Solution 5 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=4*a33 - b12 *kap
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
r418=0
r421=0
r423=0
r424=0
1
r427=---*r494
2
r429=0
1 2 1 2
---*a33 *r494 - ---*b12 *kap*r494
2 8
r430=-----------------------------------
2
a33
r432=0
r433=0
2 3 2
a33 *r494 - ---*b12 *kap*r494
8
r434=-------------------------------
2
b12
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r447=0
r449=0
r450=0
r452=0
1
---*b12*kap*r494
4
r453=------------------
a33
r454=0
r455=0
r456=0
1
r457=---*r494
2
r458=0
r459=0
1 2
- ---*b12 *kap*r494
8
r460=----------------------
2
a33
r462=0
r463=0
r464=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
r479=0
r482=0
r484=0
r485=0
1
---*b12*kap*r494
4
r487=------------------
a33
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r495=0
r497=0
r498=0
r499=0
r4100=0
1
---*b12*kap*r494
2
r4101=------------------
a33
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
1 2 1 2
---*a33 *r494 - ---*b12 *kap*r494
2 8
r4110=-----------------------------------
2
a33
r4112=0
r4113=0
2 1 2 2
a33 *kap*r494 - ---*b12 *kap *r494
4
r4114=------------------------------------
2
a33
r4115=0
r4116=0
1
---*b12*kap*r494
2
r4117=------------------
a33
r4118=0
1 2 2
---*b12 *kap *r494
8
r4119=--------------------
2
a33
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
a23=0
a22=0
a13=0
1 2
---*b12 *kap
2
a11=--------------
a33
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:
r494, a33, b12
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.
{b12,r494,a33}
Relevance for the application:
Modulo the following equation:
2 2
0=4*a33 - b12 *kap
the system of equations related to the Hamiltonian HAM:
1 2 2 2 2
---*u1 *b12 *kap + u1*v2*a33*b12 - u2*v1*a33*b12 + u3 *a33
2
HAM=-------------------------------------------------------------
a33
has apart from the Hamiltonian and Casimirs only the following first integral:
1 2 2 4 2 1 2 3
FI=---*u1 *u2 *b12 *kap + ---*u1 *u2*v1*a33*b12 *kap
8 2
2 2 2 2 1 4 2
+ u1 *u3 *(a33 *b12 *kap - ---*b12 *kap )
4
2 2 1 2 2 1 4 1 2 3
+ u1 *v1 *(---*a33 *b12 - ---*b12 *kap) + ---*u1*u2 *v2*a33*b12 *kap
2 8 2
2 2 1 2 3 1 2 2 4
+ u1*u2*v1*v2*a33 *b12 + ---*u1*u3 *v2*a33*b12 *kap - ---*u2 *v1 *b12 *kap
4 8
1 2 2 2 2 1 2 3
+ ---*u2 *v2 *a33 *b12 + ---*u2*u3 *v1*a33*b12 *kap
2 4
4 4 3 2 2 2 2 1 2 2 1 4
+ u3 *(a33 - ---*a33 *b12 *kap) + u3 *v1 *(---*a33 *b12 - ---*b12 *kap)
8 2 8
1 2 2 2 2
+ ---*u3 *v2 *a33 *b12
2
1 3 5 2 1 2 5 2
{HAM,FI} = ---*u1 *u3*v1*a33*b12 *kap + ---*u1 *u2*u3*v2*a33*b12 *kap
4 4
1 2 2 5 2 1 2 6 2
+ ---*u1 *u3 *v3*a33*b12 *kap + ---*u1 *u3*v1*v2*b12 *kap
4 4
1 2 6 2 1 2 6 2
+ ---*u1*u2*u3*v2 *b12 *kap + ---*u1*u3 *v2*v3*b12 *kap
4 4
1 2 5 1 3 5
+ ---*u1*u3*v1*v2 *a33*b12 *kap + ---*u2*u3*v2 *a33*b12 *kap
4 4
1 2 2 5
+ ---*u3 *v2 *v3*a33*b12 *kap
4
And again in machine readable form:
HAM=(1/2*u1**2*b12**2*kap + u1*v2*a33*b12 - u2*v1*a33*b12 + u3**2*a33**2)/a33$
FI=1/8*u1**2*u2**2*b12**4*kap**2 + 1/2*u1**2*u2*v1*a33*b12**3*kap + u1**2*u3**2*
(a33**2*b12**2*kap - 1/4*b12**4*kap**2) + u1**2*v1**2*(1/2*a33**2*b12**2 - 1/8*
b12**4*kap) + 1/2*u1*u2**2*v2*a33*b12**3*kap + u1*u2*v1*v2*a33**2*b12**2 + 1/4*
u1*u3**2*v2*a33*b12**3*kap - 1/8*u2**2*v1**2*b12**4*kap + 1/2*u2**2*v2**2*a33**2
*b12**2 + 1/4*u2*u3**2*v1*a33*b12**3*kap + u3**4*(a33**4 - 3/8*a33**2*b12**2*kap
) + u3**2*v1**2*(1/2*a33**2*b12**2 - 1/8*b12**4*kap) + 1/2*u3**2*v2**2*a33**2*
b12**2$