Solution 3 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2
0=4*a23 + a33 - b32
2 2 2 2
0=8*a23 + 2*a33 - 24*a33*c22 + b32 + 48*c22
2 2 4 2 2 4
0=256*a23 *c22 + b32 - 32*b32 *c22 + 256*c22
Expressions
The solution is given through the following expressions:
1 2 3 2
- ---*a33 *r483 + ---*a33*c22*r483 - 2*c22 *r483
4 2
r40=---------------------------------------------------
a23*b32
r41=0
1 2 5 2
- ---*a33 *r483 + ---*a33*c22*r483 - 3*c22 *r483
2 2
r42=---------------------------------------------------
a23*b32
r43=0
1 2 2
- ---*a33 *r483 + a33*c22*r483 - c22 *r483
4
r44=---------------------------------------------
a23*b32
r45=0
r46=0
r47=0
r48=0
1 2 3 2
- ---*a33 *r483 + ---*a33*c22*r483 - 2*c22 *r483
4 2
r49=---------------------------------------------------
a23*b32
r410=0
1 2 2
- ---*a33 *r483 + a33*c22*r483 - c22 *r483
4
r411=---------------------------------------------
a23*b32
r412=0
r413=0
r415=0
1
- ---*a33*r483 + c22*r483
2
r416=----------------------------
a23
r417=0
2 1 2
2*a23 *r483 + a33*c22*r483 - ---*b32 *r483
4
r418=--------------------------------------------
a23*a33 - 4*a23*c22
r419=0
r420=0
r421=0
r422=0
1
- ---*a33*r483 + c22*r483
2
r423=----------------------------
a23
r424=0
1 2
- ---*a33 *r483
4
r425=------------------
a23*b32
r426=0
2 1 2 1 2 1 2
a23 *a33*r483 + ---*a33 *c22*r483 - ---*a33*b32 *r483 + ---*b32 *c22*r483
2 4 2
r427=---------------------------------------------------------------------------
a23*a33*b32 - 4*a23*b32*c22
r428=0
r429=0
1
- ---*b32*r483
4
r430=-----------------
a23
r431=0
r432=0
r433=0
r434=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
- a33*r483
r445=-------------
b32
r446=0
2 1 2
4*a23 *r483 + 2*a33*c22*r483 - ---*b32 *r483
2
r447=----------------------------------------------
a33*b32 - 4*b32*c22
r448=0
r449=0
r451=0
r452=0
r453=0
r454=0
1 2 1 2
---*a33 *r483 - ---*b32 *r483
4 4
r455=-------------------------------
a23*b32
r456=0
2 1 2 1 2
- a23 *a33*r483 - ---*a33 *c22*r483 + ---*b32 *c22*r483
2 2
r457=----------------------------------------------------------
a23*a33*b32 - 4*a23*b32*c22
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
r465=0
r466=0
r467=0
r468=0
r469=0
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r478=0
r479=0
r480=0
r481=0
r482=0
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=0
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
1 2 1 2
---*a33 *r483 - ---*b32 *r483
4 4
r4105=-------------------------------
a23*b32
r4106=0
2 1 2 1 2
- a23 *a33*r483 - ---*a33 *c22*r483 + ---*b32 *c22*r483
2 2
r4107=----------------------------------------------------------
a23*a33*b32 - 4*a23*b32*c22
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
1
c11= - ---*a33 + 2*c22
2
b33=0
b31=0
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:
r483, a33, b32, 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.
{c22,a23,r483,b32,a33 - 4*c22}
Relevance for the application:
Modulo the following equations:
2 2 2
0=4*a23 + a33 - b32
2 2 2 2
0=8*a23 + 2*a33 - 24*a33*c22 + b32 + 48*c22
2 2 4 2 2 4
0=256*a23 *c22 + b32 - 32*b32 *c22 + 256*c22
the system of equations related to the Hamiltonian HAM:
2 2 1 2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v2*b32 + v1 *( - ---*a33 + 2*c22) + v2 *c22
2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 1 2 1 2
FI=u1 *v2 *( - a23 *a33 - ---*a33 *c22 + ---*b32 *c22)
2 2
2 2 1 3 2 1 2 2
+ u1 *v3 *(---*a33 - a33 *c22 - ---*a33*b32 + b32 *c22)
