Solution 4 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=a23 + c22
2 2 2
0=4*a23 + a33 - b31
Expressions
The solution is given through the following expressions:
r40=0
r41=0
r42=0
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
r49=0
r410=0
r411=0
r412=0
r413=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
1 1
- ---*a23*a33*r481 - ---*a23*c22*r481
4 2
r425=----------------------------------------
b31*c22
r426=0
1 1
- ---*a23*a33*r481 + ---*a23*c22*r481
4 2
r427=----------------------------------------
b31*c22
r428=0
r429=0
r430=0
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
1
---*a33*r481 + c22*r481
2
r445=-------------------------
b31
r446=0
1
- ---*a33*r481 + c22*r481
2
r447=----------------------------
b31
r448=0
r449=0
r451=0
r452=0
r453=0
r454=0
1 2 1 2 1 2
r455=( - ---*a23 *a33*r481 - ---*a23 *c22*r481 - ---*a33 *c22*r481
4 2 4
1 2 1 2
- ---*a33*c22 *r481 + ---*b31 *c22*r481)/(a23*b31*c22)
2 4
r456=0
1 2 1 2 1 2
r457=( - ---*a23 *a33*r481 + ---*a23 *c22*r481 + ---*a33 *c22*r481
4 2 4
1 2 1 2
- ---*a33*c22 *r481 - ---*b31 *c22*r481)/(a23*b31*c22)
2 4
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
r482=0
r483=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 1 2
r4105=( - ---*a23 *a33*r481 - ---*a23 *c22*r481 - ---*a33 *c22*r481
4 2 4
1 2 1 2
- ---*a33*c22 *r481 + ---*b31 *c22*r481)/(a23*b31*c22)
2 4
r4106=0
1 2 1 2 1 2
r4107=( - ---*a23 *a33*r481 + ---*a23 *c22*r481 + ---*a33 *c22*r481
4 2 4
1 2 1 2
- ---*a33*c22 *r481 - ---*b31 *c22*r481)/(a23*b31*c22)
2 4
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 1
c11=---*a33 + ---*c22
4 2
b33=0
b32=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:
r481, a33, 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,r481,a23,c22}
Relevance for the application:
Modulo the following equations:
2 2
0=a23 + c22
2 2 2
0=4*a23 + a33 - b31
the system of equations related to the Hamiltonian HAM:
2 2 1 1 2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v1*b31 + v1 *(---*a33 + ---*c22) + v2 *c22
4 2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 1 2 1 2 1 2 1 2
FI=u1 *v2 *( - ---*a23 *a33 + ---*a23 *c22 + ---*a33 *c22 - ---*a33*c22
4 2 4 2
1 2 2 2 1 2 1 2
- ---*b31 *c22) + u1 *v3 *( - ---*a23 *a33 - ---*a23 *c22
4 4 2
1 2 1 2 1 2
- ---*a33 *c22 - ---*a33*c22 + ---*b31 *c22) + u1*u3*v2*v3*a23*b31*c22 +
4 2 4
2 2 1 2 1 2 1 2 1 2
u2 *v2 *( - ---*a23 *a33 + ---*a23 *c22 + ---*a33 *c22 - ---*a33*c22
4 2 4 2
1 2 2 2 1 2 1 2
- ---*b31 *c22) + u2 *v3 *( - ---*a23 *a33 - ---*a23 *c22
4 4 2
1 2 1 2 1 2
- ---*a33 *c22 - ---*a33*c22 + ---*b31 *c22)
4 2 4
2 1 2
+ u2*u3*v2 *( - ---*a23*a33*c22 + a23*c22 )
2
2 1 2
+ u2*u3*v3 *(---*a23*a33*c22 + a23*c22 )
2
2 2 1 2 1 2
+ u3 *v2 *( - ---*a23 *a33 + ---*a23 *c22)
4 2
2 2 1 2 1 2
+ u3 *v3 *( - ---*a23 *a33 - ---*a23 *c22)
4 2
1
= a product of the elements of: { - ---,
4
2 2 2 2 2 2 2
u1 *v2 *(a23 *a33 - 2*a23 *c22 - a33 *c22 + 2*a33*c22 + b31 *c22)
2 2 2 2 2 2 2
+ u1 *v3 *(a23 *a33 + 2*a23 *c22 + a33 *c22 + 2*a33*c22 - b31 *c22)
- 4*u1*u3*v2*v3*a23*b31*c22
2 2 2 2 2 2 2
+ u2 *v2 *(a23 *a33 - 2*a23 *c22 - a33 *c22 + 2*a33*c22 + b31 *c22)
2 2 2 2 2 2 2
+ u2 *v3 *(a23 *a33 + 2*a23 *c22 + a33 *c22 + 2*a33*c22 - b31 *c22)
2 2
+ u2*u3*v2 *(2*a23*a33*c22 - 4*a23*c22 )
2 2
+ u2*u3*v3 *( - 2*a23*a33*c22 - 4*a23*c22 )
2 2 2 2 2 2 2 2
+ u3 *v2 *(a23 *a33 - 2*a23 *c22) + u3 *v3 *(a23 *a33 + 2*a23 *c22)}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v1*b31 + v1**2*(1/4*a33 + 1/2*c22) + v2**2*c22$
FI=u1**2*v2**2*( - 1/4*a23**2*a33 + 1/2*a23**2*c22 + 1/4*a33**2*c22 - 1/2*a33*
c22**2 - 1/4*b31**2*c22) + u1**2*v3**2*( - 1/4*a23**2*a33 - 1/2*a23**2*c22 - 1/4
*a33**2*c22 - 1/2*a33*c22**2 + 1/4*b31**2*c22) + u1*u3*v2*v3*a23*b31*c22 + u2**2
*v2**2*( - 1/4*a23**2*a33 + 1/2*a23**2*c22 + 1/4*a33**2*c22 - 1/2*a33*c22**2 - 1
/4*b31**2*c22) + u2**2*v3**2*( - 1/4*a23**2*a33 - 1/2*a23**2*c22 - 1/4*a33**2*
c22 - 1/2*a33*c22**2 + 1/4*b31**2*c22) + u2*u3*v2**2*( - 1/2*a23*a33*c22 + a23*
c22**2) + u2*u3*v3**2*(1/2*a23*a33*c22 + a23*c22**2) + u3**2*v2**2*( - 1/4*a23**
2*a33 + 1/2*a23**2*c22) + u3**2*v3**2*( - 1/4*a23**2*a33 - 1/2*a23**2*c22)$