Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2 2
0=a13 + a23 + a33 + b12 *kap
Expressions
The solution is given through the following expressions:
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r312= - r338
r314=0
r315= - r338
- 2*a13*r338
r317=---------------
b12
2*a23*r338
r318=------------
b12
r319=kap*r338
r320=0
r321=r338
r322=0
r323=0
r324=0
r325=0
r327=0
- 2*a33*r338
r328=---------------
b12
r329=0
r330=0
- 2*a13*r338
r331=---------------
b12
r332=0
2 2 2 2
- 2*a13 *r338 - 2*a23 *r338 - 2*a33 *r338 - b12 *kap*r338
r333=------------------------------------------------------------
2
b12
r334=0
r335=0
r336=0
r337=0
r339=0
r340=0
2*a33*r338
r342=------------
b12
r343=0
r344=0
r345=0
2*a23*r338
r346=------------
b12
- 2*a13*r338
r347=---------------
b12
r348=0
r349=0
r350=0
r351=0
2*a23*r338
r352=------------
b12
2 2 2 2
- 2*a13 *r338 - 2*a23 *r338 - 2*a33 *r338 - b12 *kap*r338
r353=------------------------------------------------------------
2
b12
r354=0
r355=0
a22= - a33
a11= - a33
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:
r338, a33, a23, a13, b12
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.
{a23,b12,r338}
Relevance for the application:
Modulo the following equation:
2 2 2 2
0=a13 + a23 + a33 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2
HAM= - u1 *a33 + 2*u1*u3*a13 + u1*v2*b12 - u2 *a33 + 2*u2*u3*a23 - u2*v1*b12
2
+ u3 *a33
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 2 2
FI=u1 *u3*( - 2*a13 - 2*a23 - 2*a33 - b12 *kap) + 2*u1 *v1*a23*b12
- 2*u1*u2*v1*a13*b12 + 2*u1*u2*v2*a23*b12 + 2*u1*u3*v2*a33*b12
2 2 2 2 2 2
+ u1*v1*v3*b12 + u2 *u3*( - 2*a13 - 2*a23 - 2*a33 - b12 *kap)
2 2 3 2
- 2*u2 *v2*a13*b12 - 2*u2*u3*v1*a33*b12 + u2*v2*v3*b12 + u3 *b12 *kap
2 2 2 2 2 2
+ 2*u3 *v1*a23*b12 - 2*u3 *v2*a13*b12 - u3*v1 *b12 - u3*v2 *b12
3 2 3 2
{HAM,FI} = u1 *v1*(4*a13 *b12 + b12 *kap) + 8*u1 *u2*v1*a13*a23*b12
2 2 3 2
+ u1 *u2*v2*(4*a13 *b12 + b12 *kap) + 8*u1 *u3*v1*a13*a33*b12
2 2 3 2 2
+ u1 *u3*v3*(4*a13 *b12 + b12 *kap) + 2*u1 *v1*v2*a33*b12
2 2 2 2 3
- 2*u1 *v1*v3*a23*b12 + u1*u2 *v1*(4*a23 *b12 + b12 *kap)
2
+ 8*u1*u2 *v2*a13*a23*b12 + 8*u1*u2*u3*v1*a23*a33*b12
+ 8*u1*u2*u3*v2*a13*a33*b12 + 8*u1*u2*u3*v3*a13*a23*b12
2 2 2
- 2*u1*u2*v1 *a33*b12 + 2*u1*u2*v1*v3*a13*b12
2 2 2
+ 2*u1*u2*v2 *a33*b12 - 2*u1*u2*v2*v3*a23*b12
2 2 2 2
+ u1*u3 *v1*( - 4*a13 *b12 - 4*a23 *b12) + 8*u1*u3 *v3*a13*a33*b12
2 2 2
+ 4*u1*u3*v1 *a23*b12 - 4*u1*u3*v1*v2*a13*b12
2 2 2 3 3
+ 2*u1*u3*v2*v3*a33*b12 - 2*u1*u3*v3 *a23*b12 - u1*v1 *b12
2 3 3 2 3
- u1*v1*v2 *b12 + u2 *v2*(4*a23 *b12 + b12 *kap)
2 2 2 3
+ 8*u2 *u3*v2*a23*a33*b12 + u2 *u3*v3*(4*a23 *b12 + b12 *kap)
2 2 2 2
- 2*u2 *v1*v2*a33*b12 + 2*u2 *v2*v3*a13*b12
2 2 2 2
+ u2*u3 *v2*( - 4*a13 *b12 - 4*a23 *b12) + 8*u2*u3 *v3*a23*a33*b12
2 2
+ 4*u2*u3*v1*v2*a23*b12 - 2*u2*u3*v1*v3*a33*b12
2 2 2 2 2 3
- 4*u2*u3*v2 *a13*b12 + 2*u2*u3*v3 *a13*b12 - u2*v1 *v2*b12
3 3 3 2 2
- u2*v2 *b12 + u3 *v3*( - 4*a13 *b12 - 4*a23 *b12)
2 2 2 2 2 3
+ 4*u3 *v1*v3*a23*b12 - 4*u3 *v2*v3*a13*b12 - u3*v1 *v3*b12
2 3
- u3*v2 *v3*b12
And again in machine readable form:
HAM= - u1**2*a33 + 2*u1*u3*a13 + u1*v2*b12 - u2**2*a33 + 2*u2*u3*a23 - u2*v1*b12
+ u3**2*a33$
FI=u1**2*u3*( - 2*a13**2 - 2*a23**2 - 2*a33**2 - b12**2*kap) + 2*u1**2*v1*a23*
b12 - 2*u1*u2*v1*a13*b12 + 2*u1*u2*v2*a23*b12 + 2*u1*u3*v2*a33*b12 + u1*v1*v3*
b12**2 + u2**2*u3*( - 2*a13**2 - 2*a23**2 - 2*a33**2 - b12**2*kap) - 2*u2**2*v2*
a13*b12 - 2*u2*u3*v1*a33*b12 + u2*v2*v3*b12**2 + u3**3*b12**2*kap + 2*u3**2*v1*
a23*b12 - 2*u3**2*v2*a13*b12 - u3*v1**2*b12**2 - u3*v2**2*b12**2$