Solution 7 to problem over

Equations

The following unsolved equations remain:

     2      2
0=a11  + b12 *kap

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


r310=r312


r311=0


r313=0


r314=0


r315=r312


      b12*r327 + m3*r487
r316=--------------------
             b12


r317=0


r318=0


r320=0


r321=0


r322=0


r323=0


r324=0


r325=0


r326=0


r328= - r342


r329=0


r330=0


r331=0


r332=0


      a11*r342 + b12*kap*r312
r333=-------------------------
                b12


r334=0


r335=0


r336=0


r337=0


r338=0


r339=0


r340=0


r341=0


r343=r327


r344=0


r345=0


r346=0


r347=0


r348=0


r349=0


r350=0


r351=0


r352=0


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


r354=0


r355=0


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=r488


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=r488


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


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


m2=0


m1=0


n2=0


n1=0


a33=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, r312, r319, r327, r342, r434, r487, r488, r4110, 
n3, a11, m3, 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.

 
                                                 2
{{r488,r434,r487,2*r4110,r4110,a11*b12*r487 - b12 *kap*r4110},b12,a11}

Relevance for the application:

Modulo the following equation:

     2      2
0=a11  + b12 *kap


the system of equations related to the Hamiltonian HAM:

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

has apart from the Hamiltonian and Casimirs the following 10 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 the program can not factorize further.

{HAM,FI} = 0



        2           2        3
FI=u1*u3 *v1 + u2*u3 *v2 + u3 *v3

  = a product of the elements of: {u3,

   u3,

   u1*v1 + u2*v2 + u3*v3}

{HAM,FI} = 0



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

  = a product of the elements of: {u3,

   u3,

     2                     2
   u1 *a11 + u1*v2*b12 + u2 *a11 - u2*v1*b12 + v3*m3}

{HAM,FI} = 0



     4
FI=u3

  = a product of the elements of: {u3,u3,u3,u3}

{HAM,FI} = 0



     2                                      2
FI=u1 *u3*a11 + u1*u3*v2*b12 - u1*v1*m3 + u2 *u3*a11 - u2*u3*v1*b12 - u2*v2*m3

which the program can not factorize further.

{HAM,FI} = 0



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

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

{HAM,FI} = 0



     3
FI=u3

  = a product of the elements of: {u3,u3,u3}

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

{HAM,FI} = 0



     2
FI=u3

  = a product of the elements of: {u3,u3}

{HAM,FI} = 0



FI=u3

which the program can not factorize further.

{HAM,FI} = 0





And again in machine readable form:



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

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*u3**2*v1 + u2*u3**2*v2 + u3**3*v3$

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

FI=u3**4$

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

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

FI=u3**3$

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

FI=u3**2$

FI=u3$