Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=a11 + a23
Expressions
The solution is given through the following expressions:
1 2 2 1 3 2
---*a11 *a23*b23 *r213 + ---*a23 *b23 *r213
4 8
r20=---------------------------------------------
5 3 2
a11 - a11 *a23
1 2 2 2 1 4 2
---*a11 *a23 *b23 *r213 + ---*a23 *b23 *r213
2 4
r21=----------------------------------------------
6 4 2 2 4
a11 - 2*a11 *a23 + a11 *a23
r23=0
r24=0
3 6 2 3 4 2 2 1 2 4 2
r25=(---*a11 *b23 *r213 + ---*a11 *a23 *b23 *r213 + ---*a11 *a23 *b23 *r213
8 4 8
1 6 2 7 5 3 3 5
- ---*a23 *b23 *r213)/(a11 *a23 - 2*a11 *a23 + a11 *a23 )
8
1 4 1 4
- ---*a11 *b23*r213 - ---*a23 *b23*r213
2 2
r26=------------------------------------------
3 2 4
a11 *a23 - a11*a23
1
---*a23*b23*r213
2
r27=------------------
2 2
a11 - a23
r28=0
1
- ---*a23*r213
2
r29=-----------------
a11
1 4 1 4
---*a11 *b23*r213 + ---*a23 *b23*r213
4 4
r210=---------------------------------------
2 3 5
a11 *a23 - a23
r212=0
1
- ---*a11*r213
2
r214=-----------------
a23
r215=0
r216=0
1 4 1 2 2 1 4
- ---*a11 *b23*r213 - ---*a11 *a23 *b23*r213 - ---*a23 *b23*r213
2 2 2
r217=-------------------------------------------------------------------
3 2 4
a11 *a23 - a11*a23
r218=0
r219=0
6 1 6 2 4 3 3 4 2 2
c33=( - a11 *a23*c23 - ---*a11 *b23 + 2*a11 *a23 *c23 - ---*a11 *a23 *b23
2 2
2 5 1 2 4 2 1 6 2 7 5 2
- a11 *a23 *c23 - ---*a11 *a23 *b23 + ---*a23 *b23 )/(a11 - 2*a11 *a23
2 4
3 4
+ a11 *a23 )
8 1 8 2 6 3 1 6 2 2
c22=( - a11 *a23*c23 + ---*a11 *b23 + 2*a11 *a23 *c23 - ---*a11 *a23 *b23
4 4
4 5 7 4 4 2 3 2 6 2 1 8 2
- a11 *a23 *c23 - ---*a11 *a23 *b23 - ---*a11 *a23 *b23 + ---*a23 *b23 )
4 4 4
7 2 5 4 3 6
/(a11 *a23 - 2*a11 *a23 + a11 *a23 )
c13=0
c12=0
- 2*a11*a23*b23
b33=------------------
2 2
a11 - a23
b31=0
b21=0
2 3
- 2*a11 *a23*b23 - a23 *b23
b11=------------------------------
3 2
a11 - a11*a23
2 2
a11 - a23
a33=-------------
a11
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:
r213, c23, b23, a11, 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.
2
{16*a23*c23 + 3*b23 ,b23,r213,a11,a23,a33,a11 - a23,a11 + a23}
Relevance for the application:
Modulo the following equation:
2 2
0=a11 + a23
the system of equations related to the Hamiltonian HAM:
2 8 2 6 4 4 6
HAM=(u1 *(a11 *a23 - 2*a11 *a23 + a11 *a23 )
6 3 4 5 2 7
+ u1*v1*( - 2*a11 *a23 *b23 + a11 *a23 *b23 + a11 *a23 *b23)
7 3 5 5 3 7
+ u2*u3*(2*a11 *a23 - 4*a11 *a23 + 2*a11 *a23 )
7 2 5 4 3 6
+ u2*v3*(a11 *a23 *b23 - 2*a11 *a23 *b23 + a11 *a23 *b23)
2 8 2 6 4 4 6 2 8
+ u3 *(a11 *a23 - 3*a11 *a23 + 3*a11 *a23 - a11 *a23 )
6 3 4 5 2 8
+ u3*v3*( - 2*a11 *a23 *b23 + 2*a11 *a23 *b23) + v2 *( - a11 *a23*c23
1 8 2 6 3 1 6 2 2 4 5
+ ---*a11 *b23 + 2*a11 *a23 *c23 - ---*a11 *a23 *b23 - a11 *a23 *c23
4 4
7 4 4 2 3 2 6 2 1 8 2
- ---*a11 *a23 *b23 - ---*a11 *a23 *b23 + ---*a23 *b23 )
4 4 4
7 2 5 4 3 6 2
+ v2*v3*(2*a11 *a23 *c23 - 4*a11 *a23 *c23 + 2*a11 *a23 *c23) + v3 *(
