Solution 4 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
1
---*b33*r435
2
r40=--------------
a23
r41=0
1
---*b33*r442
2
r42=--------------
a23
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
1
---*b33*r442
2
r49=--------------
a23
r410=0
r411=0
r412=0
r413=0
r414=0
1 1 2
---*a23*a33*r435 + ---*b33 *r461
2 4
r415=----------------------------------
2
a23
r416=0
1
---*a33*r442
2
r417=--------------
a23
r418=0
r419=0
r420=0
r421=0
1
---*a33*r442
2
r422=--------------
a23
r423=0
r424=0
1
---*a33*b33*r461
2
r425=------------------
2
a23
r426=0
r427=0
r428=0
r429=0
r430=0
1 2
---*a33 *r461
4
r431=---------------
2
a23
r432=0
r433=0
r434=0
r436=0
r437=r442
r438=0
r439=0
r440=0
r441=0
r443=0
r444=0
b33*r461
r445=----------
a23
r446=0
r447=0
r448=0
r449=0
r450=0
a33*r461
r451=----------
a23
r452=0
r453=0
r454=0
r455=0
r456=0
r457=0
r458=0
r459=0
r460=0
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
r478=0
r479=0
r480=0
r481=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
r4105=0
r4106=0
r4107=0
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
c11=c22
b32=0
b31=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:
r435, r461, r442, b33, a33, 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.
{r442,c22,a23}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v3*b33 + v1 *c22 + v2 *c22
has apart from the Hamiltonian and Casimirs the following 3 first integrals:
2 2 1 2 2 1 2 2
FI=u2*u3*v1 *v3*a23 + u2*u3*v2 *v3*a23 + ---*u3 *v1 *v3*a33 + ---*u3 *v2 *v3*a33
2 2
1 2 2 1 2 2
+ ---*u3*v1 *v3 *b33 + ---*u3*v2 *v3 *b33
2 2
1
= a product of the elements of: {---,
2
v3,
2 2
v1 + v2 ,
u3,
2*u2*a23 + u3*a33 + v3*b33}
{HAM,FI} = 0
2 2 2 3 2 2 1 4 2
FI=u2 *u3 *v3*a23 + u2*u3 *v3*a23*a33 + u2*u3 *v3 *a23*b33 + ---*u3 *v3*a33
4
1 3 2 1 2 3 2
+ ---*u3 *v3 *a33*b33 + ---*u3 *v3 *b33
2 4
1
= a product of the elements of: {---,
4
u3,
u3,
2*u2*a23 + u3*a33 + v3*b33,
2*u2*a23 + u3*a33 + v3*b33,
v3}
{HAM,FI} = 0
3 1 2 3 1 4
FI=u2*u3*v3 *a23 + ---*u3 *v3 *a33 + ---*u3*v3 *b33
2 2
1
= a product of the elements of: {---,
2
v3,
v3,
v3,
u3,
2*u2*a23 + u3*a33 + v3*b33}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v3*b33 + v1**2*c22 + v2**2*c22$
FI=u2*u3*v1**2*v3*a23 + u2*u3*v2**2*v3*a23 + 1/2*u3**2*v1**2*v3*a33 + 1/2*u3**2*
v2**2*v3*a33 + 1/2*u3*v1**2*v3**2*b33 + 1/2*u3*v2**2*v3**2*b33$
FI=u2**2*u3**2*v3*a23**2 + u2*u3**3*v3*a23*a33 + u2*u3**2*v3**2*a23*b33 + 1/4*u3
**4*v3*a33**2 + 1/2*u3**3*v3**2*a33*b33 + 1/4*u3**2*v3**3*b33**2$
FI=u2*u3*v3**3*a23 + 1/2*u3**2*v3**3*a33 + 1/2*u3*v3**4*b33$