Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
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Equations
The following unsolved equations remain:
2 2 2
0=a11 - 2*a11*a22 + a22 + a23
Expressions
The solution is given through the following expressions:
1 3 5 2 2 1 2 2
r10=( - ---*a11 *a22*b33*r13 + ---*a11 *a22 *b33*r13 + ---*a11 *a23 *b33*r13
2 4 4
3 1 2 1 4
- a11*a22 *b33*r13 - ---*a11*a22*a23 *b33*r13 + ---*a22 *b33*r13
2 4
1 2 2 3 2
+ ---*a22 *a23 *b33*r13)/(a11 *a23 )
4
1 3 2 1 3 2 3 2 3
r11=( - ---*a11 *a22 *b33*r13 - ---*a11 *a23 *b33*r13 + ---*a11 *a22 *b33*r13
4 4 4
1 2 2 3 4
+ ---*a11 *a22*a23 *b33*r13 - ---*a11*a22 *b33*r13
4 4
1 2 2 1 5 1 3 2
- ---*a11*a22 *a23 *b33*r13 + ---*a22 *b33*r13 + ---*a22 *a23 *b33*r13)/(
4 4 4
3 3
a11 *a23 )
r12=0
- a11*r13 + a22*r13
r14=----------------------
a23
r15=0
1 3 3 2 3 2 2
m2=( - ---*a11 *a22*b33*n3 - a11 *a23 *m3 + ---*a11 *a22 *b33*n3
2 2
2 2 3 3 1 2
+ a11 *a22*a23 *m3 - ---*a11*a22 *b33*n3 - ---*a11*a22*a23 *b33*n3
2 2
1 4 1 2 2 2 3
+ ---*a22 *b33*n3 + ---*a22 *a23 *b33*n3)/(a11 *a23 )
2 2
m1=0
- a11*n3 + a22*n3
n2=--------------------
a23
n1=0
6 4 1 5 4 2 1 5 2 2 2
c23=( - a11 *a23 *c33 - ----*a11 *a22 *b33 - ---*a11 *a22 *a23 *b33
16 8
5 4 1 5 4 2 5 4 5 2
+ a11 *a22*a23 *c33 - ----*a11 *a23 *b33 + ----*a11 *a22 *b33
16 16
3 4 3 2 2 1 4 4 2 5 3 6 2
+ ---*a11 *a22 *a23 *b33 + ----*a11 *a22*a23 *b33 - ---*a11 *a22 *b33
8 16 8
1 3 4 2 2 1 3 2 4 2 5 2 7 2
- ---*a11 *a22 *a23 *b33 - ---*a11 *a22 *a23 *b33 + ---*a11 *a22 *b33
2 8 8
1 2 5 2 2 1 2 3 4 2 5 8 2
+ ---*a11 *a22 *a23 *b33 + ---*a11 *a22 *a23 *b33 - ----*a11*a22 *b33
2 8 16
3 6 2 2 1 4 4 2 1 9 2
- ---*a11*a22 *a23 *b33 - ----*a11*a22 *a23 *b33 + ----*a22 *b33
8 16 16
1 7 2 2 1 5 4 2 5 5
+ ---*a22 *a23 *b33 + ----*a22 *a23 *b33 )/(a11 *a23 )
8 16
7 4 1 6 4 2 1 6 2 2 2
c22=(a11 *a23 *c33 + ----*a11 *a22 *b33 + ---*a11 *a22 *a23 *b33
16 8
6 4 1 6 4 2 3 5 5 2
- 2*a11 *a22*a23 *c33 + ----*a11 *a23 *b33 - ---*a11 *a22 *b33
16 8
1 5 3 2 2 5 2 4 1 5 4 2
- ---*a11 *a22 *a23 *b33 + a11 *a22 *a23 *c33 - ---*a11 *a22*a23 *b33
4 8
15 4 6 2 3 4 4 2 2
+ ----*a11 *a22 *b33 - ----*a11 *a22 *a23 *b33
16 16
