Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
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Equations
The following unsolved equations remain:
4 2 2 2 2
0=4*a23 + 4*a23 *c22 - b31 *c22
2 2 4 2 2
0=4*b31 *c22 - b33 + 4*b33 *c22
2 4 4 2 2 4 2 2 2 6 4 2
0=16*a23 *b33 + 4*b31 *c22 - b31 *b33 - 8*b31 *b33 *c22 - b33 + 4*b33 *c22
Expressions
The solution is given through the following expressions:
1 2 2 1 4 1 2 2
---*b31 *c22 *r354 - ---*b33 *r354 + ---*b33 *c22 *r354
2 8 2
r30=---------------------------------------------------------
3
a23 *b33
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
1 4 2 1 2 4 1 2 2 2
- ---*b31 *c22 *r354 + ---*b31 *b33 *r354 - ---*b31 *b33 *c22 *r354
2 8 2
r37=----------------------------------------------------------------------
3 3
a23 *b33
r38=0
r39=0
1 4 4 5 2 4 2 3 8
r310=(---*b31 *c22 *r354 - ---*b31 *b33 *c22 *r354 + ----*b33 *r354
2 8 32
1 6 2 1 4 4 3 4
- ---*b33 *c22 *r354 - ---*b33 *c22 *r354)/(a23 *b33 *c22)
4 2
r311=0
1 4 3 1 2 2 3
- ---*b31 *c22 *r354 + ---*b31 *b33 *c22 *r354
2 2
r312=-------------------------------------------------
3 4
a23 *b33
3 3 2 3 4 3 2 2
- ---*b31 *c22 *r354 + ----*b31*b33 *r354 - ---*b31*b33 *c22 *r354
4 16 4
r313=---------------------------------------------------------------------
3
a23 *b33*c22
r314=0
3 4 2 3 2 4 1 2 2 2
- ---*b31 *c22 *r354 + ----*b31 *b33 *r354 - ---*b31 *b33 *c22 *r354
8 32 4
r315=-----------------------------------------------------------------------
3 2
a23 *b33 *c22
1 2 4 3 4 2 3 2 4
r316=(---*a23 *b33 *r354 - ---*b31 *c22 *r354 + ----*b31 *b33 *r354
2 4 16
3 6 3 4 2 3 3
- ----*b33 *r354 + ---*b33 *c22 *r354)/(a23 *b33 )
16 4
r317=0
1 2 4 3 5 2 3 3 4
r318=(---*a23 *b31*b33 *r354 - ---*b31 *c22 *r354 + ----*b31 *b33 *r354
2 4 16
3 6 3 4 2 3 4
- ----*b31*b33 *r354 + ---*b31*b33 *c22 *r354)/(a23 *b33 )
16 4
1 2 2 4 1 2 6 3 6 3
r319=(---*a23 *b31 *b33 *c22*r354 - ---*a23 *b33 *c22*r354 - ---*b31 *c22 *r354
2 2 8
3 4 4 3 4 2 3
+ ----*b31 *b33 *c22*r354 + ---*b31 *b33 *c22 *r354
32 8
3 2 6 3 2 4 3 3 8
- ----*b31 *b33 *c22*r354 + ---*b31 *b33 *c22 *r354 + ----*b33 *c22*r354
16 8 32
3 6 3 3 6
