Solution 7 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*a22*c12*r212
r21=-------------------
a23*b33
3 2 2 2
r22=( - a22 *a23*b33*c12*c23*r212 - a22 *a23 *b12*c12 *r212
1 2 2 1 2 4
- ---*a22 *a23*b12*b33 *c23*r212 + ----*a22 *b12*b33 *r212
4 16
3 4 2
+ a22*a23 *b33*c12*c23*r212 + a23 *b12*c12 *r212
1 3 2 1 2 4 3 2
- ---*a23 *b12*b33 *c23*r212 + ----*a23 *b12*b33 *r212)/(a22*a23 *b33 *c23
4 16
4
+ a23 *b12*b33*c12)
1
---*b33*r212
2
r23=--------------
a23
r24=0
1 2 2 2 2 3 2 4
r25=( - ---*a22 *a23 *b12*b33 *c23 *r212 + ----*a22 *a23*b12*b33 *c23*r212
2 16
1 2 6 4 2
- ----*a22 *b12*b33 *r212 + a22*a23 *b33*c12*c23 *r212
64
1 3 2 1 3 3
- ---*a22*a23 *b12 *b33*c12*c23*r212 - ---*a22*a23 *b33 *c12*c23*r212
4 4
1 2 2 3 5 2
+ ----*a22*a23 *b12 *b33 *c12*r212 + a23 *b12*c12 *c23*r212
16
1 4 2 2 1 4 2 2
- ---*a23 *b12*b33 *c12 *r212 - ---*a23 *b12*b33 *c23 *r212
4 4
1 3 4 1 2 6
+ ---*a23 *b12*b33 *c23*r212 - ----*a23 *b12*b33 *r212)/(
8 64
4 2 2 1 3 4 5
a22*a23 *b33 *c23 - ---*a22*a23 *b33 *c23 + a23 *b12*b33*c12*c23
4
1 4 3
- ---*a23 *b12*b33 *c12)
4
r26=0
r27=0
- a22*r212
r28=-------------
a23
r29=0
r210=0
r213=0
a22*r212
r215=----------
a23
r216= - r212
r217=0
r218=0
r219=0
r220=0
3 2 1 2 2 3
c33=( - a22 *b33*c23 - ---*a22 *b12*b33 *c12 + a23 *b12*c12*c23
4
1 2 2 2 2
- ---*a23 *b12*b33 *c12)/(a22 *a23*b33*c23 + a22*a23 *b12*c12)
4
1 2 2 1 2 2 3 3
c22=(---*a22 *a23*b12 *b33*c23 - ----*a22 *b12 *b33 + a22*a23 *b12*c12*c23
4 16
1 2 2 4 2 1 3 2
- ---*a22*a23 *b12*b33 *c12 - a23 *b33*c23 + ---*a23 *b12 *b33*c23
4 4
1 3 3 1 2 2 3 3
+ ---*a23 *b33 *c23 - ----*a23 *b12 *b33 )/(a22*a23 *b33*c23
4 16
4
+ a23 *b12*c12)
- a22*c12
c13=------------
a23
- a23*b33
b32=------------
a22
- a22*b12
b13=------------
a23
b11= - b33
2
a23
a33=------
a22
2 1 2 3 3
b31=( - a22 *a23*b12*b33*c23 + ---*a22 *b12*b33 - a23 *b12*b33*c23
4
1 2 3 2 3
+ ---*a23 *b12*b33 )/(a22*a23 *b33*c23 + a23 *b12*c12)
4
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:
c12, r212, c23, b33, 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.
