Solution 5 to problem h
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem h
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
r423=0
r424=0
r426=0
2
b12 *kap*r4110
r427=----------------
2
a11
r428=0
r429=0
r430= - r4110
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
r445=0
2
- 2*b12 *kap*r4110
r446=---------------------
2
a11
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
- 2*b12*kap*r4110
r453=--------------------
a11
r454=0
r455=0
r456=0
2
- b12 *kap*r4110
r457=-------------------
2
a11
r458=0
r459=0
2 2
- a11 *r4110 - b12 *kap*r4110
r460=--------------------------------
2
a11
r461=0
r462=0
r463=0
2 2
- b12 *kap *r4110
r464=--------------------
2
a11
r465=0
r466=0
2 3 2
- 2*a11 *b12*kap*r4110 - 2*b12 *kap *r4110
r467=---------------------------------------------
3
a11
r468=0
2 2 2 4 3
- a11 *b12 *kap *r4110 - b12 *kap *r4110
r469=-------------------------------------------
4
a11
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r479=0
r480=0
r482=0
r483=2*r4110
r484=0
r485=0
r486=0
2*b12*kap*r4110
r487=-----------------
a11
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=2*r4110
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
2 3 2
2*a11 *b12*kap*r4110 + 2*b12 *kap *r4110
r4101=------------------------------------------
3
a11
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4114=kap*r4110
r4115=0
r4116=0
r4117=0
r4118=0
2 2 2
a11 *kap*r4110 + b12 *kap *r4110
r4119=----------------------------------
2
a11
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
a33=0
a23=0
2
- b12 *kap
a22=-------------
a11
a13=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:
r4110, a11, 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
{a11,a11 + b12 *kap,b12,r4110,a22}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2
u1 *a11 + u1*v2*a11*b12 - u2 *b12 *kap - u2*v1*a11*b12
HAM=---------------------------------------------------------
a11
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 4 2 2 2 2 2 4 2 2 4
FI=u1 *u2 *(a11 *kap + a11 *b12 *kap ) + u1 *u3 *a11 *kap + u1 *v1 *a11
2 3 3 2 4
+ u1*u2 *v2*(2*a11 *b12*kap + 2*a11*b12 *kap ) + 2*u1*u2*v1*v2*a11
2 3 4
+ 2*u1*u3 *v2*a11 *b12*kap + 2*u1*u3*v1*v3*a11
4 2 2 2 4 3
+ u2 *( - a11 *b12 *kap - b12 *kap )
3 3 3 2 2 2 2 2 2
+ u2 *v1*( - 2*a11 *b12*kap - 2*a11*b12 *kap ) - u2 *u3 *a11 *b12 *kap
2 2 4 2 2 2 2 2 2
+ u2 *v1 *( - a11 - a11 *b12 *kap) - u2 *v2 *a11 *b12 *kap
2 3 2 2 2 2 4
- 2*u2*u3 *v1*a11 *b12*kap - 2*u2*u3*v2*v3*a11 *b12 *kap - u3 *v1 *a11
2 2 2 2
+ u3 *v2 *a11 *b12 *kap
which is not further factorizable.
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a11**2 + u1*v2*a11*b12 - u2**2*b12**2*kap - u2*v1*a11*b12)/a11$
FI=u1**2*u2**2*(a11**4*kap + a11**2*b12**2*kap**2) + u1**2*u3**2*a11**4*kap + u1
**2*v1**2*a11**4 + u1*u2**2*v2*(2*a11**3*b12*kap + 2*a11*b12**3*kap**2) + 2*u1*
u2*v1*v2*a11**4 + 2*u1*u3**2*v2*a11**3*b12*kap + 2*u1*u3*v1*v3*a11**4 + u2**4*(
- a11**2*b12**2*kap**2 - b12**4*kap**3) + u2**3*v1*( - 2*a11**3*b12*kap - 2*a11*
b12**3*kap**2) - u2**2*u3**2*a11**2*b12**2*kap**2 + u2**2*v1**2*( - a11**4 - a11
**2*b12**2*kap) - u2**2*v2**2*a11**2*b12**2*kap - 2*u2*u3**2*v1*a11**3*b12*kap -
2*u2*u3*v2*v3*a11**2*b12**2*kap - u3**2*v1**2*a11**4 + u3**2*v2**2*a11**2*b12**
2*kap$