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