Solution 1 to problem over

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

Expressions

The solution is given through the following expressions:


r10=0


r11=0


r12=0


r14=0


r15=0


r20=0


r21=0


r23=0


r24=0


r25=0


     m3*r353
r26=---------
       a22


r27=0


r28=0


r210=0


r212=0


r213=0


r214=0


r215=0


r216=0


r217=0


r218=0


r219=0


r30=0


r31=0


r32=0


r33=0


r34=0


r35=0


r36=0


r37=0


r38=0


r39=0


r311=0


      a22*r310 - c33*r353
r312=---------------------
              a22


r313=0


r314=0


      a22*r310 - c33*r353
r315=---------------------
              a22


      a22*r327 + b33*r353
r316=---------------------
              a22


r317=0


r318=0


r320=0


r321=0


r322=0


r323=0


r324=0


r325=0


r326=0


r328=0


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


r342=0


r343=r327


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


c23=0


c22=0


c13=0


c12=0


c11=0


b32=0


b31=0


b23=0


b22=0


b21=0


b13=0


b12=0


b11=0


a33=0


a23=0


a13=0


a12=0


a11=a22

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:

 r29, r13, r310, r353, r327, r319, c33, m3, n3, b33, a22

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.

 
{a22}

Relevance for the application:

The new Hamiltonian in form of a list of vanishing expressions: 

{a11 - a22,
a12,
a13,
a23,
a33,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
c11,
c12,
c13,
c22,
c23,
n1,
n2,
m1,
m2}$

The system of equations related to the Hamiltonian HAM:

      2         2                             2
HAM=u1 *a22 + u2 *a22 + u3*v3*b33 + u3*n3 + v3 *c33 + v3*m3

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

     3
FI=u3

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

{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



     2            2            2               2            2
FI=u1 *u3*a22 + u2 *u3*a22 + u3 *v3*b33 - u3*v1 *c33 - u3*v2 *c33 + u3*v3*m3

  = a product of the elements of: {u3,

     2         2                     2         2
   u1 *a22 + u2 *a22 + u3*v3*b33 - v1 *c33 - v2 *c33 + v3*m3}

{HAM,FI} = 0



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

                                     2     2     2
  = a product of the elements of: {v1  + v2  + v3 ,u3}

{HAM,FI} = 0



FI=u3

which the program can not factorize further.

{HAM,FI} = 0



     2
FI=u3

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

{HAM,FI} = 0





And again in machine readable form:



HAM=u1**2*a22 + u2**2*a22 + u3*v3*b33 + u3*n3 + v3**2*c33 + v3*m3$

FI=u3**3$

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

FI=u1**2*u3*a22 + u2**2*u3*a22 + u3**2*v3*b33 - u3*v1**2*c33 - u3*v2**2*c33 + u3
*v3*m3$

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

FI=u3$

FI=u3**2$