Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
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Equations
The following unsolved equations remain:
5 4 4 3 3 2 2 2
0= - 64*a23 *c23 + 32*a23 *a33*b22*c23 - 4*a23 *a33 *b22 *c23
3 4 3 2 4 3 2 4 3 6
- 32*a23 *a33*b33*c23 - 16*a23 *b22 *c23 + 16*a23 *b33 *c23 + 64*a23 *c23
2 2 3 2 3 3 2 2 3
+ 16*a23 *a33 *b22*b33*c23 + 8*a23 *a33*b22 *c23 - 8*a23 *a33*b22*b33 *c23
2 5 3 2 2 2 4 2
- 64*a23 *a33*b22*c23 - 2*a23*a33 *b22 *b33*c23 - a23*a33 *b22 *c23
2 2 2 2 2 2 4 3 3 3
+ a23*a33 *b22 *b33 *c23 + 20*a23*a33 *b22 *c23 - 2*a33 *b22 *c23
Expressions
The solution is given through the following expressions:
2 2 1
r00=( - a23 *c23 *n3*r10 + ---*a23*a33*b22*c23*n3*r10
4
1 1 2 3
+ ---*a23*b22*b33*c23*n3*r10 - ----*a33*b22 *b33*n3*r10)/(a23 *b22*c23
4 16
1 2 2 2 2 2 2 1 2
- ---*a23 *a33*b22 + a23 *a33*c23 - a23 *b33*c23 - ---*a23*a33 *b22*c23
4 4
1 1 2 2
+ ---*a23*a33*b22*b33*c23 - ----*a33 *b22 *b33)
2 16
1 2 2 2 3 1 3
r11=(---*a23 *b22 *c23*r10 - 2*a23 *c23 *r10 - ---*a23*a33*b22 *r10
2 8
3 2 1 2 2 3
+ ---*a23*a33*b22*c23 *r10 - ---*a33 *b22 *c23*r10)/(a23 *b22*c23
2 4
1 2 2 2 2 2 2 1 2
- ---*a23 *a33*b22 + a23 *a33*c23 - a23 *b33*c23 - ---*a23*a33 *b22*c23
4 4
1 1 2 2
+ ---*a23*a33*b22*b33*c23 - ----*a33 *b22 *b33)
2 16
r12=0
2 2 1 2
r13=( - a23 *a33*c23 *r10 + ---*a23*a33 *b22*c23*r10
4
1 1 2 2 3
+ ---*a23*a33*b22*b33*c23*r10 - ----*a33 *b22 *b33*r10)/(a23 *b22*c23
4 16
1 2 2 2 2 2 2 1 2
- ---*a23 *a33*b22 + a23 *a33*c23 - a23 *b33*c23 - ---*a23*a33 *b22*c23
4 4
1 1 2 2
+ ---*a23*a33*b22*b33*c23 - ----*a33 *b22 *b33)
2 16
r15=0
2
- 4*a23*c23 *n3 + b22*b33*c23*n3
m3=---------------------------------------
2 2
a23*b22 - 4*a23*c23 + 2*a33*b22*c23
b12=0
2 2 2
4*a23 *b22*c23 + 4*a23*a33*c23 - 4*a23*b33*c23 + a33*b22*b33*c23
c33=--------------------------------------------------------------------
2 2
a23*b22 - 4*a23*c23 + 2*a33*b22*c23
3 2 1 2 1 2
r14=( - 2*a23 *c23 *r10 + ---*a23 *a33*b22*c23*r10 + ---*a23 *b22*b33*c23*r10
2 2
1 2 3 1 2 2
- ---*a23*a33*b22 *b33*r10)/(a23 *b22*c23 - ---*a23 *a33*b22
8 4
2 2 2 2 1 2
+ a23 *a33*c23 - a23 *b33*c23 - ---*a23*a33 *b22*c23
4
1 1 2 2
+ ---*a23*a33*b22*b33*c23 - ----*a33 *b22 *b33)
2 16
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:
r10, b33, n3, a33, c23, b22, 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.