1 4 2 4 2 1 4 6 2 5 3 7 2
+ ----*a11 *a22 *a23 *b33 - ----*a11 *a23 *b33 - ---*a11 *a22 *b33
16 16 4
3 3 5 2 2 1 3 3 4 2 15 2 8 2
+ ---*a11 *a22 *a23 *b33 + ---*a11 *a22 *a23 *b33 + ----*a11 *a22 *b33
4 2 16
1 2 6 2 2 17 2 4 4 2
- ---*a11 *a22 *a23 *b33 - ----*a11 *a22 *a23 *b33
2 16
1 2 2 6 2 3 9 2 5 5 4 2
- ---*a11 *a22 *a23 *b33 - ---*a11*a22 *b33 + ---*a11*a22 *a23 *b33
8 8 8
1 3 6 2 1 10 2 1 8 2 2
+ ---*a11*a22 *a23 *b33 + ----*a22 *b33 + ----*a22 *a23 *b33
4 16 16
1 6 4 2 1 4 6 2 5 6
- ----*a22 *a23 *b33 - ----*a22 *a23 *b33 )/(a11 *a23 )
16 16
c13=0
c12=0
2 2
- a11*a22*b33 + a22 *b33 + a23 *b33
b32=--------------------------------------
a11*a23
b31=0
b21=0
1 2 2 1 2 2 3 1 4
b11=( - ---*a11 *a22 *b33 + ---*a11 *a23 *b33 + a11*a22 *b33 - ---*a22 *b33
2 2 2
1 2 2 2 2
- ---*a22 *a23 *b33)/(a11 *a23 )
2
2 2
a11 - a11*a22 - a23
a33=-----------------------
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:
r13, c33, m3, n3, b33, a11, 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.
{r13,
a11,
a22,
a23,
a11 - a22,
2 3 3 3
{a11*a22 *a23*m3 - a11*a23 *m3 - a22 *a23*m3 - a22*a23 *m3,
m3,
a11*n3 - a22*n3,
n3}}
Relevance for the application:
Modulo the following equation:
2 2 2
0=a11 - 2*a11*a22 + a22 + a23
the system of equations related to the Hamiltonian HAM:
2 7 6 6 6 1 6 2 4
HAM=(u1 *(a11 *a23 - a11 *a22*a23 ) + u1*v1*( - ---*a11 *a22 *a23 *b33
2
1 6 6 3 5 3 4 1 5 6
+ ---*a11 *a23 *b33 + ---*a11 *a22 *a23 *b33 - ---*a11 *a22*a23 *b33
2 2 2
3 4 4 4 1 4 2 6
- ---*a11 *a22 *a23 *b33 - ---*a11 *a22 *a23 *b33
2 2
1 3 5 4 1 3 3 6
+ ---*a11 *a22 *a23 *b33 + ---*a11 *a22 *a23 *b33)
2 2
2 6 6 5 2 6
+ u2 *(a11 *a22*a23 - a11 *a22 *a23 )
6 7 5 7
+ u2*u3*(2*a11 *a23 - 2*a11 *a22*a23 )
7 5 6 5 5 2 5
+ u2*( - a11 *a23 *n3 + 2*a11 *a22*a23 *n3 - a11 *a22 *a23 *n3)
2 7 6 6 6 5 8
+ u3 *(a11 *a23 - a11 *a22*a23 - a11 *a23 ) + u3*v2*(
6 5 5 2 5 5 7
- a11 *a22*a23 *b33 + 2*a11 *a22 *a23 *b33 + a11 *a23 *b33
4 3 5 4 7
- a11 *a22 *a23 *b33 - a11 *a22*a23 *b33)
6 6 5 6
+ u3*v3*(a11 *a23 *b33 - a11 *a22*a23 *b33)
6 6 5 6 2 8 4
+ u3*(a11 *a23 *n3 - a11 *a22*a23 *n3) + v2 *(a11 *a23 *c33
1 7 4 2 1 7 2 2 2 7 4
+ ----*a11 *a22 *b33 + ---*a11 *a22 *a23 *b33 - 3*a11 *a22*a23 *c33