{r212,a22,a23}
Relevance for the application:
Modulo the following equation:
2 2
0=a22 + a23
the system of equations related to the Hamiltonian HAM:
2 3 2 4
HAM=(u1*v1*( - a22 *a23 *b33 *c23 - a22*a23 *b12*b33*c12)
2 3 4 2
+ u1*v2*(a22 *a23 *b12*b33*c23 + a22*a23 *b12 *c12)
3 2 2 3 2
+ u1*v3*( - a22 *a23 *b12*b33*c23 - a22 *a23 *b12 *c12)
2 3 3 2 4
+ u2 *(a22 *a23 *b33*c23 + a22 *a23 *b12*c12)
2 4 5
+ u2*u3*(2*a22 *a23 *b33*c23 + 2*a22*a23 *b12*c12)
2 5 6 3 2
+ u3 *(a22*a23 *b33*c23 + a23 *b12*c12) + u3*v1*( - a22 *a23 *b12*b33*c23
1 3 3 4 1 3 3
+ ---*a22 *a23*b12*b33 - a22*a23 *b12*b33*c23 + ---*a22*a23 *b12*b33 )
4 4
4 2 5
+ u3*v2*( - a22*a23 *b33 *c23 - a23 *b12*b33*c12)
2 3 2 4
+ u3*v3*(a22 *a23 *b33 *c23 + a22*a23 *b12*b33*c12)
2 3 4 2
+ v1*v2*(2*a22 *a23 *b33*c12*c23 + 2*a22*a23 *b12*c12 )
3 2 2 3 2 2
+ v1*v3*( - 2*a22 *a23 *b33*c12*c23 - 2*a22 *a23 *b12*c12 ) + v2 *(
1 3 2 1 3 2 3 2 3
---*a22 *a23*b12 *b33*c23 - ----*a22 *b12 *b33 + a22 *a23 *b12*c12*c23
4 16
1 2 2 2 4 2
- ---*a22 *a23 *b12*b33 *c12 - a22*a23 *b33*c23
4
1 3 2 1 3 3
+ ---*a22*a23 *b12 *b33*c23 + ---*a22*a23 *b33 *c23
4 4
1 2 2 3
- ----*a22*a23 *b12 *b33 )
16
2 3 2 4 2
+ v2*v3*(2*a22 *a23 *b33*c23 + 2*a22*a23 *b12*c12*c23) + v3 *(
3 2 2 1 2 2 2 5
- a22 *a23 *b33*c23 - ---*a22 *a23 *b12*b33 *c12 + a23 *b12*c12*c23
4
1 4 2 2 3 4
- ---*a23 *b12*b33 *c12))/(a22 *a23 *b33*c23 + a22*a23 *b12*c12)
4
has apart from the Hamiltonian and Casimirs only the following first integral:
3 2 4
FI=u1*v2*( - a22*a23 *b33 *c23 - a23 *b12*b33*c12)
2 2 2 3
+ u1*v3*(a22 *a23 *b33 *c23 + a22*a23 *b12*b33*c12)
3 2 4
+ u2*v1*(a22*a23 *b33 *c23 + a23 *b12*b33*c12)
2 2 2 3 2
+ u3*v1*( - a22 *a23 *b33 *c23 - a22*a23 *b12*b33*c12) + v1 *(
1 2 2 1 2 4 3
- ---*a22 *a23*b12*b33 *c23 + ----*a22 *b12*b33 + a22*a23 *b33*c12*c23
2 16
1 2 2 4 2 1 3 2
- ---*a22*a23 *b12 *b33*c12 + a23 *b12*c12 - ---*a23 *b12*b33 *c23
4 4
1 2 4
+ ----*a23 *b12*b33 )
16
1 2 3 1 3 2 2
+ v1*v3*(---*a22*a23 *b33 *c23 + ---*a23 *b12*b33 *c12) + v2 *(
2 2
3 2 2 2 1 2 2
- a22 *a23*b33*c12*c23 - a22 *a23 *b12*c12 - ---*a22 *a23*b12*b33 *c23
4
1 2 4 3 4 2
+ ----*a22 *b12*b33 + a22*a23 *b33*c12*c23 + a23 *b12*c12
16
1 3 2 1 2 4
- ---*a23 *b12*b33 *c23 + ----*a23 *b12*b33 )
4 16
2 2 3 2
+ v2*v3*( - 2*a22 *a23 *b33*c12*c23 - 2*a22*a23 *b12*c12 )
3 5 6 2 2
{HAM,FI} = v2 *( - 4*a22*a23 *b12*b33*c12*c23 - 2*a23 *b12 *c12