4 4
+ u1*u3*v1*v3*(a23*a33*b32 - 4*a23*b32*c22)
2 2 2 1 2 1 2
+ u2 *v2 *( - a23 *a33 - ---*a33 *c22 + ---*b32 *c22)
2 2
2 2 1 3 2 1 2 2
+ u2 *v3 *(---*a33 - a33 *c22 - ---*a33*b32 + b32 *c22)
4 4
2 3 1 2
+ u2*u3*v2 *(4*a23 + 2*a23*a33*c22 - ---*a23*b32 )
2
2 2
+ u2*u3*v3 *( - a23*a33 + 4*a23*a33*c22)
2 2 1 2 2
+ u3 *v1 *( - ---*a33*b32 + b32 *c22)
4
2 2 2 1 2 1 2 1 2
+ u3 *v2 *(a23 *a33 + ---*a33 *c22 - ---*a33*b32 + ---*b32 *c22)
2 4 2
2 2 1 3 2
+ u3 *v3 *( - ---*a33 + a33 *c22)
4
2 1 2 2
+ u3*v1 *v2*( - ---*a33 *b32 + 3*a33*b32*c22 - 4*b32*c22 )
2
3 2 1 3
+ u3*v2 *(2*a23 *b32 + a33*b32*c22 - ---*b32 )
4
2 1 2 2
+ u3*v2*v3 *( - ---*a33 *b32 + 3*a33*b32*c22 - 4*b32*c22 )
2
2 2 1 3 2 2 3
+ v1 *v2 *( - ---*a33 + 2*a33 *c22 - 5*a33*c22 + 4*c22 )
4
2 2 1 3 5 2 2 3
+ v1 *v3 *( - ---*a33 + ---*a33 *c22 - 8*a33*c22 + 8*c22 )
4 2
4 1 3 2 2 3
+ v2 *( - ---*a33 + 2*a33 *c22 - 5*a33*c22 + 4*c22 )
4
2 2 1 3 9 2 2 3
+ v2 *v3 *( - ---*a33 + ---*a33 *c22 - 13*a33*c22 + 12*c22 )
2 2
4 1 3 5 2 2 3
+ v3 *( - ---*a33 + ---*a33 *c22 - 8*a33*c22 + 8*c22 )
4 2
1
= a product of the elements of: {---,
4
2 2 2 2 2
u1 *v2 *( - 4*a23 *a33 - 2*a33 *c22 + 2*b32 *c22)
2 2 3 2 2 2
+ u1 *v3 *(a33 - 4*a33 *c22 - a33*b32 + 4*b32 *c22)
+ u1*u3*v1*v3*(4*a23*a33*b32 - 16*a23*b32*c22)
2 2 2 2 2
+ u2 *v2 *( - 4*a23 *a33 - 2*a33 *c22 + 2*b32 *c22)
2 2 3 2 2 2
+ u2 *v3 *(a33 - 4*a33 *c22 - a33*b32 + 4*b32 *c22)
2 3 2
+ u2*u3*v2 *(16*a23 + 8*a23*a33*c22 - 2*a23*b32 )
2 2
+ u2*u3*v3 *( - 4*a23*a33 + 16*a23*a33*c22)
2 2 2 2
+ u3 *v1 *( - a33*b32 + 4*b32 *c22)
2 2 2 2 2 2
+ u3 *v2 *(4*a23 *a33 + 2*a33 *c22 - a33*b32 + 2*b32 *c22)
2 2 3 2
+ u3 *v3 *( - a33 + 4*a33 *c22)
2 2 2
+ u3*v1 *v2*( - 2*a33 *b32 + 12*a33*b32*c22 - 16*b32*c22 )
3 2 3
+ u3*v2 *(8*a23 *b32 + 4*a33*b32*c22 - b32 )
2 2 2
+ u3*v2*v3 *( - 2*a33 *b32 + 12*a33*b32*c22 - 16*b32*c22 )
2 2 3 2 2 3
+ v1 *v2 *( - a33 + 8*a33 *c22 - 20*a33*c22 + 16*c22 )
2 2 3 2 2 3
+ v1 *v3 *( - a33 + 10*a33 *c22 - 32*a33*c22 + 32*c22 )
4 3 2 2 3
+ v2 *( - a33 + 8*a33 *c22 - 20*a33*c22 + 16*c22 )
2 2 3 2 2 3
+ v2 *v3 *( - 2*a33 + 18*a33 *c22 - 52*a33*c22 + 48*c22 )
4 3 2 2 3
+ v3 *( - a33 + 10*a33 *c22 - 32*a33*c22 + 32*c22 )}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v2*b32 + v1**2*( - 1/2*a33 + 2*c22) + v2**2*c22
$
FI=u1**2*v2**2*( - a23**2*a33 - 1/2*a33**2*c22 + 1/2*b32**2*c22) + u1**2*v3**2*(
1/4*a33**3 - a33**2*c22 - 1/4*a33*b32**2 + b32**2*c22) + u1*u3*v1*v3*(a23*a33*
b32 - 4*a23*b32*c22) + u2**2*v2**2*( - a23**2*a33 - 1/2*a33**2*c22 + 1/2*b32**2*
c22) + u2**2*v3**2*(1/4*a33**3 - a33**2*c22 - 1/4*a33*b32**2 + b32**2*c22) + u2*
u3*v2**2*(4*a23**3 + 2*a23*a33*c22 - 1/2*a23*b32**2) + u2*u3*v3**2*( - a23*a33**
2 + 4*a23*a33*c22) + u3**2*v1**2*( - 1/4*a33*b32**2 + b32**2*c22) + u3**2*v2**2*
(a23**2*a33 + 1/2*a33**2*c22 - 1/4*a33*b32**2 + 1/2*b32**2*c22) + u3**2*v3**2*(
- 1/4*a33**3 + a33**2*c22) + u3*v1**2*v2*( - 1/2*a33**2*b32 + 3*a33*b32*c22 - 4*
b32*c22**2) + u3*v2**3*(2*a23**2*b32 + a33*b32*c22 - 1/4*b32**3) + u3*v2*v3**2*(
- 1/2*a33**2*b32 + 3*a33*b32*c22 - 4*b32*c22**2) + v1**2*v2**2*( - 1/4*a33**3 +
2*a33**2*c22 - 5*a33*c22**2 + 4*c22**3) + v1**2*v3**2*( - 1/4*a33**3 + 5/2*a33
**2*c22 - 8*a33*c22**2 + 8*c22**3) + v2**4*( - 1/4*a33**3 + 2*a33**2*c22 - 5*a33
*c22**2 + 4*c22**3) + v2**2*v3**2*( - 1/2*a33**3 + 9/2*a33**2*c22 - 13*a33*c22**
2 + 12*c22**3) + v3**4*( - 1/4*a33**3 + 5/2*a33**2*c22 - 8*a33*c22**2 + 8*c22**3
)$