Solution 3 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=4*a23 + a33
2 2
0=a11 + 4*a23
Expressions
The solution is given through the following expressions:
3 2 2 1 2 2 1 3 2
----*a11 *a33*b33 *r213 - ---*a11*a33 *b33 *r213 + ----*a33 *b33 *r213
32 8 32
r20=------------------------------------------------------------------------
2 3
a11 *a23
3 2 2 1 2 1 2 2
----*a11 *b33 *r213 - ---*a11*a33*b33 *r213 + ----*a33 *b33 *r213
16 4 16
r21=-------------------------------------------------------------------
2 2
a11 *a23
r23=0
r24=0
1 4 2 2 1 3 2 2
r25=(----*a11 *a33 *b33 *r213 + ---*a11 *a23 *a33*b33 *r213
32 8
1 3 3 2 9 2 4 2
- ----*a11 *a33 *b33 *r213 + ----*a11 *a23 *b33 *r213
32 32
1 2 2 2 2 1 4 2 2 3 5
- ----*a11 *a23 *a33 *b33 *r213 - ----*a23 *a33 *b33 *r213)/(a11 *a23 )
16 32
1 2 1 2
---*a23 *b33*r213 + ---*a33 *b33*r213
2 4
r26=---------------------------------------
3
a23
1 1
- ---*a11*b33*r213 + ---*a33*b33*r213
4 4
r27=----------------------------------------
2
a23
r28=0
1 1
- ---*a11*r213 + ---*a33*r213
2 2
r29=--------------------------------
a23
1 2 1 2
- ---*a11*a23 *b33*r213 - ---*a11*a33 *b33*r213
4 8
r210=--------------------------------------------------
4
a23
r212=0
1
- ---*a11*r213
2
r214=-----------------
a23
r215=0
r216=0
1 2 3 2 1 2
---*a11 *a33*b33*r213 + ---*a11*a23 *b33*r213 - ---*a23 *a33*b33*r213
4 4 4
r217=-----------------------------------------------------------------------
3
a11*a23
r218=0
r219=0
1 3 2 9 2 2 2 1 2 2 2
- ---*a11 *a33*b33 - ----*a11 *a23 *b33 + ----*a23 *a33 *b33
8 16 16
c33=------------------------------------------------------------------
3 2
a11 *a23
c23=0
1 3 2 2 1 3 3 2 9 2 4 2
c22=(---*a11 *a23 *a33*b33 + ----*a11 *a33 *b33 - ----*a11 *a23 *b33
8 16 16
1 4 2 2 3 4
+ ----*a23 *a33 *b33 )/(a11 *a23 )
16
c13=0
c12=0
b31=0
1
- ---*a33*b33
2
b23=----------------
a23
b21=0
3 1
---*a11*b33 - ---*a33*b33
2 2
b11=---------------------------
a11
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:
r213, b33, a33, a11, 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.
{a33,r213,a11,a23,b33}
Relevance for the application:
Modulo the following equations:
2 2
0=4*a23 + a33
2 2
0=a11 + 4*a23
the system of equations related to the Hamiltonian HAM:
2 4 4 3 3 4 1 2 4
HAM=(u1 *a11 *a23 + u1*v1*(---*a11 *a23 *b33 - ---*a11 *a23 *a33*b33)
2 2
3 5 1 3 3 2 3 4
+ 2*u2*u3*a11 *a23 - ---*u2*v3*a11 *a23 *a33*b33 + u3 *a11 *a23 *a33
2
3 4 2 1 3 2 2 1 3 3 2
