Solution 4 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2 2 4 2 2
0=16*a23 *c23 - 8*a23*b22*c23*c33 + b22 *c33 - 16*c23 - 4*c23 *c33
Expressions
The solution is given through the following expressions:
1 1 2
- ---*a23*c23*c33*n3*r26 + ----*b22*c33 *n3*r26
4 16
r10=--------------------------------------------------
3
a23*c23
r11=0
r12=0
r13=0
r14=0
r15=0
1 2
- ---*c33 *r26
4
r20=-----------------
2
c23
2 2 1 2 2 4
r21=(2*a23 *c23 *r26 - a23*b22*c23*c33*r26 + ---*b22 *c33 *r26 - 2*c23 *r26
8
2 2 3
- c23 *c33 *r26)/(c23 *c33)
r22=0
r23=0
r24=0
1
- ---*c33*r26
2
r27=----------------
c23
r28=0
r29=0
r210=0
1
a23*c23*r26 - ---*b22*c33*r26
4
r211=-------------------------------
2
c23
r212=0
r214=0
r215=0
r216=0
1
a23*c23*r26 - ---*b22*c33*r26
4
r217=-------------------------------
2
c23
r218=0
r219=0
1
a23*c23*n3 - ---*b22*c33*n3
4
m3=-----------------------------
a23*c23
2 3
4*a23 *c23 - a23*b22*c33 - 4*c23
b33=-----------------------------------
c23*c33
b12=0
a33=0
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:
r00, r26, n3, c33, b22, c23, 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.
{c23,a23,b22,r26,c33}
Relevance for the application:
Modulo the following equation:
2 2 2 2 4 2 2
0=16*a23 *c23 - 8*a23*b22*c23*c33 + b22 *c33 - 16*c23 - 4*c23 *c33
the system of equations related to the Hamiltonian HAM:
2
HAM=(u1*v1*a23*b22*c23*c33 + 2*u2*u3*a23 *c23*c33 + u2*v2*a23*b22*c23*c33
2 2
+ 2*u2*v3*a23 *c23*c33 + 2*u3*v2*a23*c23 *c33
3 2 3
+ u3*v3*(4*a23 *c23 - a23 *b22*c33 - 4*a23*c23 ) + u3*a23*c23*c33*n3
2 2 2
+ 2*v2*v3*a23*c23 *c33 + v3 *a23*c23*c33
1 2
+ v3*(a23*c23*c33*n3 - ---*b22*c33 *n3))/(a23*c23*c33)
4
has apart from the Hamiltonian and Casimirs the following 2 first integrals:
2 2 1 2
FI=u1*v1*(a23 *c23 *c33 - ---*a23*b22*c23*c33 )
4
2 2 1 2 1 2 2
+ u2*v2*(a23 *c23 *c33 - ---*a23*b22*c23*c33 ) - ---*u3*v2*a23*c23 *c33
4 2
3 3 2 2
+ u3*v3*a23*c23 *c33 + v2*v3*(2*a23 *c23 - a23 *b22*c23*c33
1 2 2 4 2 2 1 2 3
+ ---*a23*b22 *c33 - 2*a23*c23 - a23*c23 *c33 ) - ---*v3 *a23*c23*c33
8 4
1 2 1 3
+ v3*( - ---*a23*c23*c33 *n3 + ----*b22*c33 *n3)
4 16
1
= a product of the elements of: {----,
16
2 2 2
u1*v1*(16*a23 *c23 *c33 - 4*a23*b22*c23*c33 )
2 2 2 2 2
+ u2*v2*(16*a23 *c23 *c33 - 4*a23*b22*c23*c33 ) - 8*u3*v2*a23*c23 *c33
3 3 2 2
+ 16*u3*v3*a23*c23 *c33 + v2*v3*(32*a23 *c23 - 16*a23 *b22*c23*c33
2 2 4 2 2 2 3
+ 2*a23*b22 *c33 - 32*a23*c23 - 16*a23*c23 *c33 ) - 4*v3 *a23*c23*c33
2 3
+ v3*( - 4*a23*c23*c33 *n3 + b22*c33 *n3)}
{HAM,FI} = 0
FI=1
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1*v1*a23*b22*c23*c33 + 2*u2*u3*a23**2*c23*c33 + u2*v2*a23*b22*c23*c33 + 2*
u2*v3*a23**2*c23*c33 + 2*u3*v2*a23*c23**2*c33 + u3*v3*(4*a23**3*c23 - a23**2*b22
*c33 - 4*a23*c23**3) + u3*a23*c23*c33*n3 + 2*v2*v3*a23*c23**2*c33 + v3**2*a23*
c23*c33**2 + v3*(a23*c23*c33*n3 - 1/4*b22*c33**2*n3))/(a23*c23*c33)$
FI=u1*v1*(a23**2*c23**2*c33 - 1/4*a23*b22*c23*c33**2) + u2*v2*(a23**2*c23**2*c33
- 1/4*a23*b22*c23*c33**2) - 1/2*u3*v2*a23*c23**2*c33**2 + u3*v3*a23*c23**3*c33
+ v2*v3*(2*a23**3*c23**2 - a23**2*b22*c23*c33 + 1/8*a23*b22**2*c33**2 - 2*a23*
c23**4 - a23*c23**2*c33**2) - 1/4*v3**2*a23*c23*c33**3 + v3*( - 1/4*a23*c23*c33
**2*n3 + 1/16*b22*c33**3*n3)$
FI=1$