Solution 12 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*r35
r21=--------
     2*c12


r22=0


     i*m2*r35
r23=----------
      2*c12


r24=0


r27=0


r28=0


r210=0


      2*i*c12*r26 - i*n3*r35
r212=------------------------
              2*c12


r213=0


r214=0


r215=0


       - 2*i*c12*r26 + i*n3*r35
r216=---------------------------
                2*c12


r217=0


r218=0


r219=0


r220=0


r31=0


r33=0


r34=0


r36=0


r37=r32 + i*r35


r38=0


r39=0


r310=i*r336 + r338


r311=0


r312=0


r313=0


r314=0


r315=0


      a33*r35
r316=---------
       2*c12


r317=0


r318=0


r319=0


r320=0


r321=r338


r322=0


r323= - r336


r324=0


r325=0


r326=0


r327=0


r328=0


r329=0


r330=0


r331=0


r332=0


r333=0


r334=0


r335=0


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= - i*c12


c23=0


c22= - 2*i*c12


c13=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:

 r30, r20, r10, r32, r26, r338, r336, r35, m2, n3, 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.

 
{a33,r336}

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

The system of equations related to the Hamiltonian HAM:

      2                                             2                   2
HAM=u3 *a33 + u3*n3 + 2*v1*v2*c12 + i*v1*m2 - 2*i*v2 *c12 + v2*m2 - i*v3 *c12

has apart from the Hamiltonian and Casimirs the following 8 first integrals: 

                               2                2
FI=i*u1*v2*n3 - i*u2*v1*n3 + u3 *v3*a33 + 2*i*v1 *v3*c12 + 2*v1*v2*v3*c12

    + i*v1*v3*m2 + v2*v3*m2

which the program can not factorize further.

{HAM,FI} = 0



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

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

{HAM,FI} = 0



                              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= - i*u1*v2 + i*u2*v1 + u3*v3

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

{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





And again in machine readable form:



HAM=u3**2*a33 + u3*n3 + 2*v1*v2*c12 + i*v1*m2 - 2*i*v2**2*c12 + v2*m2 - i*v3**2*
c12$

FI=i*u1*v2*n3 - i*u2*v1*n3 + u3**2*v3*a33 + 2*i*v1**2*v3*c12 + 2*v1*v2*v3*c12 + 
i*v1*v3*m2 + v2*v3*m2$

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

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

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

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

FI=v3$

FI=v3**2$

FI=v3**3$