16 8
1 7 4 2 7 6 5 2 3 6 3 2 2
+ ----*a11 *a23 *b33 - ----*a11 *a22 *b33 - ---*a11 *a22 *a23 *b33
16 16 8
6 2 4 3 6 4 2 21 5 6 2
+ 3*a11 *a22 *a23 *c33 - ----*a11 *a22*a23 *b33 + ----*a11 *a22 *b33
16 16
1 5 4 2 2 5 3 4
+ ----*a11 *a22 *a23 *b33 - a11 *a22 *a23 *c33
16
3 5 2 4 2 1 5 6 2 35 4 7 2
+ ----*a11 *a22 *a23 *b33 - ----*a11 *a23 *b33 - ----*a11 *a22 *b33
16 16 16
15 4 5 2 2 7 4 3 4 2
+ ----*a11 *a22 *a23 *b33 + ----*a11 *a22 *a23 *b33
16 16
1 4 6 2 35 3 8 2
+ ----*a11 *a22*a23 *b33 + ----*a11 *a22 *b33
16 16
5 3 6 2 2 25 3 4 4 2
- ---*a11 *a22 *a23 *b33 - ----*a11 *a22 *a23 *b33
4 16
1 3 2 6 2 21 2 9 2
- ---*a11 *a22 *a23 *b33 - ----*a11 *a22 *b33
8 16
1 2 7 2 2 27 2 5 4 2
+ ---*a11 *a22 *a23 *b33 + ----*a11 *a22 *a23 *b33
2 16
3 2 3 6 2 7 10 2
+ ---*a11 *a22 *a23 *b33 + ----*a11*a22 *b33
8 16
1 8 2 2 11 6 4 2
+ ----*a11*a22 *a23 *b33 - ----*a11*a22 *a23 *b33
16 16
5 4 6 2 1 11 2 1 9 2 2
- ----*a11*a22 *a23 *b33 - ----*a22 *b33 - ----*a22 *a23 *b33
16 16 16
1 7 4 2 1 5 6 2
+ ----*a22 *a23 *b33 + ----*a22 *a23 *b33 ) + v2*v3*(
16 16
7 5 1 6 4 2 1 6 2 3 2
- 2*a11 *a23 *c33 - ---*a11 *a22 *a23*b33 - ---*a11 *a22 *a23 *b33
8 4
6 5 1 6 5 2 3 5 5 2
+ 4*a11 *a22*a23 *c33 - ---*a11 *a23 *b33 + ---*a11 *a22 *a23*b33
8 4
5 3 3 2 5 2 5 1 5 5 2
+ a11 *a22 *a23 *b33 - 2*a11 *a22 *a23 *c33 + ---*a11 *a22*a23 *b33
4
15 4 6 2 7 4 4 3 2
- ----*a11 *a22 *a23*b33 - ---*a11 *a22 *a23 *b33
8 4
3 4 2 5 2 5 3 7 2
- ---*a11 *a22 *a23 *b33 + ---*a11 *a22 *a23*b33
8 2
3 5 3 2 1 3 3 5 2
+ 2*a11 *a22 *a23 *b33 + ---*a11 *a22 *a23 *b33
2
15 2 8 2 7 2 6 3 2
- ----*a11 *a22 *a23*b33 - ---*a11 *a22 *a23 *b33
8 4
3 2 4 5 2 3 9 2 7 3 2
- ---*a11 *a22 *a23 *b33 + ---*a11*a22 *a23*b33 + a11*a22 *a23 *b33
8 4
1 5 5 2 1 10 2 1 8 3 2
+ ---*a11*a22 *a23 *b33 - ---*a22 *a23*b33 - ---*a22 *a23 *b33
4 8 4
1 6 5 2 1 7 3 7 5
- ---*a22 *a23 *b33 ) + v2*( - ---*a11 *a22*a23 *b33*n3 - a11 *a23 *m3
8 2
6 2 3 6 5
+ 2*a11 *a22 *a23 *b33*n3 + 2*a11 *a22*a23 *m3
5 3 3 5 2 5
- 3*a11 *a22 *a23 *b33*n3 - a11 *a22 *a23 *m3
1 5 5 4 4 3
- ---*a11 *a22*a23 *b33*n3 + 2*a11 *a22 *a23 *b33*n3
2
4 2 5 1 3 5 3
+ a11 *a22 *a23 *b33*n3 - ---*a11 *a22 *a23 *b33*n3
