Solution 7 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r41=0
r43=0
2 2 2
- c11 *r40 + c11 *r42 + 2*c11*c22*r40 - c11*c22*r42 - c22 *r40
r44=-----------------------------------------------------------------
2
c11
r45=0
r46=0
r47=0
r48=0
- 2*c11*r40 + c11*r42 + 2*c22*r40
r49=------------------------------------
c22
r410=0
2 2 2
- 2*c11 *r40 + c11 *r42 + 4*c11*c22*r40 - c11*c22*r42 - 2*c22 *r40
r411=---------------------------------------------------------------------
c11*c22
r412=0
r413=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
r426=0
c11*r425 - c22*r425
r427=---------------------
c11
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
r446=0
c11*r445 - c22*r445
r447=---------------------
c11
r448=0
r449=0
r451=0
r452=0
r453=0
r454=0
1
a23*r425 - ---*a33*r445
2
r455=-------------------------
a23
r456=0
1 1
a23*c11*r425 - a23*c22*r425 - ---*a33*c11*r445 + ---*a33*c22*r445
2 2
r457=-------------------------------------------------------------------
a23*c11
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*r425 - ---*a33*r445
2
r4105=-------------------------
a23
r4106=0
1 1
a23*c11*r425 - a23*c22*r425 - ---*a33*c11*r445 + ---*a33*c22*r445
2 2
r4107=-------------------------------------------------------------------
a23*c11
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
b33=0
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:
r40, r425, r445, r42, a33, c11, 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.
{c22,a23,c11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=2*u2*u3*a23 + u3 *a33 + v1 *c11 + v2 *c22
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 2 2 2 2 2 4 2
FI=v1 *v2 *(c11 - c11*c22) + v1 *v3 *c11 + v2 *(c11*c22 - c22 )
2 2
+ v2 *v3 *c11*c22
2 2 2 2
= a product of the elements of: {v2 *(c11 - c22) + v3 *c11,v1 *c11 + v2 *c22}
{HAM,FI} = 0
2 2 1 1 1 2 2
FI=u1 *v2 *( - ---*a33*c11 + ---*a33*c22) - ---*u1 *v3 *a33*c11
2 2 2
2 2 1 1 1 2 2
+ u2 *v2 *( - ---*a33*c11 + ---*a33*c22) - ---*u2 *v3 *a33*c11
2 2 2
2 2
+ u2*u3*v2 *(a23*c11 - a23*c22) + u2*u3*v3 *a23*c11
1
= a product of the elements of: {---,
2
2 2
v2 *(c11 - c22) + v3 *c11,
2 2
- u1 *a33 - u2 *a33 + 2*u2*u3*a23}
{HAM,FI} = 0
2 2 2 2 2 2 2 2
FI=u1 *v2 *(c11 - c22) + u1 *v3 *c11 + u2 *v2 *(c11 - c22) + u2 *v3 *c11
2 2 2 2
+ u3 *v2 *(c11 - c22) + u3 *v3 *c11
2 2 2 2 2
= a product of the elements of: {u1 + u2 + u3 ,v2 *(c11 - c22) + v3 *c11}
{HAM,FI} = 0
2 2 3 2 2
FI=v1 *v2 *( - 2*c11 + 4*c11 *c22 - 2*c11*c22 )
2 2 3 2 4 2 2 3
+ v1 *v3 *( - 2*c11 + 2*c11 *c22) + v2 *( - c11 *c22 + 2*c11*c22 - c22 )
4 2
+ v3 *c11 *c22
2 2
= a product of the elements of: {v2 *( - c11 + c22) - v3 *c11,
2 2 2 2 2
v1 *(2*c11 - 2*c11*c22) + v2 *(c11*c22 - c22 ) - v3 *c11*c22}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + v1**2*c11 + v2**2*c22$
FI=v1**2*v2**2*(c11**2 - c11*c22) + v1**2*v3**2*c11**2 + v2**4*(c11*c22 - c22**2
) + v2**2*v3**2*c11*c22$
FI=u1**2*v2**2*( - 1/2*a33*c11 + 1/2*a33*c22) - 1/2*u1**2*v3**2*a33*c11 + u2**2*
v2**2*( - 1/2*a33*c11 + 1/2*a33*c22) - 1/2*u2**2*v3**2*a33*c11 + u2*u3*v2**2*(
a23*c11 - a23*c22) + u2*u3*v3**2*a23*c11$
FI=u1**2*v2**2*(c11 - c22) + u1**2*v3**2*c11 + u2**2*v2**2*(c11 - c22) + u2**2*
v3**2*c11 + u3**2*v2**2*(c11 - c22) + u3**2*v3**2*c11$
FI=v1**2*v2**2*( - 2*c11**3 + 4*c11**2*c22 - 2*c11*c22**2) + v1**2*v3**2*( - 2*
c11**3 + 2*c11**2*c22) + v2**4*( - c11**2*c22 + 2*c11*c22**2 - c22**3) + v3**4*
c11**2*c22$