+ u3*v3*a11 *a23 *b33 + v2 *(---*a11 *a23 *a33*b33 + ----*a11 *a33 *b33
8 16
9 2 4 2 1 4 2 2 2
- ----*a11 *a23 *b33 + ----*a23 *a33 *b33 ) + v3
16 16
1 3 2 2 9 2 4 2 1 4 2 2
*( - ---*a11 *a23 *a33*b33 - ----*a11 *a23 *b33 + ----*a23 *a33 *b33 ))/(
8 16 16
3 4
a11 *a23 )
has apart from the Hamiltonian and Casimirs only the following first integral:
1 4 2 3 3 4 1 2 4
FI=u1*v1*(---*a11 *a23 *a33*b33 + ---*a11 *a23 *b33 - ---*a11 *a23 *a33*b33)
4 4 4
1 2 4 4 3 5
- ---*u2 *a11 *a23 + u2*u3*a11 *a23
2
1 4 3 1 4 2
+ u2*v3*( - ---*a11 *a23 *b33 - ---*a11 *a23*a33 *b33)
4 8
2 1 4 4 1 3 4
+ u3 *( - ---*a11 *a23 + ---*a11 *a23 *a33)
2 2
1 4 3 1 3 3
+ u3*v2*( - ---*a11 *a23 *b33 + ---*a11 *a23 *a33*b33)
4 4
1 3 4 1 3 2 2 2
+ u3*v3*(---*a11 *a23 *b33 + ---*a11 *a23 *a33 *b33) + v1 *(
2 4
1 4 2 2 1 3 2 2 1 3 3 2
----*a11 *a33 *b33 + ---*a11 *a23 *a33*b33 - ----*a11 *a33 *b33
32 8 32
9 2 4 2 1 2 2 2 2 1 4 2 2
+ ----*a11 *a23 *b33 - ----*a11 *a23 *a33 *b33 - ----*a23 *a33 *b33 ) +
32 16 32
v2*v3
3 3 3 2 1 2 3 2 1 3 2 2
*(----*a11 *a23 *b33 - ---*a11 *a23 *a33*b33 + ----*a11*a23 *a33 *b33 ) +
16 4 16
2 3 3 2 2 1 2 2 2 2
v3 *(----*a11 *a23 *a33*b33 - ---*a11 *a23 *a33 *b33
32 8
1 2 3 2
+ ----*a11*a23 *a33 *b33 )
32
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a11**4*a23**4 + u1*v1*(3/2*a11**3*a23**4*b33 - 1/2*a11**2*a23**4*a33*
b33) + 2*u2*u3*a11**3*a23**5 - 1/2*u2*v3*a11**3*a23**3*a33*b33 + u3**2*a11**3*
a23**4*a33 + u3*v3*a11**3*a23**4*b33 + v2**2*(1/8*a11**3*a23**2*a33*b33**2 + 1/
16*a11**3*a33**3*b33**2 - 9/16*a11**2*a23**4*b33**2 + 1/16*a23**4*a33**2*b33**2)
+ v3**2*( - 1/8*a11**3*a23**2*a33*b33**2 - 9/16*a11**2*a23**4*b33**2 + 1/16*a23
**4*a33**2*b33**2))/(a11**3*a23**4)$
FI=u1*v1*(1/4*a11**4*a23**2*a33*b33 + 3/4*a11**3*a23**4*b33 - 1/4*a11**2*a23**4*
a33*b33) - 1/2*u2**2*a11**4*a23**4 + u2*u3*a11**3*a23**5 + u2*v3*( - 1/4*a11**4*
a23**3*b33 - 1/8*a11**4*a23*a33**2*b33) + u3**2*( - 1/2*a11**4*a23**4 + 1/2*a11
**3*a23**4*a33) + u3*v2*( - 1/4*a11**4*a23**3*b33 + 1/4*a11**3*a23**3*a33*b33) +
u3*v3*(1/2*a11**3*a23**4*b33 + 1/4*a11**3*a23**2*a33**2*b33) + v1**2*(1/32*a11
**4*a33**2*b33**2 + 1/8*a11**3*a23**2*a33*b33**2 - 1/32*a11**3*a33**3*b33**2 + 9
/32*a11**2*a23**4*b33**2 - 1/16*a11**2*a23**2*a33**2*b33**2 - 1/32*a23**4*a33**2
*b33**2) + v2*v3*(3/16*a11**3*a23**3*b33**2 - 1/4*a11**2*a23**3*a33*b33**2 + 1/
16*a11*a23**3*a33**2*b33**2) + v3**2*(3/32*a11**3*a23**2*a33*b33**2 - 1/8*a11**2
*a23**2*a33**2*b33**2 + 1/32*a11*a23**2*a33**3*b33**2)$