Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2
0=a11 - 2*a11*a22 + a22 + a23
Expressions
The solution is given through the following expressions:
1
- ---*a23*b33*r14
4
r10=--------------------
2
a11 - a11*a22
1
---*b33*r14
4
r11=-------------
a11
r12=0
- a23*r14
r13=------------
a11 - a22
r15=0
- a23*m2
m3=-----------
a11 - a22
m1=0
- a23*n2
n3=-----------
a11 - a22
n1=0
1 2 2 1 2 2 1 2 2
c33=( - ----*a11 *b33 + ---*a11*a22*b33 + a11*a23 *c22 - ----*a22 *b33
16 8 16
1 2 2 3 2 2
+ ----*a23 *b33 )/(a11 - 2*a11 *a22 + a11*a22 )
16
1 2
- a11*a23*c22 - ----*a23*b33
16
c23=--------------------------------
2
a11 - a11*a22
c13=0
c12=0
a23*b33
b32=-----------
a11 - a22
b31=0
b21=0
1
b11=---*b33
2
2 2
a11 - a11*a22 - a23
a33=-----------------------
a11 - a22
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, c22, m2, n2, b33, a11, a22, 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,a22,a11,r14,a11 - a22,{n2,m2}}
Relevance for the application:
Modulo the following equation:
2 2 2
0=a11 - 2*a11*a22 + a22 + a23
the system of equations related to the Hamiltonian HAM:
2 4 3 2 2
HAM=(u1 *(a11 - 2*a11 *a22 + a11 *a22 )
1 3 2 1 2
+ u1*v1*(---*a11 *b33 - a11 *a22*b33 + ---*a11*a22 *b33)
2 2
2 3 2 2 3
+ u2 *(a11 *a22 - 2*a11 *a22 + a11*a22 )
3 2 2
+ u2*u3*(2*a11 *a23 - 4*a11 *a22*a23 + 2*a11*a22 *a23)
3 2 2
+ u2*(a11 *n2 - 2*a11 *a22*n2 + a11*a22 *n2)
2 4 3 2 2 2 2 2
+ u3 *(a11 - 2*a11 *a22 + a11 *a22 - a11 *a23 + a11*a22*a23 )
2
+ u3*v2*(a11 *a23*b33 - a11*a22*a23*b33)
3 2 2
+ u3*v3*(a11 *b33 - 2*a11 *a22*b33 + a11*a22 *b33)
2
+ u3*( - a11 *a23*n2 + a11*a22*a23*n2)
2 3 2 2
+ v2 *(a11 *c22 - 2*a11 *a22*c22 + a11*a22 *c22) + v2*v3*(
2 1 2
- 2*a11 *a23*c22 + 2*a11*a22*a23*c22 - ---*a11*a23*b33
8
1 2 3 2 2 2
+ ---*a22*a23*b33 ) + v2*(a11 *m2 - 2*a11 *a22*m2 + a11*a22 *m2) + v3 *
8
1 2 2 1 2 2 1 2 2
( - ----*a11 *b33 + ---*a11*a22*b33 + a11*a23 *c22 - ----*a22 *b33
16 8 16
1 2 2 2 3
+ ----*a23 *b33 ) + v3*( - a11 *a23*m2 + a11*a22*a23*m2))/(a11
16
2 2
- 2*a11 *a22 + a11*a22 )
has apart from the Hamiltonian and Casimirs only the following first integral:
2 1 1
FI=u2*(a11 - a11*a22) - u3*a11*a23 + v2*(---*a11*b33 - ---*a22*b33)
4 4
1
- ---*v3*a23*b33
4
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*(a11**4 - 2*a11**3*a22 + a11**2*a22**2) + u1*v1*(1/2*a11**3*b33 - a11
**2*a22*b33 + 1/2*a11*a22**2*b33) + u2**2*(a11**3*a22 - 2*a11**2*a22**2 + a11*
a22**3) + u2*u3*(2*a11**3*a23 - 4*a11**2*a22*a23 + 2*a11*a22**2*a23) + u2*(a11**
3*n2 - 2*a11**2*a22*n2 + a11*a22**2*n2) + u3**2*(a11**4 - 2*a11**3*a22 + a11**2*
a22**2 - a11**2*a23**2 + a11*a22*a23**2) + u3*v2*(a11**2*a23*b33 - a11*a22*a23*
b33) + u3*v3*(a11**3*b33 - 2*a11**2*a22*b33 + a11*a22**2*b33) + u3*( - a11**2*
a23*n2 + a11*a22*a23*n2) + v2**2*(a11**3*c22 - 2*a11**2*a22*c22 + a11*a22**2*c22
) + v2*v3*( - 2*a11**2*a23*c22 + 2*a11*a22*a23*c22 - 1/8*a11*a23*b33**2 + 1/8*
a22*a23*b33**2) + v2*(a11**3*m2 - 2*a11**2*a22*m2 + a11*a22**2*m2) + v3**2*( - 1
/16*a11**2*b33**2 + 1/8*a11*a22*b33**2 + a11*a23**2*c22 - 1/16*a22**2*b33**2 + 1
/16*a23**2*b33**2) + v3*( - a11**2*a23*m2 + a11*a22*a23*m2))/(a11**3 - 2*a11**2*
a22 + a11*a22**2)$
FI=u2*(a11**2 - a11*a22) - u3*a11*a23 + v2*(1/4*a11*b33 - 1/4*a22*b33) - 1/4*v3*
a23*b33$