Solution 9 to problem over

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

Expressions

The solution is given through the following expressions:


r11=0


r12=0


r13=0


r14=0


r15=0


     m2*r316
r21=---------
       a33


r22=0


      - i*m2*r316
r23=--------------
         a33


r24=0


r27=0


r28=0


r210=0


       - i*a33*r26 + i*n3*r316
r212=--------------------------
                a33


r213=0


r214=0


r215=0


      i*a33*r26 - i*n3*r316
r216=-----------------------
               a33


r217=0


r218=0


r219=0


r220=0


     2*i*c13*r316
r31=--------------
         a33


r33=0


     2*c13*r316
r34=------------
        a33


r35=0


r36=0


r37=r32


r38=0


r39=0


r311=0


       - 2*i*a33*c13*r26 + 2*i*c13*n3*r316
r312=--------------------------------------
                     a33*m2


r313=0


r314=0


       - 2*i*a33*c13*r26 + 2*i*c13*n3*r316
r315=--------------------------------------
                     a33*m2


r317=0


r318=0


r319=0


r320=0


       - 2*i*a33*c13*r26 + a33*m2*r310 + 2*i*c13*n3*r316
r321=----------------------------------------------------
                            a33*m2


r322=0


      2*a33*c13*r26 - 2*c13*n3*r316
r323=-------------------------------
                 a33*m2


r324=0


r325=0


r326=0


r327=0


r328=0


r329=0


r330=0


r331=0


r332=0


r333=0


r334=0


r335=0


       - 2*a33*c13*r26 + 2*c13*n3*r316
r336=----------------------------------
                   a33*m2


r337=0


r339=0


r340=0


r341=0


r342=0


r343=0


r344=0


r345=0


r346=0


r347=0


r348=0


r349=0


r350=0


r351=0


r352=0


r353=0


r354=0


r355=0


m3=0


m1= - i*m2


n2=0


n1=0


c33=0


c23=i*c13


c22=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


       - 2*i*a33*c13*r26 + a33*m2*r310 + 2*i*c13*n3*r316
r338=----------------------------------------------------
                            a33*m2

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:

 r26, r30, r20, r10, r32, r316, r310, n3, m2, 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.

 
{m2,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,
c12,
c22,
 - i*c13 + c23,
c33,
n1,
n2,
m1 + i*m2,
m3}$

The system of equations related to the Hamiltonian HAM:

      2
HAM=u3 *a33 + u3*n3 + 2*v1*v3*c13 - i*v1*m2 + 2*i*v2*v3*c13 + v2*m2

has apart from the Hamiltonian and Casimirs the following 7 first integrals: 

                              2
FI=u1*v1*v3 + u2*v2*v3 + u3*v3

  = a product of the elements of: {v3,u1*v1 + u2*v2 + u3*v3}

{HAM,FI} = 0



FI=2*i*u1*v1*v3*c13*n3 + 2*u1*v2*v3*c13*n3 - i*u1*v2*m2*n3 - 2*u2*v1*v3*c13*n3

                                              2                      2
    + i*u2*v1*m2*n3 + 2*i*u2*v2*v3*c13*n3 + u3 *v3*a33*m2 + 2*i*u3*v1 *c13*n3

               2                 2                    2            2
    + 2*i*u3*v2 *c13*n3 + 2*v1*v3 *c13*m2 - i*v1*v3*m2  + 2*i*v2*v3 *c13*m2

              2
    + v2*v3*m2

which the program can not factorize further.

{HAM,FI} = 0



     2        2
FI=v1 *v3 + v2 *v3

  = a product of the elements of: {v3,v1 - i*v2,v1 + i*v2}

{HAM,FI} = 0



FI=v3

which the program can not factorize further.

{HAM,FI} = 0



     2
FI=v3

  = a product of the elements of: {v3,v3}

{HAM,FI} = 0



     3
FI=v3

  = a product of the elements of: {v3,v3,v3}

{HAM,FI} = 0



FI= - 2*i*u1*v1*v3*c13 - 2*u1*v2*v3*c13 + i*u1*v2*m2 + 2*u2*v1*v3*c13

                                               2                2
    - i*u2*v1*m2 - 2*i*u2*v2*v3*c13 - 2*i*u3*v1 *c13 - 2*i*u3*v2 *c13 + u3*v3*m2

  = a product of the elements of: { - 2*i,

                                     - u1*v2*m2                      u2*v1*m2
   u1*v1*v3*c13 - i*u1*v2*v3*c13 + ------------- + i*u2*v1*v3*c13 + ----------
                                         2                              2

                          2            2        i*u3*v3*m2
    + u2*v2*v3*c13 + u3*v1 *c13 + u3*v2 *c13 + ------------}
                                                    2

{HAM,FI} = 0





And again in machine readable form:



HAM=u3**2*a33 + u3*n3 + 2*v1*v3*c13 - i*v1*m2 + 2*i*v2*v3*c13 + v2*m2$

FI=u1*v1*v3 + u2*v2*v3 + u3*v3**2$

FI=2*i*u1*v1*v3*c13*n3 + 2*u1*v2*v3*c13*n3 - i*u1*v2*m2*n3 - 2*u2*v1*v3*c13*n3 +
 i*u2*v1*m2*n3 + 2*i*u2*v2*v3*c13*n3 + u3**2*v3*a33*m2 + 2*i*u3*v1**2*c13*n3 + 2
*i*u3*v2**2*c13*n3 + 2*v1*v3**2*c13*m2 - i*v1*v3*m2**2 + 2*i*v2*v3**2*c13*m2 + 
v2*v3*m2**2$

FI=v1**2*v3 + v2**2*v3$

FI=v3$

FI=v3**2$

FI=v3**3$

FI= - 2*i*u1*v1*v3*c13 - 2*u1*v2*v3*c13 + i*u1*v2*m2 + 2*u2*v1*v3*c13 - i*u2*v1*
m2 - 2*i*u2*v2*v3*c13 - 2*i*u3*v1**2*c13 - 2*i*u3*v2**2*c13 + u3*v3*m2$