Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
2 1 2 1 2 1 2
a23 *r27 - ---*b12 *r27 - ---*b22 *r27 + ---*b33 *r27
4 4 4
r20=-------------------------------------------------------
a23*b22
r21=r27
r22=0
- b12*r27
r23=------------
b22
r24=0
1 2 1 2
- ---*b12 *r27 - ---*b22 *r27
2 2
r26=--------------------------------
a23*b22
r29=0
r210=0
1
- ---*b33*r27
2
r211=----------------
a23
1
---*b12*b33*r27
2
r212=-----------------
a23*b22
r214=0
r215=0
1
- ---*b12*b33*r27
2
r216=--------------------
a23*b22
1
- ---*b33*r27
2
r217=----------------
a23
r218=0
r219=0
c33=0
c23=0
2 1 2 1 2 1 2
- 2*a23 - ---*b12 - ---*b22 + ---*b33
2 2 2
a33=--------------------------------------------
b33
- b12*r27
r28=------------
b22
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:
r27, b22, b33, b12, 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.
{b33,a23}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
HAM=(u1*v1*b22*b33 + u1*v2*b12*b33 + 2*u2*u3*a23*b33 - u2*v1*b12*b33
+ u2*v2*b22*b33 + 2*u2*v3*a23*b33
2 2 1 2 1 2 1 2 2
+ u3 *( - 2*a23 - ---*b12 - ---*b22 + ---*b33 ) + u3*v3*b33 )/b33
2 2 2
has apart from the Hamiltonian and Casimirs only the following first integral:
1 1 1
FI= - ---*u1*v1*b22*b33 - ---*u1*v2*b12*b33 + ---*u2*v1*b12*b33
2 2 2
1
- ---*u2*v2*b22*b33 - u3*v1*a23*b12 + u3*v2*a23*b22
2
1 2 1 2
+ u3*v3*( - ---*b12 - ---*b22 ) - v1*v3*a23*b12 + v2*v3*a23*b22
2 2
2 2 1 2 1 2 1 2
+ v3 *(a23 - ---*b12 - ---*b22 + ---*b33 )
4 4 4
1
= a product of the elements of: {---,
4
- 2*u1*v1*b22*b33 - 2*u1*v2*b12*b33 + 2*u2*v1*b12*b33 - 2*u2*v2*b22*b33
2 2
- 4*u3*v1*a23*b12 + 4*u3*v2*a23*b22 + u3*v3*( - 2*b12 - 2*b22 )
2 2 2 2 2
- 4*v1*v3*a23*b12 + 4*v2*v3*a23*b22 + v3 *(4*a23 - b12 - b22 + b33 )}
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1*v1*b22*b33 + u1*v2*b12*b33 + 2*u2*u3*a23*b33 - u2*v1*b12*b33 + u2*v2*b22
*b33 + 2*u2*v3*a23*b33 + u3**2*( - 2*a23**2 - 1/2*b12**2 - 1/2*b22**2 + 1/2*b33
**2) + u3*v3*b33**2)/b33$
FI= - 1/2*u1*v1*b22*b33 - 1/2*u1*v2*b12*b33 + 1/2*u2*v1*b12*b33 - 1/2*u2*v2*b22*
b33 - u3*v1*a23*b12 + u3*v2*a23*b22 + u3*v3*( - 1/2*b12**2 - 1/2*b22**2) - v1*v3
*a23*b12 + v2*v3*a23*b22 + v3**2*(a23**2 - 1/4*b12**2 - 1/4*b22**2 + 1/4*b33**2)
$