Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2
0=a22 - 2*a22*a33 + 4*a23 + a33
Expressions
The solution is given through the following expressions:
2
- 2*a22 *c22*r28 + 2*a22*a33*c22*r28
r21=---------------------------------------
3*a22*a23*b31 + a23*a33*b31
3 2 2 2 2
r22=( - 4*a22 *c22*r28 + 8*a22 *a33*c22*r28 + a22 *b31 *r28 - 4*a22*a33 *c22*r28
2 2 2
- a22*a33*b31 *r28)/(12*a22*a23 *b31 + 4*a23 *a33*b31)
r23=0
r24=0
3 2 2 2
r25=( - 4*a22 *c22*r28 + 8*a22 *a33*c22*r28 + a22 *b31 *r28
2 2 2
- 12*a22*a23 *c22*r28 - 4*a22*a33 *c22*r28 - a22*a33*b31 *r28
2 2 2
- 4*a23 *a33*c22*r28)/(12*a22*a23 *b31 + 4*a23 *a33*b31)
r26=0
r27=0
4 3 2 2 2 2
r29=( - a22 *r28 + 2*a22 *a33*r28 - 3*a22 *a23 *r28 - a22 *a33 *r28
2 2 2 2 2
+ 2*a22*a23 *a33*r28 + a23 *a33 *r28)/(3*a22*a23 *b31 + a23 *a33*b31)
r210=0
4 3 2 2 2 2
r212=( - a22 *r28 + 2*a22 *a33*r28 - 3*a22 *a23 *r28 - a22 *a33 *r28
2 3 3
- a22*a23 *a33*r28)/(3*a22*a23 + a23 *a33)
5 4 3 2 3 2
r215=( - a22 *r28 + 2*a22 *a33*r28 - 3*a22 *a23 *r28 - a22 *a33 *r28
2 2 4 4
- a22 *a23 *a33*r28)/(3*a22*a23 + a23 *a33)
r216=
4 3 2 2 2 2 2
a22 *r28 - 2*a22 *a33*r28 + 3*a22 *a23 *r28 + a22 *a33 *r28 + a22*a23 *a33*r28
--------------------------------------------------------------------------------
3 3
3*a22*a23 + a23 *a33
r217=0
r218=0
r219=0
4 3 2 2 2
- a22 *r28 + a22 *a33*r28 - 3*a22 *a23 *r28 - a22*a23 *a33*r28
r220=-----------------------------------------------------------------
2 2
3*a22*a23 *b31 + a23 *a33*b31
c33=c22
c23=0
c13=0
c12=0
b33=0
b32=0
b13=0
b12=0
b11=0
3 2 2 2
2*a22 *r28 - 2*a22 *a33*r28 + 6*a22*a23 *r28 + 2*a23 *a33*r28
r213=---------------------------------------------------------------
3*a22*a23*b31 + a23*a33*b31
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:
r28, c22, a33, b31, a22, a23
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.
