Solution 3 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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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
r418=0
r421=0
r423=0
r424=0
1
r427=---*r494
2
r429=0
r430=0
r432=0
r433=0
1 2
- ---*a33 *r494
2
r434=------------------
2
b12
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r447=0
r449=0
r450=0
r452=0
a33*r494
r453=----------
b12
r454=0
r455=0
r456=0
1
r457=---*r494
2
r458=0
r459=0
1
r460= - ---*r494
2
r462=0
r463=0
2 1 2
- a33 *r494 + ---*b12 *kap*r494
4
r464=----------------------------------
2
b12
r465=0
r466=0
2 1 2
a33 *r494 - ---*b12 *kap*r494
4
r467=-------------------------------
a33*b12
r468=0
1 4 1 2 2 1 4 2
- ---*a33 *r494 + ---*a33 *b12 *kap*r494 - ----*b12 *kap *r494
2 4 32
r469=-----------------------------------------------------------------
2 2
a33 *b12
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r479=0
r482=0
r484=0
r485=0
a33*r494
r487=----------
b12
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r495=0
r497=0
r498=0
r499=0
r4100=0
2 1 2
a33 *r494 + ---*b12 *kap*r494
4
r4101=-------------------------------
a33*b12
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4110=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
2*a33*r494
r4117=------------
b12
r4118=0
1
r4119=---*kap*r494
2
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
a23=0
2 1 2
a33 - ---*b12 *kap
4
a22=---------------------
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:
r494, 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.
2 2 2 2
{a33,b12,r494,4*a33 + b12 *kap,4*a33 - b12 *kap}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2 1 2
HAM=(2*u1 *a33 + u1*v2*a33*b12 + u2 *(a33 - ---*b12 *kap) - u2*v1*a33*b12
4
2 2
+ u3 *a33 )/a33
has apart from the Hamiltonian and Casimirs only the following first integral:
1 2 2 2 2 2 3
FI=---*u1 *u2 *a33 *b12 *kap + 2*u1 *u2*v1*a33 *b12
2
2 3 1 3 2 2
+ u1*u2 *v2*(a33 *b12 + ---*a33*b12 *kap) + u1*u2*v1*v2*a33 *b12
4
2 3
+ u1*u3 *v2*a33 *b12
4 1 4 1 2 2 1 4 2
+ u2 *( - ---*a33 + ---*a33 *b12 *kap - ----*b12 *kap )
2 4 32
3 3 1 3
+ u2 *v1*(a33 *b12 - ---*a33*b12 *kap)
4
2 2 4 1 2 2 1 2 2 2 2
+ u2 *u3 *( - a33 + ---*a33 *b12 *kap) - ---*u2 *v1 *a33 *b12
4 2
1 2 2 2 2 2 3 1 4 4
+ ---*u2 *v2 *a33 *b12 + u2*u3 *v1*a33 *b12 - ---*u3 *a33
2 2
1 2 2 2 2
+ ---*u3 *v2 *a33 *b12
2
3 4 2 4
{HAM,FI} = 4*u1 *u3*v1*a33 *b12 + 4*u1 *u2*u3*v2*a33 *b12
2 2 4 2 3 2
+ 4*u1 *u3 *v3*a33 *b12 + 4*u1 *u3*v1*v2*a33 *b12
2 3 2 2 3 2
+ 4*u1*u2*u3*v2 *a33 *b12 + 4*u1*u3 *v2*v3*a33 *b12
2 2 3 3 2 3
+ u1*u3*v1*v2 *a33 *b12 + u2*u3*v2 *a33 *b12
2 2 2 3
+ u3 *v2 *v3*a33 *b12
And again in machine readable form:
HAM=(2*u1**2*a33**2 + u1*v2*a33*b12 + u2**2*(a33**2 - 1/4*b12**2*kap) - u2*v1*
a33*b12 + u3**2*a33**2)/a33$
FI=1/2*u1**2*u2**2*a33**2*b12**2*kap + 2*u1**2*u2*v1*a33**3*b12 + u1*u2**2*v2*(
a33**3*b12 + 1/4*a33*b12**3*kap) + u1*u2*v1*v2*a33**2*b12**2 + u1*u3**2*v2*a33**
3*b12 + u2**4*( - 1/2*a33**4 + 1/4*a33**2*b12**2*kap - 1/32*b12**4*kap**2) + u2
**3*v1*(a33**3*b12 - 1/4*a33*b12**3*kap) + u2**2*u3**2*( - a33**4 + 1/4*a33**2*
b12**2*kap) - 1/2*u2**2*v1**2*a33**2*b12**2 + 1/2*u2**2*v2**2*a33**2*b12**2 + u2
*u3**2*v1*a33**3*b12 - 1/2*u3**4*a33**4 + 1/2*u3**2*v2**2*a33**2*b12**2$