Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=4*a23 + a33
Expressions
The solution is given through the following expressions:
r41=0
1
a23*r49 - ---*a33*r442
2
r42=------------------------
a23
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
r410=0
r411=0
r412=0
r413=0
1
---*a33*r435
2
r415=--------------
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
a23*r4105 + a33*r435
r425=----------------------
a23
r426=0
r427=0
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
r434=0
r436=0
r437= - r442
r438=0
r439=0
r440=0
r441=0
r443=0
r444=0
r445=2*r435
r446=0
r447=0
r448=0
r449=0
r451=0
r452=0
r453=0
r454=0
r455=r4105
r456=0
r457=0
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= - 2*r442
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
r4106=0
r4107=0
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
c33=0
c23=0
1
b33=---*a33
2
b32=0
b23=a23
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, r49, r435, r4105, r442, a33, 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.
{b33,a33,r442,a23}
Relevance for the application:
Modulo the following equation:
2 2
0=4*a23 + a33
the system of equations related to the Hamiltonian HAM:
2 1
HAM=2*u2*u3*a23 + u2*v3*a23 + u3 *a33 + ---*u3*v3*a33
2
has apart from the Hamiltonian and Casimirs the following 5 first integrals:
2 2 1 2
FI= - 2*u1*v1*v2*v3*a23 + u2*v1 *v3*a23 - u2*v2 *v3*a23 + ---*u3*v1 *v3*a33
2
1 2 1 2 2
- ---*u3*v2 *v3*a33 - ---*v2 *v3 *a33
2 2
1
= a product of the elements of: { - ---,
2
v3,
2 2 2 2
4*u1*v1*v2*a23 - 2*u2*v1 *a23 + 2*u2*v2 *a23 - u3*v1 *a33 + u3*v2 *a33
2
+ v2 *v3*a33}
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1 *v3 + u2 *v3 + u3 *v3
2 2 2
= a product of the elements of: {v3,v3,u1 + u2 + u3 }
{HAM,FI} = 0
2 3 2 2 1 3
FI=2*u2*u3*v3 *a23 + u2*v3 *a23 + u3 *v3 *a33 + ---*u3*v3 *a33
2
1
= a product of the elements of: {---,
2
v3,
v3,
2*u3 + v3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
2 2 2 2
FI=v1 *v3 + v2 *v3
2 2
= a product of the elements of: {v3,v3,v1 + v2 }
{HAM,FI} = 0
4
FI=v3
= a product of the elements of: {v3,v3,v3,v3}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u2*v3*a23 + u3**2*a33 + 1/2*u3*v3*a33$
FI= - 2*u1*v1*v2*v3*a23 + u2*v1**2*v3*a23 - u2*v2**2*v3*a23 + 1/2*u3*v1**2*v3*
a33 - 1/2*u3*v2**2*v3*a33 - 1/2*v2**2*v3**2*a33$
FI=u1**2*v3**2 + u2**2*v3**2 + u3**2*v3**2$
FI=2*u2*u3*v3**2*a23 + u2*v3**3*a23 + u3**2*v3**2*a33 + 1/2*u3*v3**3*a33$
FI=v1**2*v3**2 + v2**2*v3**2$
FI=v3**4$