{3*a22 + a33,
3 2 2 2
a22 - a22 *a33 + 3*a22*a23 + a23 *a33,
a33,
a23,
a22,
b31}
Relevance for the application:
Modulo the following equation:
2 2 2
0=a22 - 2*a22*a33 + 4*a23 + a33
the system of equations related to the Hamiltonian HAM:
2 2 2 2
HAM=u2 *a22 + 2*u2*u3*a23 + u3 *a33 + u3*v1*b31 + v2 *c22 + v3 *c22
has apart from the Hamiltonian and Casimirs only the following first integral:
2 4 2 3 2 2 4 4
FI=u1 *( - 4*a22 *a23 + 4*a22 *a23 *a33 - 12*a22 *a23 - 4*a22*a23 *a33) + u1
4 3 2 3
*v2*(4*a22 *a23*b31 - 8*a22 *a23*a33*b31 + 12*a22 *a23 *b31
2 2 3 5
+ 4*a22 *a23*a33 *b31 + 4*a22*a23 *a33*b31) + u1*v3*( - 4*a22 *b31
4 3 2 3 2
+ 8*a22 *a33*b31 - 12*a22 *a23 *b31 - 4*a22 *a33 *b31
2 2
- 4*a22 *a23 *a33*b31)
3 3 2 3 5 5
+ u2*u3*(8*a22 *a23 - 8*a22 *a23 *a33 + 24*a22*a23 + 8*a23 *a33) + u2*v1*(
4 3 2 3
- 4*a22 *a23*b31 + 8*a22 *a23*a33*b31 - 12*a22 *a23 *b31
2 2 3 2 4 2
- 4*a22 *a23*a33 *b31 - 4*a22*a23 *a33*b31) + u3 *( - 4*a22 *a23
3 2 2 4 2 2 2 4
+ 8*a22 *a23 *a33 - 12*a22 *a23 - 4*a22 *a23 *a33 + 8*a22*a23 *a33
4 2 4 4 2
+ 4*a23 *a33 ) + u3*v1*(12*a22*a23 *b31 + 4*a23 *a33*b31) + v1 *(
3 2 2 2 2 2 2
- 4*a22 *a23 *c22 + 8*a22 *a23 *a33*c22 + a22 *a23 *b31
4 2 2 2 2
- 12*a22*a23 *c22 - 4*a22*a23 *a33 *c22 - a22*a23 *a33*b31
4 2 3 2 2 2
- 4*a23 *a33*c22) + v2 *( - 4*a22 *a23 *c22 + 8*a22 *a23 *a33*c22
2 2 2 2 2 2 2
+ a22 *a23 *b31 - 4*a22*a23 *a33 *c22 - a22*a23 *a33*b31 )
2 3 3
+ v2*v3*( - 8*a22 *a23 *c22 + 8*a22*a23 *a33*c22)
{HAM,FI} = 0
And again in machine readable form:
HAM=u2**2*a22 + 2*u2*u3*a23 + u3**2*a33 + u3*v1*b31 + v2**2*c22 + v3**2*c22$
FI=u1**2*( - 4*a22**4*a23**2 + 4*a22**3*a23**2*a33 - 12*a22**2*a23**4 - 4*a22*
a23**4*a33) + u1*v2*(4*a22**4*a23*b31 - 8*a22**3*a23*a33*b31 + 12*a22**2*a23**3*
b31 + 4*a22**2*a23*a33**2*b31 + 4*a22*a23**3*a33*b31) + u1*v3*( - 4*a22**5*b31 +
8*a22**4*a33*b31 - 12*a22**3*a23**2*b31 - 4*a22**3*a33**2*b31 - 4*a22**2*a23**2
*a33*b31) + u2*u3*(8*a22**3*a23**3 - 8*a22**2*a23**3*a33 + 24*a22*a23**5 + 8*a23
**5*a33) + u2*v1*( - 4*a22**4*a23*b31 + 8*a22**3*a23*a33*b31 - 12*a22**2*a23**3*
b31 - 4*a22**2*a23*a33**2*b31 - 4*a22*a23**3*a33*b31) + u3**2*( - 4*a22**4*a23**
2 + 8*a22**3*a23**2*a33 - 12*a22**2*a23**4 - 4*a22**2*a23**2*a33**2 + 8*a22*a23
**4*a33 + 4*a23**4*a33**2) + u3*v1*(12*a22*a23**4*b31 + 4*a23**4*a33*b31) + v1**
2*( - 4*a22**3*a23**2*c22 + 8*a22**2*a23**2*a33*c22 + a22**2*a23**2*b31**2 - 12*
a22*a23**4*c22 - 4*a22*a23**2*a33**2*c22 - a22*a23**2*a33*b31**2 - 4*a23**4*a33*
c22) + v2**2*( - 4*a22**3*a23**2*c22 + 8*a22**2*a23**2*a33*c22 + a22**2*a23**2*
b31**2 - 4*a22*a23**2*a33**2*c22 - a22*a23**2*a33*b31**2) + v2*v3*( - 8*a22**2*
a23**3*c22 + 8*a22*a23**3*a33*c22)$