Solution 3 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2
0=4*a23 + a33 + b12
Expressions
The solution is given through the following expressions:
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
2 1 2 1 2
- a23 *r313 - ---*a33 *r313 - ---*b12 *r313
4 4
r36=----------------------------------------------
2
a23
r37=0
r38=0
2 1 2 1 2
- a23 *r313 - ---*a33 *r313 - ---*b12 *r313
4 4
r39=----------------------------------------------
2
a23
1
---*b12*r313
2
r310=--------------
a23
r311=0
r312=0
r314=0
r315=0
1
---*b12*r313
2
r316=--------------
a23
r317=0
r318=r313
r319=0
r320=0
1
---*b12*r313
2
r321=--------------
a23
r322=0
1
- ---*a33*r313
2
r323=-----------------
a23
2 1 2 1 2
a23 *r313 + ---*a33 *r313 + ---*b12 *r313
4 4
r324=-------------------------------------------
2
a23
r325=0
r326=0
1
---*b12*r313
2
r327=--------------
a23
1
- ---*a33*r313
2
r328=-----------------
a23
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
r335= - r313
1
---*a33*r313
2
r336=--------------
a23
r337=0
1
---*b12*r313
2
r338=--------------
a23
r339=0
2 1 2 1 2
a23 *r313 + ---*a33 *r313 + ---*b12 *r313
4 4
r340=-------------------------------------------
2
a23
r341= - r313
1
---*a33*r313
2
r342=--------------
a23
1
---*b12*r313
2
r343=--------------
a23
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
c33=a33
b33=a33
b22= - a33
c23=a23
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:
a23, r313, b12, a33
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.
{c23,r313}
Relevance for the application:
Modulo the following equation:
2 2 2
0=4*a23 + a33 + b12
the system of equations related to the Hamiltonian HAM:
HAM= - u1*v1*a33 + u1*v2*b12 + 2*u2*u3*a23 - u2*v1*b12 - u2*v2*a33 + 2*u2*v3*a23
2 2
+ u3 *a33 + 2*u3*v2*a23 + u3*v3*a33 + 2*v2*v3*a23 + v3 *a33
has apart from the Hamiltonian and Casimirs only the following first integral:
1 1 2
FI=---*u1*u3*v1*a23*b12 + ---*u1*u3*v2*a23*a33 - u1*u3*v3*a23
2 2
2 2 1 2 1 2 1
+ u1*v1 *(a23 + ---*a33 + ---*b12 ) + ---*u1*v1*v3*a23*b12
4 4 2
1 2 2 1
+ ---*u1*v2*v3*a23*a33 - u1*v3 *a23 - ---*u2*u3*v1*a23*a33
2 2
1 2 1 2 1 2
+ ---*u2*u3*v2*a23*b12 + u2*v1*v2*(a23 + ---*a33 + ---*b12 )
2 4 4
1 1 2 2
- ---*u2*v1*v3*a23*a33 + ---*u2*v2*v3*a23*b12 + u3 *v1*a23
2 2
1 2 2 1 2
+ ---*u3 *v3*a23*b12 + u3*v1*v3*a23 + ---*u3*v3 *a23*b12
2 2
3 2 1 2 1 2
+ v1 *( - a23 - ---*a33 - ---*b12 )
4 4
2 2 1 2 1 2
+ v1*v2 *( - a23 - ---*a33 - ---*b12 )
4 4
1
= a product of the elements of: { - ---,
4
2
- 2*u1*u3*v1*a23*b12 - 2*u1*u3*v2*a23*a33 + 4*u1*u3*v3*a23
2 2 2 2
+ u1*v1 *( - 4*a23 - a33 - b12 ) - 2*u1*v1*v3*a23*b12 - 2*u1*v2*v3*a23*a33
2 2
+ 4*u1*v3 *a23 + 2*u2*u3*v1*a23*a33 - 2*u2*u3*v2*a23*b12
2 2 2
+ u2*v1*v2*( - 4*a23 - a33 - b12 ) + 2*u2*v1*v3*a23*a33
2 2 2 2
- 2*u2*v2*v3*a23*b12 - 4*u3 *v1*a23 - 2*u3 *v3*a23*b12 - 4*u3*v1*v3*a23
2 3 2 2 2
- 2*u3*v3 *a23*b12 + v1 *(4*a23 + a33 + b12 )
2 2 2 2
+ v1*v2 *(4*a23 + a33 + b12 )}
{HAM,FI} = 0
And again in machine readable form:
HAM= - u1*v1*a33 + u1*v2*b12 + 2*u2*u3*a23 - u2*v1*b12 - u2*v2*a33 + 2*u2*v3*a23
+ u3**2*a33 + 2*u3*v2*a23 + u3*v3*a33 + 2*v2*v3*a23 + v3**2*a33$
FI=1/2*u1*u3*v1*a23*b12 + 1/2*u1*u3*v2*a23*a33 - u1*u3*v3*a23**2 + u1*v1**2*(a23
**2 + 1/4*a33**2 + 1/4*b12**2) + 1/2*u1*v1*v3*a23*b12 + 1/2*u1*v2*v3*a23*a33 -
u1*v3**2*a23**2 - 1/2*u2*u3*v1*a23*a33 + 1/2*u2*u3*v2*a23*b12 + u2*v1*v2*(a23**2
+ 1/4*a33**2 + 1/4*b12**2) - 1/2*u2*v1*v3*a23*a33 + 1/2*u2*v2*v3*a23*b12 + u3**
2*v1*a23**2 + 1/2*u3**2*v3*a23*b12 + u3*v1*v3*a23**2 + 1/2*u3*v3**2*a23*b12 + v1
**3*( - a23**2 - 1/4*a33**2 - 1/4*b12**2) + v1*v2**2*( - a23**2 - 1/4*a33**2 - 1
/4*b12**2)$