Solution 8 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r40=r44
r41=0
r42=2*r44
r43=0
r45=0
r46=0
r47=0
r48=0
r49=r411
r410=0
r412=0
r413=0
r415=0
r416=0
r417=0
r418=0
1
---*b31*r447
2
r419=--------------
a23
r420=0
1
---*b31*r447
2
r421=--------------
a23
r422=0
r423=0
r424=0
r425=r427
r426=0
r428=0
r429=0
r430=0
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=r447
r446=0
r448=0
r449=0
r451=0
r452=0
r453=0
r454=0
1
a23*r427 - ---*a33*r447
2
r455=-------------------------
a23
r456=0
1
a23*r427 - ---*a33*r447
2
r457=-------------------------
a23
r458=0
r459=0
r460=0
r461=0
r462=0
r463=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
1
a23*r427 - ---*a33*r447
2
r4105=-------------------------
a23
r4106=0
1
a23*r427 - ---*a33*r447
2
r4107=-------------------------
a23
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
c22=0
b33=0
b32=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:
r44, r427, r447, r411, b31, a33, c11, 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.
{a23,c11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v1*b31 + v1 *c11
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 2 2 2
FI=v1 *v2 + v1 *v3
2 2
= a product of the elements of: {v1,v1,v2 + v3 }
{HAM,FI} = 0
1 2 2 1 2 2 1 2 2 1 2 2
FI= - ---*u1 *v2 *a33 - ---*u1 *v3 *a33 - ---*u2 *v2 *a33 - ---*u2 *v3 *a33
2 2 2 2
2 2 1 2 1 2
+ u2*u3*v2 *a23 + u2*u3*v3 *a23 + ---*u3*v1*v2 *b31 + ---*u3*v1*v3 *b31
2 2
1
= a product of the elements of: {---,
2
2 2
v2 + v3 ,
2 2
- u1 *a33 - u2 *a33 + 2*u2*u3*a23 + u3*v1*b31}
{HAM,FI} = 0
2 2 2 2 2 2 2 2 2 2 2 2
FI=u1 *v2 + u1 *v3 + u2 *v2 + u2 *v3 + u3 *v2 + u3 *v3
2 2 2 2 2
= a product of the elements of: {v2 + v3 ,u1 + u2 + u3 }
{HAM,FI} = 0
4 2 2 4
FI=v2 + 2*v2 *v3 + v3
2 2 2 2
= a product of the elements of: {v2 + v3 ,v2 + v3 }
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v1*b31 + v1**2*c11$
FI=v1**2*v2**2 + v1**2*v3**2$
FI= - 1/2*u1**2*v2**2*a33 - 1/2*u1**2*v3**2*a33 - 1/2*u2**2*v2**2*a33 - 1/2*u2**
2*v3**2*a33 + u2*u3*v2**2*a23 + u2*u3*v3**2*a23 + 1/2*u3*v1*v2**2*b31 + 1/2*u3*
v1*v3**2*b31$
FI=u1**2*v2**2 + u1**2*v3**2 + u2**2*v2**2 + u2**2*v3**2 + u3**2*v2**2 + u3**2*
v3**2$
FI=v2**4 + 2*v2**2*v3**2 + v3**4$