3    1         2
             - 2*a33*b12*c12 + ----*b11  - ---*b11*b12
         2                      16          4
     + v3 *---------------------------------------------
                              a33*b11

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

                                       1           2
FI= - u1*v2*a33*b11 + u2*v1*a33*b11 + ---*v1*v2*b11
                                       4

        2               1
    + v3 *(2*a33*c12 + ---*b11*b12)
                        4

which the program can not factorize further.

{HAM,FI} = 0





And again in machine readable form:



HAM=u1*v1*b11 + u1*v2*b12 - u2*v1*b12 + u3**2*a33 + 1/2*u3*v3*b11 + 2*v1*v2*c12 
+ v3**2*( - 2*a33*b12*c12 + 1/16*b11**3 - 1/4*b11*b12**2)/(a33*b11)$

FI= - u1*v2*a33*b11 + u2*v1*a33*b11 + 1/4*v1*v2*b11**2 + v3**2*(2*a33*c12 + 1/4*
b11*b12)$