- ---*b33 *c22 *r354)/(a23 *b33 )
8
2 2 1 4 2 2
b31 *c22 *r354 - ---*b33 *r354 + b33 *c22 *r354
4
r320=-------------------------------------------------
2 2
a23 *b33
r321=0
2 2
- b31 *c22 *r354
r322=-------------------
2 2
a23 *b33
1
---*b31*b33*r354
4
r323=------------------
2
a23
r324=0
r325=0
3 2 2 3 4 3 2 2
- ---*b31 *c22 *r354 + ---*b33 *r354 - ---*b33 *c22 *r354
2 8 2
r326=------------------------------------------------------------
2
a23 *b33*c22
r327=0
3 3 2 3 4 3 2 2
- ---*b31 *c22 *r354 + ---*b31*b33 *r354 - ---*b31*b33 *c22 *r354
2 8 2
r328=--------------------------------------------------------------------
2 2
a23 *b33 *c22
2 4 3 4 2 3 2 4 3 6
r329=(a23 *b33 *r354 - ---*b31 *c22 *r354 + ---*b31 *b33 *r354 - ---*b33 *r354
2 8 8
3 4 2 2 4
+ ---*b33 *c22 *r354)/(a23 *b33 )
2
1
---*b33*r354
2
r330=--------------
a23
r331=0
1
---*b31*r354
2
r332=--------------
a23
2 2 3 4 2 2
- b31 *c22 *r354 + ---*b33 *r354 - 2*b33 *c22 *r354
8
r333=------------------------------------------------------
2
a23*b33 *c22
r334=r354
r335=0
1
---*b31*c22*r354
2
r336=------------------
2
a23
r337=0
r338=0
1 2
---*b31 *c22*r354
2
r339=-------------------
2
a23 *b33
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
1
---*b33*r354
2
r350=--------------
a23
r351=0
1
---*b31*r354
2
r352=--------------
a23
1 2 1 2
---*b31 *c22*r354 - ---*b33 *c22*r354
2 2
r353=---------------------------------------
2
a23*b33
r355=0
2 2
b31 *c22 + b33 *c22
c11=---------------------
2
b33
b32=0
2 2
b31 *c22 - b33 *c22
a33=---------------------
2
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:
r354, b33, b31, c22, 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.
{b31,c22,a23,r354,b33}
Relevance for the application:
Modulo the following equations:
4 2 2 2 2
0=4*a23 + 4*a23 *c22 - b31 *c22
2 2 4 2 2
0=4*b31 *c22 - b33 + 4*b33 *c22
2 4 4 2 2 4 2 2 2 6 4 2
0=16*a23 *b33 + 4*b31 *c22 - b31 *b33 - 8*b31 *b33 *c22 - b33 + 4*b33 *c22
the system of equations related to the Hamiltonian HAM:
2 2 2 2 2 3
HAM=(2*u2*u3*a23*b33 + u3 *(b31 *c22 - b33 *c22) + u3*v1*b31*b33 + u3*v3*b33
2 2 2 2 2 2
+ v1 *(b31 *c22 + b33 *c22) + v2 *b33 *c22)/b33
