Solution 4 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
r13=0
r14=0
r15=0
2
m3 *r460
r20=----------
2
b12
r21=0
2
m3 *r460
r22=----------
2
b12
r23=0
r24=0
- 2*m3*n2*r460
r27=-----------------
2
b12
r28=0
2
n2 *r460
r29=----------
2
b12
2*m3*n2*r460
r210=--------------
2
b12
r211=0
r212=0
r213=0
2
n2 *r460
r214=----------
2
b12
r215=0
r217=0
r218=0
r219=0
2 2
kap*m3 *r460 + n2 *r460
r220=-------------------------
2
b12
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=0
r311=0
r312=0
r313=0
2*m3*r460
r314=-----------
b12
r315=0
4*a33*m3*r460
r316=---------------
2
b12
r317=0
- 2*n2*r460
r318=--------------
b12
r319=0
r320=0
r321=0
r322=0
- 2*m3*r460
r323=--------------
b12
r324=0
r325=0
r326=0
r327=0
r328=0
4*a33*n2*r460
r329=---------------
2
b12
4*a33*m3*r460
r330=---------------
2
b12
r331=0
- 2*n2*r460
r332=--------------
b12
r333=0
4*a33*n2*r460
r334=---------------
2
b12
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
2*n2*r460
r341=-----------
b12
r342=0
4*a33*m3*r460
r343=---------------
2
b12
r344=0
r345=0
2*n2*r460
r346=-----------
b12
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
4*a33*n2*r460
r354=---------------
2
b12
r355=0
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
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r423=0
r424=0
r426=0
r427=0
r428=0
r429=0
r430=r460
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=0
r446=0
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
- 4*a33*r460
r453=---------------
b12
r454=0
r455=0
r456=0
r457=0
r458=0
r459=0
r461=0
r462=0
r463=0
2
4*a33 *r460
r464=-------------
2
b12
r465=0
r466=0
- 4*a33*r460
r467=---------------
b12
r468=0
2
4*a33 *r460
r469=-------------
2
b12
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r479=0
r480=0
r482=0
r483= - 2*r460
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494= - 2*r460
r495=0
4*a33*r460
r496=------------
b12
r497=0
r498=0
r499=0
r4100=0
4*a33*r460
r4101=------------
b12
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4110= - r460
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4118=0
2
4*a33 *r460
r4119=-------------
2
b12
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
n3=0
n1=0
a23=0
a13=0
2 1 2
a33 - ---*b12 *kap
4
a11=---------------------
a33
a22=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:
a33, r460, m3, n2, 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
{a22 - 4*a22*a33 - b12 *kap,r460,b12,a22,a33}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 1 2 2 2
HAM=(u1 *(a33 - ---*b12 *kap) + u1*v2*a33*b12 + 2*u2 *a33 - u2*v1*a33*b12
4
2 2
+ u2*a33*n2 + u3 *a33 + v3*a33*m3)/a33
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 2 2 2 2 2 2
FI=4*u1 *u2 *a33 + 4*u1 *u2*a33*n2 - u1 *v1 *b12 + u1 *(kap*m3 + n2 )
2 2
+ 4*u1*u2 *v2*a33*b12 + 4*u1*u2*u3*v3*a33*b12 - 2*u1*u2*v1*v2*b12
2
+ 2*u1*u2*v2*b12*n2 - 2*u1*u3*v1*v3*b12 + 4*u1*u3*v1*a33*m3
4 2 3 3
+ 2*u1*u3*v3*b12*n2 + 4*u2 *a33 - 4*u2 *v1*a33*b12 + 4*u2 *a33*n2
2 2 2 2 2 2 2 2
+ 4*u2 *u3 *a33 + u2 *v1 *b12 - 2*u2 *v1*b12*n2 + 4*u2 *v3*a33*m3
2 2 2 2
+ u2 *n2 - 4*u2*u3 *v1*a33*b12 + 4*u2*u3 *a33*n2 - 2*u2*v1*v3*b12*m3
2 2 2 2 2 2 2
+ 2*u2*v3*m3*n2 + u3 *v1 *b12 - 2*u3 *v1*b12*n2 + 4*u3 *v3*a33*m3 + u3 *n2
2 2 2 2
+ 2*u3*v1*v2*b12*m3 - 2*u3*v2*m3*n2 + v2 *m3 + v3 *m3
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*(a33**2 - 1/4*b12**2*kap) + u1*v2*a33*b12 + 2*u2**2*a33**2 - u2*v1*
a33*b12 + u2*a33*n2 + u3**2*a33**2 + v3*a33*m3)/a33$
FI=4*u1**2*u2**2*a33**2 + 4*u1**2*u2*a33*n2 - u1**2*v1**2*b12**2 + u1**2*(kap*m3
**2 + n2**2) + 4*u1*u2**2*v2*a33*b12 + 4*u1*u2*u3*v3*a33*b12 - 2*u1*u2*v1*v2*b12
**2 + 2*u1*u2*v2*b12*n2 - 2*u1*u3*v1*v3*b12**2 + 4*u1*u3*v1*a33*m3 + 2*u1*u3*v3*
b12*n2 + 4*u2**4*a33**2 - 4*u2**3*v1*a33*b12 + 4*u2**3*a33*n2 + 4*u2**2*u3**2*
a33**2 + u2**2*v1**2*b12**2 - 2*u2**2*v1*b12*n2 + 4*u2**2*v3*a33*m3 + u2**2*n2**
2 - 4*u2*u3**2*v1*a33*b12 + 4*u2*u3**2*a33*n2 - 2*u2*v1*v3*b12*m3 + 2*u2*v3*m3*
n2 + u3**2*v1**2*b12**2 - 2*u3**2*v1*b12*n2 + 4*u3**2*v3*a33*m3 + u3**2*n2**2 +
2*u3*v1*v2*b12*m3 - 2*u3*v2*m3*n2 + v2**2*m3**2 + v3**2*m3**2$