2
1 3 3 5 2 6 6 5 6
- ---*a11 *a22 *a23 *b33*n3) + v3 *(a11 *a23 *c33 - a11 *a22*a23 *c33)
2
6 6 5 6 6 6 5 6
+ v3*(a11 *a23 *m3 - a11 *a22*a23 *m3))/(a11 *a23 - a11 *a22*a23 )
has apart from the Hamiltonian and Casimirs only the following first integral:
4 2 3 2 3 3 1 3 2
FI=u2*( - a11 *a23 + a11 *a22*a23 ) + u3*a11 *a23 + v2*( - ---*a11 *a22 *b33
4
1 3 2 3 2 3 1 2 2
- ---*a11 *a23 *b33 + ---*a11 *a22 *b33 + ---*a11 *a22*a23 *b33
4 4 4
3 4 1 2 2 1 5
- ---*a11*a22 *b33 - ---*a11*a22 *a23 *b33 + ---*a22 *b33
4 4 4
1 3 2 1 3
+ ---*a22 *a23 *b33) + v3*( - ---*a11 *a22*a23*b33
4 2
5 2 2 1 2 3 3
+ ---*a11 *a22 *a23*b33 + ---*a11 *a23 *b33 - a11*a22 *a23*b33
4 4
1 3 1 4 1 2 3
- ---*a11*a22*a23 *b33 + ---*a22 *a23*b33 + ---*a22 *a23 *b33)
2 4 4
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*(a11**7*a23**6 - a11**6*a22*a23**6) + u1*v1*( - 1/2*a11**6*a22**2*a23
**4*b33 + 1/2*a11**6*a23**6*b33 + 3/2*a11**5*a22**3*a23**4*b33 - 1/2*a11**5*a22*
a23**6*b33 - 3/2*a11**4*a22**4*a23**4*b33 - 1/2*a11**4*a22**2*a23**6*b33 + 1/2*
a11**3*a22**5*a23**4*b33 + 1/2*a11**3*a22**3*a23**6*b33) + u2**2*(a11**6*a22*a23
**6 - a11**5*a22**2*a23**6) + u2*u3*(2*a11**6*a23**7 - 2*a11**5*a22*a23**7) + u2
*( - a11**7*a23**5*n3 + 2*a11**6*a22*a23**5*n3 - a11**5*a22**2*a23**5*n3) + u3**
2*(a11**7*a23**6 - a11**6*a22*a23**6 - a11**5*a23**8) + u3*v2*( - a11**6*a22*a23
**5*b33 + 2*a11**5*a22**2*a23**5*b33 + a11**5*a23**7*b33 - a11**4*a22**3*a23**5*
b33 - a11**4*a22*a23**7*b33) + u3*v3*(a11**6*a23**6*b33 - a11**5*a22*a23**6*b33)
+ u3*(a11**6*a23**6*n3 - a11**5*a22*a23**6*n3) + v2**2*(a11**8*a23**4*c33 + 1/
16*a11**7*a22**4*b33**2 + 1/8*a11**7*a22**2*a23**2*b33**2 - 3*a11**7*a22*a23**4*
c33 + 1/16*a11**7*a23**4*b33**2 - 7/16*a11**6*a22**5*b33**2 - 3/8*a11**6*a22**3*
a23**2*b33**2 + 3*a11**6*a22**2*a23**4*c33 - 3/16*a11**6*a22*a23**4*b33**2 + 21/
16*a11**5*a22**6*b33**2 + 1/16*a11**5*a22**4*a23**2*b33**2 - a11**5*a22**3*a23**
4*c33 + 3/16*a11**5*a22**2*a23**4*b33**2 - 1/16*a11**5*a23**6*b33**2 - 35/16*a11
**4*a22**7*b33**2 + 15/16*a11**4*a22**5*a23**2*b33**2 + 7/16*a11**4*a22**3*a23**
4*b33**2 + 1/16*a11**4*a22*a23**6*b33**2 + 35/16*a11**3*a22**8*b33**2 - 5/4*a11