has apart from the Hamiltonian and Casimirs only the following first integral:
2 3 6
FI=u1 *u2*u3*a23 *b33 *c22
2 2 1 2 2 4 2 1 2 6 2
+ u1 *u3 *(---*a23 *b31 *b33 *c22 - ---*a23 *b33 *c22 )
2 2
1 2 2 6 1 2 2 7
+ ---*u1 *u3*v1*a23 *b31*b33 *c22 + ---*u1 *u3*v3*a23 *b33 *c22
2 2
1 2 5 2 1 6 2
+ ---*u1*u3*v1*v2*a23*b31 *b33 *c22 + ---*u1*u3*v2*v3*a23*b31*b33 *c22
2 2
3 3 6
+ u2 *u3*a23 *b33 *c22
2 2 2 2 4 2 3 2 8 2 6 2
+ u2 *u3 *( - a23 *b31 *b33 *c22 + ---*a23 *b33 - 2*a23 *b33 *c22 )
8
1 2 2 6 1 2 2 7 3
+ ---*u2 *u3*v1*a23 *b31*b33 *c22 + ---*u2 *u3*v3*a23 *b33 *c22 + u2*u3 *(
2 2
3 6 3 4 2 3 3 2 6
a23 *b33 *c22 - ---*a23*b31 *b33 *c22 + ---*a23*b31 *b33 *c22
2 8
3 8 3 6 3 2
- ---*a23*b33 *c22 + ---*a23*b33 *c22 ) + u2*u3 *v1
8 2
3 3 4 2 3 8 3 6 2
*( - ---*a23*b31 *b33 *c22 + ---*a23*b31*b33 - ---*a23*b31*b33 *c22 )
2 8 2
2 3 2 5 2 3 9 3 7 2
+ u2*u3 *v3*( - ---*a23*b31 *b33 *c22 + ---*a23*b33 - ---*a23*b33 *c22 )
2 8 2
1 7 2 2 4 3
+ ---*u2*u3*v1*v3*a23*b31*b33 *c22 - u2*u3*v2 *a23*b31 *b33 *c22
4
2 2 4 3 1 8 6 3 4
+ u2*u3*v3 *(a23*b31 *b33 *c22 - ---*a23*b33 *c22 + a23*b33 *c22 ) + u3 *(
4
1 2 2 4 2 1 2 6 2 3 6 4
---*a23 *b31 *b33 *c22 - ---*a23 *b33 *c22 - ---*b31 *c22
2 2 8
3 4 4 2 3 4 2 4 3 2 6 2
+ ----*b31 *b33 *c22 + ---*b31 *b33 *c22 - ----*b31 *b33 *c22
32 8 16
3 2 4 4 3 8 2 3 6 4 3
+ ---*b31 *b33 *c22 + ----*b33 *c22 - ---*b33 *c22 ) + u3 *v1*(
8 32 8
1 2 6 3 5 2 3 3 3 6
---*a23 *b31*b33 *c22 - ---*b31 *b33 *c22 + ----*b31 *b33 *c22
2 4 16
3 8 3 6 3 3 1 2 7
- ----*b31*b33 *c22 + ---*b31*b33 *c22 ) + u3 *v3*(---*a23 *b33 *c22
16 4 2
3 4 3 3 3 2 7 3 9 3 7 3
- ---*b31 *b33 *c22 + ----*b31 *b33 *c22 - ----*b33 *c22 + ---*b33 *c22
4 16 16 4
2 2 3 4 4 2 3 2 8 1 2 6 2
) + u3 *v1 *( - ---*b31 *b33 *c22 + ----*b31 *b33 - ---*b31 *b33 *c22 )
8 32 4
2 3 3 5 2 3 9 3 7 2
+ u3 *v1*v3*( - ---*b31 *b33 *c22 + ----*b31*b33 - ---*b31*b33 *c22 )
4 16 4
2 2 1 4 2 4 1 2 4 4 2 2
+ u3 *v2 *( - ---*b31 *b33 *c22 + ---*b31 *b33 *c22 ) + u3 *v3 *(
2 2
1 4 2 4 5 2 6 2 3 10 1 8 2
---*b31 *b33 *c22 - ---*b31 *b33 *c22 + ----*b33 - ---*b33 *c22
2 8 32 4
1 6 4
- ---*b33 *c22 )
2
2 1 4 3 3 1 2 7 1 2 5 3
