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 2
a23 *r14 - ---*b22 *r14 + ---*b33 *r14 - c23 *r14
4 4
r10=---------------------------------------------------
a23*b33 + b22*c23
a23*b22*r14 + b33*c23*r14
r11=---------------------------
a23*b33 + b22*c23
r12=0
2 1 2 1 2 2
- a23 *r14 - ---*b22 *r14 + ---*b33 *r14 + c23 *r14
4 4
r13=------------------------------------------------------
a23*b33 + b22*c23
r15=0
2 1 2 1 2 3
2*a23 *c23 - ---*b22 *c23 + ---*b33 *c23 - 2*c23
2 2
c33=---------------------------------------------------
a23*b22 + b33*c23
b12=0
3 1 2 1 2 2
- 2*a23 - ---*a23*b22 + ---*a23*b33 + 2*a23*c23
2 2
a33=------------------------------------------------------
a23*b33 + b22*c23
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:
r14, c23, b33, 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.
{a23*b22 + b33*c23,b22,a23,r14,a23*b33 + b22*c23,4*a23*c23 - b22*b33}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 3 2 2 2
HAM=(u1*v1*(a23 *b22 *b33 + a23*b22 *c23 + a23*b22*b33 *c23 + b22 *b33*c23 ) +
u2*u3
3 2 2 2 2 2
*(2*a23 *b22*b33 + 2*a23 *b22 *c23 + 2*a23 *b33 *c23 + 2*a23*b22*b33*c23 )
2 2 3 2 2 2
+ u2*v2*(a23 *b22 *b33 + a23*b22 *c23 + a23*b22*b33 *c23 + b22 *b33*c23 )
+ u2*v3
3 2 2 2 2 2
*(2*a23 *b22*b33 + 2*a23 *b22 *c23 + 2*a23 *b33 *c23 + 2*a23*b22*b33*c23 )
2 4 3 1 2 3 1 2 2
+ u3 *( - 2*a23 *b22 - 2*a23 *b33*c23 - ---*a23 *b22 + ---*a23 *b22*b33
2 2
2 2 1 2 1 3
+ 2*a23 *b22*c23 - ---*a23*b22 *b33*c23 + ---*a23*b33 *c23
2 2
3
+ 2*a23*b33*c23 ) + u3*v2
2 2 2 2 2 3
*(2*a23 *b22*b33*c23 + 2*a23*b22 *c23 + 2*a23*b33 *c23 + 2*b22*b33*c23 )
2 2 2 3 2 2
+ u3*v3*(a23 *b22*b33 + a23*b22 *b33*c23 + a23*b33 *c23 + b22*b33 *c23 )
+ v2*v3
2 2 2 2 2 3
*(2*a23 *b22*b33*c23 + 2*a23*b22 *c23 + 2*a23*b33 *c23 + 2*b22*b33*c23 )
2 3 2 2 1 2
+ v3 *(2*a23 *b33*c23 + 2*a23 *b22*c23 - ---*a23*b22 *b33*c23
2
1 3 3 1 3 2
+ ---*a23*b33 *c23 - 2*a23*b33*c23 - ---*b22 *c23
2 2
1 2 2 4 2 2
+ ---*b22*b33 *c23 - 2*b22*c23 ))/(a23 *b22*b33 + a23*b22 *c23
2
2 2
+ a23*b33 *c23 + b22*b33*c23 )
has apart from the Hamiltonian and Casimirs only the following first integral:
FI=u2*u3*(a23*b33 + b22*c23) + u2*v3*(a23*b33 + b22*c23)
2 2 1 2 1 2 2
+ u3 *( - a23 - ---*b22 + ---*b33 + c23 ) + u3*v2*(a23*b22 + b33*c23)
4 4
1 2 1 2
+ u3*v3*( - ---*b22 + ---*b33 ) + v2*v3*(a23*b22 + b33*c23)
2 2
2 2 1 2 1 2 2
+ v3 *(a23 - ---*b22 + ---*b33 - c23 )
4 4
1
= a product of the elements of: { - ---,
4
u3 + v3,
2 2 2 2
u2*( - 4*a23*b33 - 4*b22*c23) + u3*(4*a23 + b22 - b33 - 4*c23 )
2 2 2 2
+ v2*( - 4*a23*b22 - 4*b33*c23) + v3*( - 4*a23 + b22 - b33 + 4*c23 )}
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1*v1*(a23**2*b22**2*b33 + a23*b22**3*c23 + a23*b22*b33**2*c23 + b22**2*b33
*c23**2) + u2*u3*(2*a23**3*b22*b33 + 2*a23**2*b22**2*c23 + 2*a23**2*b33**2*c23 +
2*a23*b22*b33*c23**2) + u2*v2*(a23**2*b22**2*b33 + a23*b22**3*c23 + a23*b22*b33
**2*c23 + b22**2*b33*c23**2) + u2*v3*(2*a23**3*b22*b33 + 2*a23**2*b22**2*c23 + 2
*a23**2*b33**2*c23 + 2*a23*b22*b33*c23**2) + u3**2*( - 2*a23**4*b22 - 2*a23**3*
b33*c23 - 1/2*a23**2*b22**3 + 1/2*a23**2*b22*b33**2 + 2*a23**2*b22*c23**2 - 1/2*
a23*b22**2*b33*c23 + 1/2*a23*b33**3*c23 + 2*a23*b33*c23**3) + u3*v2*(2*a23**2*
b22*b33*c23 + 2*a23*b22**2*c23**2 + 2*a23*b33**2*c23**2 + 2*b22*b33*c23**3) + u3
*v3*(a23**2*b22*b33**2 + a23*b22**2*b33*c23 + a23*b33**3*c23 + b22*b33**2*c23**2
) + v2*v3*(2*a23**2*b22*b33*c23 + 2*a23*b22**2*c23**2 + 2*a23*b33**2*c23**2 + 2*
b22*b33*c23**3) + v3**2*(2*a23**3*b33*c23 + 2*a23**2*b22*c23**2 - 1/2*a23*b22**2
*b33*c23 + 1/2*a23*b33**3*c23 - 2*a23*b33*c23**3 - 1/2*b22**3*c23**2 + 1/2*b22*
b33**2*c23**2 - 2*b22*c23**4))/(a23**2*b22*b33 + a23*b22**2*c23 + a23*b33**2*c23
+ b22*b33*c23**2)$
FI=u2*u3*(a23*b33 + b22*c23) + u2*v3*(a23*b33 + b22*c23) + u3**2*( - a23**2 - 1/
4*b22**2 + 1/4*b33**2 + c23**2) + u3*v2*(a23*b22 + b33*c23) + u3*v3*( - 1/2*b22
**2 + 1/2*b33**2) + v2*v3*(a23*b22 + b33*c23) + v3**2*(a23**2 - 1/4*b22**2 + 1/4
*b33**2 - c23**2)$