6 3 1 6 2 2 4 5
- a11 *a23 *c23 - ---*a11 *a23 *b23 + 2*a11 *a23 *c23
2
3 4 4 2 2 7 1 2 6 2
- ---*a11 *a23 *b23 - a11 *a23 *c23 - ---*a11 *a23 *b23
2 2
1 8 2 7 2 5 4 3 6
+ ---*a23 *b23 ))/(a11 *a23 - 2*a11 *a23 + a11 *a23 )
4
has apart from the Hamiltonian and Casimirs only the following first integral:
1 8 1 2 7
FI=u1*v1*( - ---*a11 *a23*b23 + ---*a11 *a23 *b23)
2 2
2 1 8 2 6 4 1 4 6
+ u2 *( - ---*a11 *a23 + a11 *a23 - ---*a11 *a23 )
2 2
7 3 5 5 3 7
+ u2*u3*(a11 *a23 - 2*a11 *a23 + a11 *a23 ) + u2*v3
1 9 1 7 2 1 5 4 1 3 6
*(---*a11 *b23 - ---*a11 *a23 *b23 + ---*a11 *a23 *b23 - ---*a11 *a23 *b23)
4 4 4 4
2 1 6 4 4 6 1 2 8
+ u3 *( - ---*a11 *a23 + a11 *a23 - ---*a11 *a23 )
2 2
1 5 4 1 3 6 1 8
+ u3*v2*(---*a11 *a23 *b23 - ---*a11 *a23 *b23) + u3*v3*( - ---*a11 *a23*b23
2 2 2
1 6 3 1 4 5 1 2 7 2
+ ---*a11 *a23 *b23 - ---*a11 *a23 *b23 + ---*a11 *a23 *b23) + v1 *(
2 2 2
3 6 2 2 3 4 4 2 1 2 6 2
---*a11 *a23 *b23 + ---*a11 *a23 *b23 + ---*a11 *a23 *b23
8 4 8
1 8 2 1 3 5 2 1 7 2
- ---*a23 *b23 ) + v2*v3*(---*a11 *a23 *b23 + ---*a11*a23 *b23 )
8 2 4
2 1 4 4 2 1 2 6 2 1 8 2
+ v3 *(---*a11 *a23 *b23 - ---*a11 *a23 *b23 - ---*a23 *b23 )
4 8 8
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*(a11**8*a23**2 - 2*a11**6*a23**4 + a11**4*a23**6) + u1*v1*( - 2*a11**
6*a23**3*b23 + a11**4*a23**5*b23 + a11**2*a23**7*b23) + u2*u3*(2*a11**7*a23**3 -
4*a11**5*a23**5 + 2*a11**3*a23**7) + u2*v3*(a11**7*a23**2*b23 - 2*a11**5*a23**4
*b23 + a11**3*a23**6*b23) + u3**2*(a11**8*a23**2 - 3*a11**6*a23**4 + 3*a11**4*
a23**6 - a11**2*a23**8) + u3*v3*( - 2*a11**6*a23**3*b23 + 2*a11**4*a23**5*b23) +
v2**2*( - a11**8*a23*c23 + 1/4*a11**8*b23**2 + 2*a11**6*a23**3*c23 - 1/4*a11**6
*a23**2*b23**2 - a11**4*a23**5*c23 - 7/4*a11**4*a23**4*b23**2 - 3/4*a11**2*a23**
6*b23**2 + 1/4*a23**8*b23**2) + v2*v3*(2*a11**7*a23**2*c23 - 4*a11**5*a23**4*c23
+ 2*a11**3*a23**6*c23) + v3**2*( - a11**6*a23**3*c23 - 1/2*a11**6*a23**2*b23**2
+ 2*a11**4*a23**5*c23 - 3/2*a11**4*a23**4*b23**2 - a11**2*a23**7*c23 - 1/2*a11
**2*a23**6*b23**2 + 1/4*a23**8*b23**2))/(a11**7*a23**2 - 2*a11**5*a23**4 + a11**
3*a23**6)$
FI=u1*v1*( - 1/2*a11**8*a23*b23 + 1/2*a11**2*a23**7*b23) + u2**2*( - 1/2*a11**8*
a23**2 + a11**6*a23**4 - 1/2*a11**4*a23**6) + u2*u3*(a11**7*a23**3 - 2*a11**5*
a23**5 + a11**3*a23**7) + u2*v3*(1/4*a11**9*b23 - 1/4*a11**7*a23**2*b23 + 1/4*
a11**5*a23**4*b23 - 1/4*a11**3*a23**6*b23) + u3**2*( - 1/2*a11**6*a23**4 + a11**
4*a23**6 - 1/2*a11**2*a23**8) + u3*v2*(1/2*a11**5*a23**4*b23 - 1/2*a11**3*a23**6
*b23) + u3*v3*( - 1/2*a11**8*a23*b23 + 1/2*a11**6*a23**3*b23 - 1/2*a11**4*a23**5
*b23 + 1/2*a11**2*a23**7*b23) + v1**2*(3/8*a11**6*a23**2*b23**2 + 3/4*a11**4*a23
**4*b23**2 + 1/8*a11**2*a23**6*b23**2 - 1/8*a23**8*b23**2) + v2*v3*(1/2*a11**3*
a23**5*b23**2 + 1/4*a11*a23**7*b23**2) + v3**2*(1/4*a11**4*a23**4*b23**2 - 1/8*
a11**2*a23**6*b23**2 - 1/8*a23**8*b23**2)$