**3*a22**6*a23**2*b33**2 - 25/16*a11**3*a22**4*a23**4*b33**2 - 1/8*a11**3*a22**2
*a23**6*b33**2 - 21/16*a11**2*a22**9*b33**2 + 1/2*a11**2*a22**7*a23**2*b33**2 +
27/16*a11**2*a22**5*a23**4*b33**2 + 3/8*a11**2*a22**3*a23**6*b33**2 + 7/16*a11*
a22**10*b33**2 + 1/16*a11*a22**8*a23**2*b33**2 - 11/16*a11*a22**6*a23**4*b33**2
- 5/16*a11*a22**4*a23**6*b33**2 - 1/16*a22**11*b33**2 - 1/16*a22**9*a23**2*b33**
2 + 1/16*a22**7*a23**4*b33**2 + 1/16*a22**5*a23**6*b33**2) + v2*v3*( - 2*a11**7*
a23**5*c33 - 1/8*a11**6*a22**4*a23*b33**2 - 1/4*a11**6*a22**2*a23**3*b33**2 + 4*
a11**6*a22*a23**5*c33 - 1/8*a11**6*a23**5*b33**2 + 3/4*a11**5*a22**5*a23*b33**2
+ a11**5*a22**3*a23**3*b33**2 - 2*a11**5*a22**2*a23**5*c33 + 1/4*a11**5*a22*a23
**5*b33**2 - 15/8*a11**4*a22**6*a23*b33**2 - 7/4*a11**4*a22**4*a23**3*b33**2 - 3
/8*a11**4*a22**2*a23**5*b33**2 + 5/2*a11**3*a22**7*a23*b33**2 + 2*a11**3*a22**5*
a23**3*b33**2 + 1/2*a11**3*a22**3*a23**5*b33**2 - 15/8*a11**2*a22**8*a23*b33**2
- 7/4*a11**2*a22**6*a23**3*b33**2 - 3/8*a11**2*a22**4*a23**5*b33**2 + 3/4*a11*
a22**9*a23*b33**2 + a11*a22**7*a23**3*b33**2 + 1/4*a11*a22**5*a23**5*b33**2 - 1/
8*a22**10*a23*b33**2 - 1/4*a22**8*a23**3*b33**2 - 1/8*a22**6*a23**5*b33**2) + v2
*( - 1/2*a11**7*a22*a23**3*b33*n3 - a11**7*a23**5*m3 + 2*a11**6*a22**2*a23**3*
b33*n3 + 2*a11**6*a22*a23**5*m3 - 3*a11**5*a22**3*a23**3*b33*n3 - a11**5*a22**2*
a23**5*m3 - 1/2*a11**5*a22*a23**5*b33*n3 + 2*a11**4*a22**4*a23**3*b33*n3 + a11**
4*a22**2*a23**5*b33*n3 - 1/2*a11**3*a22**5*a23**3*b33*n3 - 1/2*a11**3*a22**3*a23
**5*b33*n3) + v3**2*(a11**6*a23**6*c33 - a11**5*a22*a23**6*c33) + v3*(a11**6*a23
**6*m3 - a11**5*a22*a23**6*m3))/(a11**6*a23**6 - a11**5*a22*a23**6)$
FI=u2*( - a11**4*a23**2 + a11**3*a22*a23**2) + u3*a11**3*a23**3 + v2*( - 1/4*a11
**3*a22**2*b33 - 1/4*a11**3*a23**2*b33 + 3/4*a11**2*a22**3*b33 + 1/4*a11**2*a22*
a23**2*b33 - 3/4*a11*a22**4*b33 - 1/4*a11*a22**2*a23**2*b33 + 1/4*a22**5*b33 + 1
/4*a22**3*a23**2*b33) + v3*( - 1/2*a11**3*a22*a23*b33 + 5/4*a11**2*a22**2*a23*
b33 + 1/4*a11**2*a23**3*b33 - a11*a22**3*a23*b33 - 1/2*a11*a22*a23**3*b33 + 1/4*
a22**4*a23*b33 + 1/4*a22**2*a23**3*b33)$