{4*a23*c23 - a33*b22,c23,b22,a23}
Relevance for the application:
Modulo the following equation:
5 4 4 3 3 2 2 2
0= - 64*a23 *c23 + 32*a23 *a33*b22*c23 - 4*a23 *a33 *b22 *c23
3 4 3 2 4 3 2 4 3 6
- 32*a23 *a33*b33*c23 - 16*a23 *b22 *c23 + 16*a23 *b33 *c23 + 64*a23 *c23
2 2 3 2 3 3 2 2 3
+ 16*a23 *a33 *b22*b33*c23 + 8*a23 *a33*b22 *c23 - 8*a23 *a33*b22*b33 *c23
2 5 3 2 2 2 4 2
- 64*a23 *a33*b22*c23 - 2*a23*a33 *b22 *b33*c23 - a23*a33 *b22 *c23
2 2 2 2 2 2 4 3 3 3
+ a23*a33 *b22 *b33 *c23 + 20*a23*a33 *b22 *c23 - 2*a33 *b22 *c23
the system of equations related to the Hamiltonian HAM:
3 2 2
HAM=(u1*v1*(a23*b22 - 4*a23*b22*c23 + 2*a33*b22 *c23)
2 2 2 2
+ u2*u3*(2*a23 *b22 - 8*a23 *c23 + 4*a23*a33*b22*c23)
3 2 2
+ u2*v2*(a23*b22 - 4*a23*b22*c23 + 2*a33*b22 *c23)
2 2 2 2
+ u2*v3*(2*a23 *b22 - 8*a23 *c23 + 4*a23*a33*b22*c23)
2 2 2 2
+ u3 *(a23*a33*b22 - 4*a23*a33*c23 + 2*a33 *b22*c23)
2 3 2
+ u3*v2*(2*a23*b22 *c23 - 8*a23*c23 + 4*a33*b22*c23 )
2 2
+ u3*v3*(a23*b22 *b33 - 4*a23*b33*c23 + 2*a33*b22*b33*c23)
2 2
+ u3*(a23*b22 *n3 - 4*a23*c23 *n3 + 2*a33*b22*c23*n3)
2 3 2
+ v2*v3*(2*a23*b22 *c23 - 8*a23*c23 + 4*a33*b22*c23 )
2 2 2 2
+ v3 *(4*a23 *b22*c23 + 4*a23*a33*c23 - 4*a23*b33*c23 + a33*b22*b33*c23)
2 2 2
+ v3*( - 4*a23*c23 *n3 + b22*b33*c23*n3))/(a23*b22 - 4*a23*c23
+ 2*a33*b22*c23)
has apart from the Hamiltonian and Casimirs only the following first integral:
3 2 1 2 1 2
FI=u2*u3*( - 2*a23 *c23 + ---*a23 *a33*b22*c23 + ---*a23 *b22*b33*c23
2 2
1 2 3 2
- ---*a23*a33*b22 *b33) + u2*v3*( - 2*a23 *c23
8
1 2 1 2 1 2
+ ---*a23 *a33*b22*c23 + ---*a23 *b22*b33*c23 - ---*a23*a33*b22 *b33) +
2 2 8
2 2 2 1 2 1
u3 *( - a23 *a33*c23 + ---*a23*a33 *b22*c23 + ---*a23*a33*b22*b33*c23
4 4
1 2 2 1 2 2 2 3
- ----*a33 *b22 *b33) + u3*v2*(---*a23 *b22 *c23 - 2*a23 *c23
16 2
1 3 3 2 1 2 2
- ---*a23*a33*b22 + ---*a23*a33*b22*c23 - ---*a33 *b22 *c23) + u3*v3*(
8 2 4
3 1 2 2 2 2 3
a23 *b22*c23 - ---*a23 *a33*b22 - a23 *b33*c23 + ---*a23*a33*b22*b33*c23
4 4
1 2 2 2 2 1
- ---*a33 *b22 *b33) + u3*( - a23 *c23 *n3 + ---*a23*a33*b22*c23*n3
8 4
1 1 2
+ ---*a23*b22*b33*c23*n3 - ----*a33*b22 *b33*n3) + v2*v3*(
4 16
1 2 2 2 3 1 3 3 2
---*a23 *b22 *c23 - 2*a23 *c23 - ---*a23*a33*b22 + ---*a23*a33*b22*c23
2 8 2
1 2 2 2 3 1 2 2
- ---*a33 *b22 *c23) + v3 *(a23 *b22*c23 - ---*a23 *a33*b22
4 4
2 2 2 2 1 2
+ a23 *a33*c23 - a23 *b33*c23 - ---*a23*a33 *b22*c23
4
1 1 2 2 2 2
+ ---*a23*a33*b22*b33*c23 - ----*a33 *b22 *b33) + v3*( - a23 *c23 *n3
2 16
1 1 1 2
+ ---*a23*a33*b22*c23*n3 + ---*a23*b22*b33*c23*n3 - ----*a33*b22 *b33*n3)
4 4 16
1
= a product of the elements of: {----,
16
u3 + v3,
- 4*a23*c23 + a33*b22,
2
