Solution 4 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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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
- ---*b12 *kap*r494
2
r430=----------------------
2
a22
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
r447=0
r449=0
r450=0
r452=0
b12*kap*r494
r453=--------------
a22
r454=0
r455=0
r456=0
1
r457=---*r494
2
r458=0
r459=0
1 2 1 2
- ---*a22 *r494 - ---*b12 *kap*r494
2 2
r460=--------------------------------------
2
a22
r462=0
r463=0
1
r464= - ---*kap*r494
2
r465=0
r466=0
2 2
a22 *r494 + b12 *kap*r494
r467=---------------------------
a22*b12
r468=0
1 2 1 2
- ---*a22 *r494 - ---*b12 *kap*r494
2 2
r469=--------------------------------------
2
b12
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r479=0
r482=0
r484=0
r485=0
- b12*kap*r494
r487=-----------------
a22
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r495=0
r497=0
r498=0
r499=0
r4100=0
2 2
- a22 *r494 - b12 *kap*r494
r4101=------------------------------
a22*b12
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
1 2
- ---*b12 *kap*r494
2
r4110=----------------------
2
a22
r4112=0
r4113=0
1 2 2
---*b12 *kap *r494
2
r4114=--------------------
2
a22
r4115=0
r4116=0
r4117=0
r4118=0
1 2 1 2 2
---*a22 *kap*r494 + ---*b12 *kap *r494
2 2
r4119=----------------------------------------
2
a22
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
a33=0
a23=0
a13=0
2
- b12 *kap
a11=-------------
a22
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, a22, 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.
2 2
{a22 + b12 *kap,r494,b12,a22}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2
- u1 *b12 *kap + u1*v2*a22*b12 + u2 *a22 - u2*v1*a22*b12
HAM=------------------------------------------------------------
a22
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 1 2 2 1 4 2 1 2 2 4 2
FI=u1 *u2 *(---*a22 *b12 *kap + ---*b12 *kap ) + ---*u1 *u3 *b12 *kap
2 2 2
1 2 2 4 2 3 3
- ---*u1 *v1 *b12 *kap + u1*u2 *v2*( - a22 *b12 - a22*b12 *kap)
2
2 2 2 3
+ u1*u2*v1*v2*a22 *b12 - u1*u3 *v2*a22*b12 *kap
4 1 4 1 2 2 3 3 3
+ u2 *( - ---*a22 - ---*a22 *b12 *kap) + u2 *v1*(a22 *b12 + a22*b12 *kap)
2 2
1 2 2 2 2 2 2 1 2 2 1 4
- ---*u2 *u3 *a22 *b12 *kap + u2 *v1 *( - ---*a22 *b12 - ---*b12 *kap)
2 2 2
1 2 2 2 2 2 3 1 2 2 4
+ ---*u2 *v2 *a22 *b12 + u2*u3 *v1*a22*b12 *kap - ---*u3 *v1 *b12 *kap
2 2
1 2 2 2 2
+ ---*u3 *v2 *a22 *b12
2
3 5 2 2 5 2 2 2 5 2
{HAM,FI} = u1 *u3*v1*b12 *kap + u1 *u2*u3*v2*b12 *kap + u1 *u3 *v3*b12 *kap
2 4 2 2 3
- 2*u1 *u3*v1*v2*a22*b12 *kap - u1*u2 *u3*v1*a22 *b12 *kap
2 4 2 4
+ 2*u1*u2*u3*v1 *a22*b12 *kap - 2*u1*u2*u3*v2 *a22*b12 *kap
2 4 3 5
- 2*u1*u3 *v2*v3*a22*b12 *kap - u1*u3*v1 *b12 *kap
2 2 3 3 2 3
+ u1*u3*v1*v2 *a22 *b12 - u2 *u3*v2*a22 *b12 *kap
2 2 2 3 2 4
- u2 *u3 *v3*a22 *b12 *kap + 2*u2 *u3*v1*v2*a22*b12 *kap
2 4 2 5
+ 2*u2*u3 *v1*v3*a22*b12 *kap - u2*u3*v1 *v2*b12 *kap
3 2 3 2 2 5 2 2 2 3
+ u2*u3*v2 *a22 *b12 - u3 *v1 *v3*b12 *kap + u3 *v2 *v3*a22 *b12
And again in machine readable form:
HAM=( - u1**2*b12**2*kap + u1*v2*a22*b12 + u2**2*a22**2 - u2*v1*a22*b12)/a22$
FI=u1**2*u2**2*(1/2*a22**2*b12**2*kap + 1/2*b12**4*kap**2) + 1/2*u1**2*u3**2*b12
**4*kap**2 - 1/2*u1**2*v1**2*b12**4*kap + u1*u2**2*v2*( - a22**3*b12 - a22*b12**
3*kap) + u1*u2*v1*v2*a22**2*b12**2 - u1*u3**2*v2*a22*b12**3*kap + u2**4*( - 1/2*
a22**4 - 1/2*a22**2*b12**2*kap) + u2**3*v1*(a22**3*b12 + a22*b12**3*kap) - 1/2*
u2**2*u3**2*a22**2*b12**2*kap + u2**2*v1**2*( - 1/2*a22**2*b12**2 - 1/2*b12**4*
kap) + 1/2*u2**2*v2**2*a22**2*b12**2 + u2*u3**2*v1*a22*b12**3*kap - 1/2*u3**2*v1
**2*b12**4*kap + 1/2*u3**2*v2**2*a22**2*b12**2$