6 2 2 2 5 2 2
+ 2*a23 *b33 *c23 ) + v2 *v3*(6*a22*a23 *b12 *c12
5 2 2 6 2
- 6*a22*a23 *b33 *c23 - 12*a23 *b12*b33*c12*c23) + v2*v3
5 6 2 2 6 2 2
*(12*a22*a23 *b12*b33*c12*c23 + 6*a23 *b12 *c12 - 6*a23 *b33 *c23 )
3 5 2 2 5 2 2
+ v3 *( - 2*a22*a23 *b12 *c12 + 2*a22*a23 *b33 *c23
6
+ 4*a23 *b12*b33*c12*c23)
And again in machine readable form:
HAM=(u1*v1*( - a22**2*a23**3*b33**2*c23 - a22*a23**4*b12*b33*c12) + u1*v2*(a22**
2*a23**3*b12*b33*c23 + a22*a23**4*b12**2*c12) + u1*v3*( - a22**3*a23**2*b12*b33*
c23 - a22**2*a23**3*b12**2*c12) + u2**2*(a22**3*a23**3*b33*c23 + a22**2*a23**4*
b12*c12) + u2*u3*(2*a22**2*a23**4*b33*c23 + 2*a22*a23**5*b12*c12) + u3**2*(a22*
a23**5*b33*c23 + a23**6*b12*c12) + u3*v1*( - a22**3*a23**2*b12*b33*c23 + 1/4*a22
**3*a23*b12*b33**3 - a22*a23**4*b12*b33*c23 + 1/4*a22*a23**3*b12*b33**3) + u3*v2
*( - a22*a23**4*b33**2*c23 - a23**5*b12*b33*c12) + u3*v3*(a22**2*a23**3*b33**2*
c23 + a22*a23**4*b12*b33*c12) + v1*v2*(2*a22**2*a23**3*b33*c12*c23 + 2*a22*a23**
4*b12*c12**2) + v1*v3*( - 2*a22**3*a23**2*b33*c12*c23 - 2*a22**2*a23**3*b12*c12
**2) + v2**2*(1/4*a22**3*a23*b12**2*b33*c23 - 1/16*a22**3*b12**2*b33**3 + a22**2
*a23**3*b12*c12*c23 - 1/4*a22**2*a23**2*b12*b33**2*c12 - a22*a23**4*b33*c23**2 +
1/4*a22*a23**3*b12**2*b33*c23 + 1/4*a22*a23**3*b33**3*c23 - 1/16*a22*a23**2*b12
**2*b33**3) + v2*v3*(2*a22**2*a23**3*b33*c23**2 + 2*a22*a23**4*b12*c12*c23) + v3
**2*( - a22**3*a23**2*b33*c23**2 - 1/4*a22**2*a23**2*b12*b33**2*c12 + a23**5*b12
*c12*c23 - 1/4*a23**4*b12*b33**2*c12))/(a22**2*a23**3*b33*c23 + a22*a23**4*b12*
c12)$
FI=u1*v2*( - a22*a23**3*b33**2*c23 - a23**4*b12*b33*c12) + u1*v3*(a22**2*a23**2*
b33**2*c23 + a22*a23**3*b12*b33*c12) + u2*v1*(a22*a23**3*b33**2*c23 + a23**4*b12
*b33*c12) + u3*v1*( - a22**2*a23**2*b33**2*c23 - a22*a23**3*b12*b33*c12) + v1**2
*( - 1/2*a22**2*a23*b12*b33**2*c23 + 1/16*a22**2*b12*b33**4 + a22*a23**3*b33*c12
*c23 - 1/4*a22*a23**2*b12**2*b33*c12 + a23**4*b12*c12**2 - 1/4*a23**3*b12*b33**2
*c23 + 1/16*a23**2*b12*b33**4) + v1*v3*(1/2*a22*a23**2*b33**3*c23 + 1/2*a23**3*
b12*b33**2*c12) + v2**2*( - a22**3*a23*b33*c12*c23 - a22**2*a23**2*b12*c12**2 -
1/4*a22**2*a23*b12*b33**2*c23 + 1/16*a22**2*b12*b33**4 + a22*a23**3*b33*c12*c23
+ a23**4*b12*c12**2 - 1/4*a23**3*b12*b33**2*c23 + 1/16*a23**2*b12*b33**4) + v2*
v3*( - 2*a22**2*a23**2*b33*c12*c23 - 2*a22*a23**3*b12*c12**2)$