Solution 3 to problem over

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

Expressions

The solution is given through the following expressions:


      - i*n3*r216
r10=--------------
         a33


      - i*n2*r216
r11=--------------
         a33


     n2*r216
r12=---------
       a33


r13=0


r14=0


r15=0


      - c12*r216 - i*c33*r216
r20=--------------------------
               b33


r21=0


r22=0


r23=0


r24=0


r26= - i*r216


r27=0


r28=0


r210=0


r212= - r216


r213=0


r214=0


r215=0


r217=0


r218=0


r219=0


r220=0


    b33*n3
m3=--------
     a33


n1=i*n2


c23=0


c22=2*i*c12


c13=0


c11=0


b32=0


b31=0


b23=0


b22=0


b21= - i*b33


b13=0


b12=i*b33


b11=0


a23=0


a22=0


a13=0


a12=0


a11=0


                                                             2
     - i*a33*b33*m2 - 2*a33*c12*n2 - 2*i*a33*c33*n2 + 2*i*b33 *n2
m1=---------------------------------------------------------------
                               a33*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:

 c33, r216, m2, n3, b33, n2, 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.

 
{n2,a33,r216}

Relevance for the application:

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

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

The system of equations related to the Hamiltonian HAM:

                                                    2
HAM=i*u1*v2*b33 + i*u1*n2 - i*u2*v1*b33 + u2*n2 + u3 *a33 + u3*v3*b33 + u3*n3

     + 2*v1*v2*c12

                                                                    2
            - i*a33*b33*m2 - 2*a33*c12*n2 - 2*i*a33*c33*n2 + 2*i*b33 *n2
     + v1*---------------------------------------------------------------
                                      a33*b33

             2                 2        v3*b33*n3
     + 2*i*v2 *c12 + v2*m2 + v3 *c33 + -----------
                                           a33

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

FI=u1*v2*a33*b33 - u2*v1*a33*b33 - i*u3*v3*a33*b33 + v1*b33*n2 - i*v2*b33*n2

        2
    + v3 *( - a33*c12 - i*a33*c33) - i*v3*b33*n3

which the program can not factorize further.

{HAM,FI} = 0





And again in machine readable form:



HAM=i*u1*v2*b33 + i*u1*n2 - i*u2*v1*b33 + u2*n2 + u3**2*a33 + u3*v3*b33 + u3*n3 
+ 2*v1*v2*c12 + v1*( - i*a33*b33*m2 - 2*a33*c12*n2 - 2*i*a33*c33*n2 + 2*i*b33**2
*n2)/(a33*b33) + 2*i*v2**2*c12 + v2*m2 + v3**2*c33 + (v3*b33*n3)/a33$

FI=u1*v2*a33*b33 - u2*v1*a33*b33 - i*u3*v3*a33*b33 + v1*b33*n2 - i*v2*b33*n2 + 
v3**2*( - a33*c12 - i*a33*c33) - i*v3*b33*n3$