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