Solution 12 to problem over

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

Expressions

The solution is given through the following expressions:


     2*m1*n3*r214 - i*n1*n3*r216
r10=-----------------------------
               a33*n1


      - i*m1*r214 - n1*r216
r11=------------------------
              a33


     m1*r214 - i*n1*r216
r12=---------------------
             a33


      - n3*r214
r13=------------
        a33


     i*n1*r214
r14=-----------
        a33


      - n1*r214
r15=------------
        a33


          2
      - m1 *r214 + i*m1*n1*r216
r20=----------------------------
                  2
                n1


r21=0


r22=0


r23=0


r24=0


     2*m1*r214 - i*n1*r216
r26=-----------------------
              n1


r27=0


r28=0


r210=0


r212= - r216


r213=0


r215=0


r217=0


r218=0


r219=0


r220=r214


m3=0


m2= - i*m1


n2= - i*n1


c33=0


c23=0


c22=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


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:

 r214, r216, m1, n3, n1, 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.

 
{a33,r216,n1}

Relevance for the application:

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

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

The system of equations related to the Hamiltonian HAM:

                        2
HAM=u1*n1 - i*u2*n1 + u3 *a33 + u3*n3 + v1*m1 - i*v2*m1

has apart from the Hamiltonian and Casimirs the following 2 first integrals: 

                                                         2        2
FI=u1*v2*a33*n1 - u2*v1*a33*n1 - i*u3*v3*a33*n1 - i*v1*n1  - v2*n1

          2
    + i*v3 *a33*m1 - i*v3*n1*n3

  = a product of the elements of: {i,

                                                            2          2
    - i*u1*v2*a33*n1 + i*u2*v1*a33*n1 - u3*v3*a33*n1 - v1*n1  + i*v2*n1

        2
    + v3 *a33*m1 - v3*n1*n3}

{HAM,FI} = 0



     2       2        3     2       2          3                            2
FI=u1 *a33*n1  - u1*n1  + u2 *a33*n1  + i*u2*n1  + 2*u3*v3*a33*m1*n1 - u3*n1 *n3

              2             2     2       2
    + v1*m1*n1  - i*v2*m1*n1  - v3 *a33*m1  + 2*v3*m1*n1*n3

which the program can not factorize further.

{HAM,FI} = 0





And again in machine readable form:



HAM=u1*n1 - i*u2*n1 + u3**2*a33 + u3*n3 + v1*m1 - i*v2*m1$

FI=u1*v2*a33*n1 - u2*v1*a33*n1 - i*u3*v3*a33*n1 - i*v1*n1**2 - v2*n1**2 + i*v3**
2*a33*m1 - i*v3*n1*n3$

FI=u1**2*a33*n1**2 - u1*n1**3 + u2**2*a33*n1**2 + i*u2*n1**3 + 2*u3*v3*a33*m1*n1
 - u3*n1**2*n3 + v1*m1*n1**2 - i*v2*m1*n1**2 - v3**2*a33*m1**2 + 2*v3*m1*n1*n3$