+ u3*v1 *v3*( - ---*b31 *b33 *c22 + ---*b31 *b33 *c22 - ---*b31 *b33 *c22 )
2 8 2
3 1 2 5 3 1 9 1 7 3
+ u3*v3 *(---*b31 *b33 *c22 - ---*b33 *c22 + ---*b33 *c22 )
2 8 2
1
= a product of the elements of: {----,
32
u3,
2 3 6 2 2 2 4 2 2 6 2
32*u1 *u2*a23 *b33 *c22 + u1 *u3*(16*a23 *b31 *b33 *c22 - 16*a23 *b33 *c22 )
2 2 6 2 2 7
+ 16*u1 *v1*a23 *b31*b33 *c22 + 16*u1 *v3*a23 *b33 *c22
2 5 2 6 2
+ 16*u1*v1*v2*a23*b31 *b33 *c22 + 16*u1*v2*v3*a23*b31*b33 *c22
3 3 6
+ 32*u2 *a23 *b33 *c22
2 2 2 4 2 2 8 2 6 2
+ u2 *u3*( - 32*a23 *b31 *b33 *c22 + 12*a23 *b33 - 64*a23 *b33 *c22 )
2 2 6 2 2 7 2
+ 16*u2 *v1*a23 *b31*b33 *c22 + 16*u2 *v3*a23 *b33 *c22 + u2*u3 *(
3 6 4 2 3 2 6
32*a23 *b33 *c22 - 48*a23*b31 *b33 *c22 + 12*a23*b31 *b33 *c22
8 6 3
- 12*a23*b33 *c22 + 48*a23*b33 *c22 ) + u2*u3*v1
3 4 2 8 6 2
*( - 48*a23*b31 *b33 *c22 + 12*a23*b31*b33 - 48*a23*b31*b33 *c22 )
2 5 2 9 7 2
+ u2*u3*v3*( - 48*a23*b31 *b33 *c22 + 12*a23*b33 - 48*a23*b33 *c22 )
7 2 2 4 3
+ 8*u2*v1*v3*a23*b31*b33 *c22 - 32*u2*v2 *a23*b31 *b33 *c22
2 2 4 3 8 6 3 3
+ u2*v3 *(32*a23*b31 *b33 *c22 - 8*a23*b33 *c22 + 32*a23*b33 *c22 ) + u3 *(
2 2 4 2 2 6 2 6 4
16*a23 *b31 *b33 *c22 - 16*a23 *b33 *c22 - 12*b31 *c22
4 4 2 4 2 4 2 6 2
+ 3*b31 *b33 *c22 + 12*b31 *b33 *c22 - 6*b31 *b33 *c22
2 4 4 8 2 6 4 2
+ 12*b31 *b33 *c22 + 3*b33 *c22 - 12*b33 *c22 ) + u3 *v1*(
2 6 5 2 3 3 6
16*a23 *b31*b33 *c22 - 24*b31 *b33 *c22 + 6*b31 *b33 *c22
8 6 3 2 2 7
- 6*b31*b33 *c22 + 24*b31*b33 *c22 ) + u3 *v3*(16*a23 *b33 *c22
4 3 3 2 7 9 7 3
- 24*b31 *b33 *c22 + 6*b31 *b33 *c22 - 6*b33 *c22 + 24*b33 *c22 )
2 4 4 2 2 8 2 6 2
+ u3*v1 *( - 12*b31 *b33 *c22 + 3*b31 *b33 - 8*b31 *b33 *c22 )
3 5 2 9 7 2
+ u3*v1*v3*( - 24*b31 *b33 *c22 + 6*b31*b33 - 24*b31*b33 *c22 )
2 4 2 4 2 4 4 2
+ u3*v2 *( - 16*b31 *b33 *c22 + 16*b31 *b33 *c22 ) + u3*v3 *(
4 2 4 2 6 2 10 8 2
16*b31 *b33 *c22 - 20*b31 *b33 *c22 + 3*b33 - 8*b33 *c22
6 4
- 16*b33 *c22 )
2 4 3 3 2 7 2 5 3
+ v1 *v3*( - 16*b31 *b33 *c22 + 4*b31 *b33 *c22 - 16*b31 *b33 *c22 )
3 2 5 3 9 7 3
+ v3 *(16*b31 *b33 *c22 - 4*b33 *c22 + 16*b33 *c22 )}
{HAM,FI} = 0
And again in machine readable form:
