Solution 8 to problem over

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

Expressions

The solution is given through the following expressions:


     i*n2*n3*r32
r10=-------------
      2*a33*c13


         2
     i*n2 *r32
r11=-----------
     2*a33*c13


          2
      - n2 *r32
r12=------------
     2*a33*c13


r13=0


r14=0


r15=0


      - m1*r32 - i*m2*r32
r20=----------------------
            4*c13


r21=0


r22=0


r23=0


r24=0


     i*n2*r32
r26=----------
      2*c13


r27=0


r28=0


r210=0


      n2*r32
r212=--------
      2*c13


r213=0


r214=0


r215=0


       - n2*r32
r216=-----------
        2*c13


r217=0


r218=0


r219=0


r220=0


r30=0


r31=0


r33=0


r34=0


r35=0


r36=0


r37=r32


r38=0


r39=0


r310=0


r311=0


r312=0


r313=0


r314=0


r315=0


r316=0


r317=0


r318=0


r319=0


r320=0


r323=0


r325=0


r326=0


r328=0


r329=0


r330=0


r332=0


r333=0


r334=0


r335=0


r336=0


r337=0


r338=0


r339=0


r340=0


r341=0


r342=0


r343=0


r344=0


r345=0


r347=0


r348=0


r349=0


r350=0


r351=0


r352=0


r353=0


r354=0


r355=0


m3=0


n1=i*n2


c33=i*c12


c23= - i*c13


c22=2*i*c12


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:

 r32, m1, m2, n3, n2, c12, c13, 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,r32,c13,a33}

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,
 - 2*i*c12 + c22,
i*c13 + c23,
 - i*c12 + c33,
n1 - i*n2,
m3}$

The system of equations related to the Hamiltonian HAM:

                        2
HAM=i*u1*n2 + u2*n2 + u3 *a33 + u3*n3 + 2*v1*v2*c12 + 2*v1*v3*c13 + v1*m1

             2                                   2
     + 2*i*v2 *c12 - 2*i*v2*v3*c13 + v2*m2 + i*v3 *c12

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

                                                               2
FI= - 2*u1*v2*a33*n2 + 2*u2*v1*a33*n2 + 2*i*u3*v3*a33*n2 + 4*v1 *v3*a33*c13

             2       2                       2     2
    - 2*v1*n2  + 4*v2 *v3*a33*c13 + 2*i*v2*n2  + v3 *( - a33*m1 - i*a33*m2)

    + 2*i*v3*n2*n3

  = a product of the elements of: {4,

     - u1*v2*a33*n2     u2*v1*a33*n2     i*u3*v3*a33*n2      2
   ----------------- + -------------- + ---------------- + v1 *v3*a33*c13
           2                 2                 2

               2                             2
        - v1*n2       2               i*v2*n2       2   - a33*m1 - i*a33*m2
    + ----------- + v2 *v3*a33*c13 + ---------- + v3 *----------------------
           2                             2                      4

       i*v3*n2*n3
    + ------------}
           2

{HAM,FI} = 0





And again in machine readable form:



HAM=i*u1*n2 + u2*n2 + u3**2*a33 + u3*n3 + 2*v1*v2*c12 + 2*v1*v3*c13 + v1*m1 + 2*
i*v2**2*c12 - 2*i*v2*v3*c13 + v2*m2 + i*v3**2*c12$

FI= - 2*u1*v2*a33*n2 + 2*u2*v1*a33*n2 + 2*i*u3*v3*a33*n2 + 4*v1**2*v3*a33*c13 - 
2*v1*n2**2 + 4*v2**2*v3*a33*c13 + 2*i*v2*n2**2 + v3**2*( - a33*m1 - i*a33*m2) + 
2*i*v3*n2*n3$