Solution 2 to problem h

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

Expressions

The solution is given through the following expressions:


r10=0


r11=0


r12=0


r14=0


r15=0


r20=0


r21=0


r22=0


r23=0


r24=0


r27=0


r28=0


r210=0


       - m3*r342
r211=------------
         b12


r212=0


r213=0


r214=0


r215=0


       - m3*r342
r217=------------
         b12


r218=0


r219=0


r220=0


r30=0


r31=0


r32=0


r33=0


r34=0


r35=0


r36=0


r37=0


r38=0


r39=0


       - a11*r342 + b12*r353
r310=------------------------
             b12*kap


r311=0


       - a11*r342 + b12*r353
r312=------------------------
             b12*kap


r313=0


r314=0


       - a11*r342 + b12*r353
r315=------------------------
             b12*kap


r316=r343


r317=0


r318=0


r320=0


r321=0


r322=0


r323=0


r324=0


r325=0


r326=0


r327=r343


r328= - r342


r329=0


r330=0


r331=0


r332=0


r333=r353


r334=0


r335=0


r336=0


r337=0


r338=0


r339=0


r340=0


r341=0


r344=0


r345=0


r346=0


r347=0


r348=0


r349=0


r350=0


r351=0


r352=0


r354=0


r355=0


m2=0


m1=0


n2=0


n1=0


a23=0


a22=a11


a13=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:

 r13, r29, r319, r343, r353, r342, m3, n3, a33, a11, b12

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.

 
{{m3,n3},b12}

Relevance for the application:

The system of equations related to the Hamiltonian HAM:

      2                     2                     2
HAM=u1 *a11 + u1*v2*b12 + u2 *a11 - u2*v1*b12 + u3 *a33 + u3*n3 + v3*m3

has apart from the Hamiltonian and Casimirs the following 6 first integrals: 

FI=u1*u3*v2*b12*kap - u1*v1*kap*m3 - u2*u3*v1*b12*kap - u2*v2*kap*m3

           2            2            2
    - u3*v1 *a11 - u3*v2 *a11 - u3*v3 *a11

which is not further factorizable.

{HAM,FI} = 0



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

         2     2     2      2     2
  =u3*(v2  + v3  + v1  + (u1  + u2 )*kap)

{HAM,FI} = 0



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

  =u3*(u2*v2 + u3*v3 + u1*v1)

{HAM,FI} = 0



     3
FI=u3

     3
  =u3

{HAM,FI} = 0



     2
FI=u3

     2
  =u3

{HAM,FI} = 0



FI=u3

which is not further factorizable.

{HAM,FI} = 0





And again in machine readable form:



HAM=u1**2*a11 + u1*v2*b12 + u2**2*a11 - u2*v1*b12 + u3**2*a33 + u3*n3 + v3*m3$

FI=u1*u3*v2*b12*kap - u1*v1*kap*m3 - u2*u3*v1*b12*kap - u2*v2*kap*m3 - u3*v1**2*
a11 - u3*v2**2*a11 - u3*v3**2*a11$

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

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

FI=u3**3$

FI=u3**2$

FI=u3$