HAM=(2*u2*u3*a23*b33**2 + u3**2*(b31**2*c22 - b33**2*c22) + u3*v1*b31*b33**2 +
u3*v3*b33**3 + v1**2*(b31**2*c22 + b33**2*c22) + v2**2*b33**2*c22)/b33**2$
FI=u1**2*u2*u3*a23**3*b33**6*c22 + u1**2*u3**2*(1/2*a23**2*b31**2*b33**4*c22**2
- 1/2*a23**2*b33**6*c22**2) + 1/2*u1**2*u3*v1*a23**2*b31*b33**6*c22 + 1/2*u1**2*
u3*v3*a23**2*b33**7*c22 + 1/2*u1*u3*v1*v2*a23*b31**2*b33**5*c22**2 + 1/2*u1*u3*
v2*v3*a23*b31*b33**6*c22**2 + u2**3*u3*a23**3*b33**6*c22 + u2**2*u3**2*( - a23**
2*b31**2*b33**4*c22**2 + 3/8*a23**2*b33**8 - 2*a23**2*b33**6*c22**2) + 1/2*u2**2
*u3*v1*a23**2*b31*b33**6*c22 + 1/2*u2**2*u3*v3*a23**2*b33**7*c22 + u2*u3**3*(a23
**3*b33**6*c22 - 3/2*a23*b31**4*b33**2*c22**3 + 3/8*a23*b31**2*b33**6*c22 - 3/8*
a23*b33**8*c22 + 3/2*a23*b33**6*c22**3) + u2*u3**2*v1*( - 3/2*a23*b31**3*b33**4*
c22**2 + 3/8*a23*b31*b33**8 - 3/2*a23*b31*b33**6*c22**2) + u2*u3**2*v3*( - 3/2*
a23*b31**2*b33**5*c22**2 + 3/8*a23*b33**9 - 3/2*a23*b33**7*c22**2) + 1/4*u2*u3*
v1*v3*a23*b31*b33**7*c22 - u2*u3*v2**2*a23*b31**2*b33**4*c22**3 + u2*u3*v3**2*(
a23*b31**2*b33**4*c22**3 - 1/4*a23*b33**8*c22 + a23*b33**6*c22**3) + u3**4*(1/2*
a23**2*b31**2*b33**4*c22**2 - 1/2*a23**2*b33**6*c22**2 - 3/8*b31**6*c22**4 + 3/
32*b31**4*b33**4*c22**2 + 3/8*b31**4*b33**2*c22**4 - 3/16*b31**2*b33**6*c22**2 +
3/8*b31**2*b33**4*c22**4 + 3/32*b33**8*c22**2 - 3/8*b33**6*c22**4) + u3**3*v1*(
1/2*a23**2*b31*b33**6*c22 - 3/4*b31**5*b33**2*c22**3 + 3/16*b31**3*b33**6*c22 -
3/16*b31*b33**8*c22 + 3/4*b31*b33**6*c22**3) + u3**3*v3*(1/2*a23**2*b33**7*c22 -
3/4*b31**4*b33**3*c22**3 + 3/16*b31**2*b33**7*c22 - 3/16*b33**9*c22 + 3/4*b33**
7*c22**3) + u3**2*v1**2*( - 3/8*b31**4*b33**4*c22**2 + 3/32*b31**2*b33**8 - 1/4*
b31**2*b33**6*c22**2) + u3**2*v1*v3*( - 3/4*b31**3*b33**5*c22**2 + 3/16*b31*b33
**9 - 3/4*b31*b33**7*c22**2) + u3**2*v2**2*( - 1/2*b31**4*b33**2*c22**4 + 1/2*
b31**2*b33**4*c22**4) + u3**2*v3**2*(1/2*b31**4*b33**2*c22**4 - 5/8*b31**2*b33**
6*c22**2 + 3/32*b33**10 - 1/4*b33**8*c22**2 - 1/2*b33**6*c22**4) + u3*v1**2*v3*(
- 1/2*b31**4*b33**3*c22**3 + 1/8*b31**2*b33**7*c22 - 1/2*b31**2*b33**5*c22**3)
+ u3*v3**3*(1/2*b31**2*b33**5*c22**3 - 1/8*b33**9*c22 + 1/2*b33**7*c22**3)$