Solution 5 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
2 2
- a23 *n3*r26 + c23 *n3*r26
r10=------------------------------
2
b33*c23
r11=0
r12=0
r13=0
r14=0
r15=0
4 1 2 2 2 2 1 2 2
r20=( - 2*a23 *r26 - ---*a23 *b33 *r26 + 4*a23 *c23 *r26 + ---*b33 *c23 *r26
2 2
4 2 2
- 2*c23 *r26)/(b33 *c23 )
2 2
- 2*a23 *r26 + 2*c23 *r26
r21=----------------------------
b33*c23
r22=0
r23=0
r24=0
2 2
- 2*a23 *r26 + 2*c23 *r26
r27=----------------------------
b33*c23
r28=0
r29=0
r210=0
a23*r26
r211=---------
c23
r212=0
r214=0
r215=0
r216=0
a23*r26
r217=---------
c23
r218=0
r219=0
m3=n3
2 1 2 2
2*a23 + ---*b33 - 2*c23
2
c33=----------------------------
b33
b12=0
b22=0
2 1 2 2
- 2*a23 + ---*b33 + 2*c23
2
a33=-------------------------------
b33
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, b33, 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.
{b33,a23,c23,r26,a23 - c23,a23 + c23}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 1 2 2
HAM=(2*u2*u3*a23*b33 + 2*u2*v3*a23*b33 + u3 *( - 2*a23 + ---*b33 + 2*c23 )
2
2
+ 2*u3*v2*b33*c23 + u3*v3*b33 + u3*b33*n3 + 2*v2*v3*b33*c23
2 2 1 2 2
+ v3 *(2*a23 + ---*b33 - 2*c23 ) + v3*b33*n3)/b33
2
has apart from the Hamiltonian and Casimirs the following 2 first integrals:
2 2
FI=u1*v1*a23*b33 *c23 + u2*v2*a23*b33 *c23
2 3 2 2
+ u3*v2*( - 2*a23 *b33*c23 + 2*b33*c23 ) + u3*v3*b33 *c23
2 3
+ v2*v3*( - 2*a23 *b33*c23 + 2*b33*c23 )
2 4 1 2 2 2 2 1 2 2 4
+ v3 *( - 2*a23 - ---*a23 *b33 + 4*a23 *c23 + ---*b33 *c23 - 2*c23 )
2 2
2 2
+ v3*( - a23 *b33*n3 + b33*c23 *n3)
1
= a product of the elements of: { - ---,
2
2 2
- 2*u1*v1*a23*b33 *c23 - 2*u2*v2*a23*b33 *c23
2 3 2 2
+ u3*v2*(4*a23 *b33*c23 - 4*b33*c23 ) - 2*u3*v3*b33 *c23
2 3
+ v2*v3*(4*a23 *b33*c23 - 4*b33*c23 )
2 4 2 2 2 2 2 2 4
+ v3 *(4*a23 + a23 *b33 - 8*a23 *c23 - b33 *c23 + 4*c23 )
2 2
+ v3*(2*a23 *b33*n3 - 2*b33*c23 *n3)}
{HAM,FI} = 0
FI=1
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=(2*u2*u3*a23*b33 + 2*u2*v3*a23*b33 + u3**2*( - 2*a23**2 + 1/2*b33**2 + 2*c23
**2) + 2*u3*v2*b33*c23 + u3*v3*b33**2 + u3*b33*n3 + 2*v2*v3*b33*c23 + v3**2*(2*
a23**2 + 1/2*b33**2 - 2*c23**2) + v3*b33*n3)/b33$
FI=u1*v1*a23*b33**2*c23 + u2*v2*a23*b33**2*c23 + u3*v2*( - 2*a23**2*b33*c23 + 2*
b33*c23**3) + u3*v3*b33**2*c23**2 + v2*v3*( - 2*a23**2*b33*c23 + 2*b33*c23**3) +
v3**2*( - 2*a23**4 - 1/2*a23**2*b33**2 + 4*a23**2*c23**2 + 1/2*b33**2*c23**2 -
2*c23**4) + v3*( - a23**2*b33*n3 + b33*c23**2*n3)$
FI=1$