Solution 9 to problem h

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

Expressions

The solution is given through the following expressions:


r40=0


r41=0


r42=0


r43=0


r44=0


r45=0


r46=0


r47=0


r48=0


r49=0


r410=0


r411=0


r412=0


r413=0


r415=0


r416=0


r417=0


r418=0


r419=0


r420=0


r421=0


r423=0


r424=0


r426=0


r427= - r4110


r428=0


r429=0


r430= - r4110


r431=0


r432=0


r433=0


r435=0


r436=0


r437=0


r438=0


r439=0


r440=0


r441=0


r442=0


r443=0


r444=0


r445=0


r446=2*r4110


r447=0


r448=0


r449=0


r450=0


r451=0


r452=0


r453= - r487


r454=0


r455=0


r456=0


r457=r4110


r458=0


r459=0


r460=0


r461=0


r462=0


r463=0


      a11*r487 - b12*kap*r4110
r464=--------------------------
                b12


r465=0


r466=0


r467=0


r468=0


r469=0


r470=0


r471=0


r472=0


r473=0


r474=0


r475=0


r476=0


r477=0


r479=0


r480=0


r482=0


r483=2*r4110


r484=0


r485=0


r486=0


r488=0


r489=0


r490=0


r491=0


r492=0


r493=0


r494=2*r4110


r495=0


r496=0


r497=0


r498=0


r499=0


r4100=0


r4101=0


r4102=0


r4103=0


r4104=0


r4105=0


r4106=0


r4108=0


r4109=0


r4111=0


r4112=0


r4113=0


       a11*r487 - b12*kap*r4110
r4114=--------------------------
                 b12


r4115=0


r4116=0


r4117=0


r4118=0


r4119=0


r4120=0


r4121=0


r4122=0


r4123=0


r4124=0


r4125=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:

 r434, r487, r4110, 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.

 
{b12,

 a11 - 2*a33,

 {a11*r4110 - 2*a33*r4110,

  r4110,

  r434,

  2*a11*r4110 - 4*a33*r4110,

  a11*b12*r487 - 2*a33*b12*r487,

  a11*r4110 - 2*a33*r4110,

     2                                        2                      2
  a11 *b12*r487 - 2*a11*a33*b12*r487 - a11*b12 *kap*r4110 + 2*a33*b12 *kap*r4110

  ,

  2*r4110,

  r487,

  2*r4110,

  r4110,

     2                                        2                      2
  a11 *b12*r487 - 2*a11*a33*b12*r487 - a11*b12 *kap*r4110 + 2*a33*b12 *kap*r4110

  }}

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

has apart from the Hamiltonian and Casimirs the following 3 first integrals: 

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

        2   2                     2   2     2   2
    + u2 *v2  + 2*u2*u3*v2*v3 - u3 *v1  - u3 *v2

which is not further factorizable.

{HAM,FI} = 0



     2   2            2            2   2            2
FI=u1 *u3 *a11 + u1*u3 *v2*b12 + u2 *u3 *a11 - u2*u3 *v1*b12

     2     2     2
  =u3 *((u1  + u2 )*a11 + (u1*v2 - u2*v1)*b12)

{HAM,FI} = 0



     4
FI=u3

     4
  =u3

{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$

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

FI=u1**2*u3**2*a11 + u1*u3**2*v2*b12 + u2**2*u3**2*a11 - u2*u3**2*v1*b12$

FI=u3**4$