u2*(8*a23 *c23 - 2*a23*b22*b33) + u3*(4*a23*a33*c23 - a33*b22*b33)
2 2
+ v2*( - 2*a23*b22 + 8*a23*c23 - 4*a33*b22*c23)
2
+ v3*( - 4*a23 *b22 - 4*a23*a33*c23 + 4*a23*b33*c23 - a33*b22*b33)
+ 4*a23*c23*n3 - b22*b33*n3}
4 1 3 2 2
{HAM,FI} = u1*u3*v3*(2*a23 *b22*c23 - ---*a23 *a33*b22 + a23 *a33*b22*b33*c23
2
1 2 3 1 2 2 2 3
+ ---*a23 *b22 *c23 - ---*a23 *b22*b33 *c23 - 2*a23 *b22*c23
2 2
1 2 2 1 4
- ---*a23*a33 *b22 *b33 - ---*a23*a33*b22
4 8
1 2 2 3 2 2
+ ---*a23*a33*b22 *b33 + ---*a23*a33*b22 *c23
8 2
1 2 3 2 4 1 3 2
- ---*a33 *b22 *c23) + u1*v3 *(2*a23 *b22*c23 - ---*a23 *a33*b22
4 2
2 1 2 3
+ a23 *a33*b22*b33*c23 + ---*a23 *b22 *c23
2
1 2 2 2 3 1 2 2
- ---*a23 *b22*b33 *c23 - 2*a23 *b22*c23 - ---*a23*a33 *b22 *b33
2 4
1 4 1 2 2
- ---*a23*a33*b22 + ---*a23*a33*b22 *b33
8 8
3 2 2 1 2 3
+ ---*a23*a33*b22 *c23 - ---*a33 *b22 *c23)
2 4
And again in machine readable form:
HAM=(u1*v1*(a23*b22**3 - 4*a23*b22*c23**2 + 2*a33*b22**2*c23) + u2*u3*(2*a23**2*
b22**2 - 8*a23**2*c23**2 + 4*a23*a33*b22*c23) + u2*v2*(a23*b22**3 - 4*a23*b22*
c23**2 + 2*a33*b22**2*c23) + u2*v3*(2*a23**2*b22**2 - 8*a23**2*c23**2 + 4*a23*
a33*b22*c23) + u3**2*(a23*a33*b22**2 - 4*a23*a33*c23**2 + 2*a33**2*b22*c23) + u3
*v2*(2*a23*b22**2*c23 - 8*a23*c23**3 + 4*a33*b22*c23**2) + u3*v3*(a23*b22**2*b33
- 4*a23*b33*c23**2 + 2*a33*b22*b33*c23) + u3*(a23*b22**2*n3 - 4*a23*c23**2*n3 +
2*a33*b22*c23*n3) + v2*v3*(2*a23*b22**2*c23 - 8*a23*c23**3 + 4*a33*b22*c23**2)
+ v3**2*(4*a23**2*b22*c23 + 4*a23*a33*c23**2 - 4*a23*b33*c23**2 + a33*b22*b33*
c23) + v3*( - 4*a23*c23**2*n3 + b22*b33*c23*n3))/(a23*b22**2 - 4*a23*c23**2 + 2*
a33*b22*c23)$
FI=u2*u3*( - 2*a23**3*c23**2 + 1/2*a23**2*a33*b22*c23 + 1/2*a23**2*b22*b33*c23 -
1/8*a23*a33*b22**2*b33) + u2*v3*( - 2*a23**3*c23**2 + 1/2*a23**2*a33*b22*c23 +
1/2*a23**2*b22*b33*c23 - 1/8*a23*a33*b22**2*b33) + u3**2*( - a23**2*a33*c23**2 +
1/4*a23*a33**2*b22*c23 + 1/4*a23*a33*b22*b33*c23 - 1/16*a33**2*b22**2*b33) + u3
*v2*(1/2*a23**2*b22**2*c23 - 2*a23**2*c23**3 - 1/8*a23*a33*b22**3 + 3/2*a23*a33*
b22*c23**2 - 1/4*a33**2*b22**2*c23) + u3*v3*(a23**3*b22*c23 - 1/4*a23**2*a33*b22
**2 - a23**2*b33*c23**2 + 3/4*a23*a33*b22*b33*c23 - 1/8*a33**2*b22**2*b33) + u3*
( - a23**2*c23**2*n3 + 1/4*a23*a33*b22*c23*n3 + 1/4*a23*b22*b33*c23*n3 - 1/16*
a33*b22**2*b33*n3) + v2*v3*(1/2*a23**2*b22**2*c23 - 2*a23**2*c23**3 - 1/8*a23*
a33*b22**3 + 3/2*a23*a33*b22*c23**2 - 1/4*a33**2*b22**2*c23) + v3**2*(a23**3*b22
*c23 - 1/4*a23**2*a33*b22**2 + a23**2*a33*c23**2 - a23**2*b33*c23**2 - 1/4*a23*
a33**2*b22*c23 + 1/2*a23*a33*b22*b33*c23 - 1/16*a33**2*b22**2*b33) + v3*( - a23
**2*c23**2*n3 + 1/4*a23*a33*b22*c23*n3 + 1/4*a23*b22*b33*c23*n3 - 1/16*a33*b22**
2*b33*n3)$