Solution 5 to problem h
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem h
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
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r423=0
r424=0
r426=0
r427= - r4110 - r460
r428=0
r429=0
r430= - r4110
r431=r488
r432=0
r433=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r446=2*r4110 + 2*r460
r447=0
r448=0
r449=0
r450=0
r451=0
r452=r488
- 4*a33*r460 - b12*r487
r453=--------------------------
b12
r454=0
r455=0
r456=0
r457=r4110 + r460
r458=0
r459=0
r461=0
r462=0
r463=0
2 2 2
4*a33 *r460 + 2*a33*b12*r487 - b12 *kap*r4110 - b12 *kap*r460
r464=---------------------------------------------------------------
2
b12
r465=0
r466=0
- 4*a33*r460
r467=---------------
b12
r468=0
2
4*a33 *r460
r469=-------------
2
b12
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
r489=0
r490=0
r491=0
r492=0
r493=0
r494=2*r4110
r495=0
4*a33*r460
r496=------------
b12
r497=0
r498=0
r499=0
r4100=0
4*a33*r460
r4101=------------
b12
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
2 2
4*a33 *r460 + 2*a33*b12*r487 - b12 *kap*r4110
r4114=-----------------------------------------------
2
b12
r4115=0
r4116=0
r4117=0
r4118=0
2
4*a33 *r460
r4119=-------------
2
b12
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
a23=0
a22=2*a33
a13=0
a11=2*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:
r434, r487, r4110, r460, r488, 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.
{r460,a33,b12,r488}
Relevance for the application:
Modulo the following equation:
2 2
0=4*a33 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2 2
HAM=2*u1 *a33 + u1*v2*b12 + 2*u2 *a33 - u2*v1*b12 + u3 *a33
has apart from the Hamiltonian and Casimirs the following 5 first integrals:
2 2 3
FI=u1*u3 *v1 + u2*u3 *v2 + u3 *v3
2
=u3 *(u2*v2 + u3*v3 + u1*v1)
{HAM,FI} = 0
2 2 2 2 2 2 2
FI=4*u1 *u2 *a33 + 4*u1 *u3 *a33 + 4*u1*u2 *v2*a33*b12 + 4*u1*u2*u3*v3*a33*b12
4 2 3 2 2 2 2 2 2 2
+ 4*u2 *a33 - 4*u2 *v1*a33*b12 + u2 *u3 *(4*a33 - b12 *kap) + u2 *v1 *b12
2 2 2 2 2 2 2 2
+ u2 *v2 *b12 - 4*u2*u3 *v1*a33*b12 + 2*u2*u3*v2*v3*b12 - u3 *v2 *b12
which is not further factorizable.
{HAM,FI} = 0
2 2 2 2 2 2
FI= - u1 *u3 *kap + u1 *v1 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2 *u3 *kap
2 2 2 2 2 2
+ u2 *v2 + 2*u2*u3*v2*v3 - u3 *v1 - u3 *v2
which is not further factorizable.
{HAM,FI} = 0
2 2 2 2 2 2
FI=2*u1 *u3 *a33 + u1*u3 *v2*b12 + 2*u2 *u3 *a33 - u2*u3 *v1*b12
2 2 2 1
=2*u3 *((u1 + u2 )*a33 + ---*(u1*v2 - u2*v1)*b12)
2
{HAM,FI} = 0
4
FI=u3
4
=u3
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u1**2*a33 + u1*v2*b12 + 2*u2**2*a33 - u2*v1*b12 + u3**2*a33$
FI=u1*u3**2*v1 + u2*u3**2*v2 + u3**3*v3$
FI=4*u1**2*u2**2*a33**2 + 4*u1**2*u3**2*a33**2 + 4*u1*u2**2*v2*a33*b12 + 4*u1*u2
*u3*v3*a33*b12 + 4*u2**4*a33**2 - 4*u2**3*v1*a33*b12 + u2**2*u3**2*(4*a33**2 -
b12**2*kap) + u2**2*v1**2*b12**2 + u2**2*v2**2*b12**2 - 4*u2*u3**2*v1*a33*b12 +
2*u2*u3*v2*v3*b12**2 - u3**2*v2**2*b12**2$
FI= - u1**2*u3**2*kap + u1**2*v1**2 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2**2*u3**
2*kap + u2**2*v2**2 + 2*u2*u3*v2*v3 - u3**2*v1**2 - u3**2*v2**2$
FI=2*u1**2*u3**2*a33 + u1*u3**2*v2*b12 + 2*u2**2*u3**2*a33 - u2*u3**2*v1*b12$
FI=u3**4$