Solution 2 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 *r4110
r20=--------------
2
b12
r21=0
2
- m3 *r4110
r22=--------------
2
b12
r23=0
r24=0
2*m3*n2*r4110
r27=---------------
2
b12
r28=0
2
- n2 *r4110
r29=--------------
2
b12
- 2*m3*n2*r4110
r210=------------------
2
b12
r211=0
r212=0
r213=0
2
- n2 *r4110
r214=--------------
2
b12
r215=0
r217=0
r218=0
r219=0
2 2
- kap*m3 *r4110 - n2 *r4110
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*r4110
r314=---------------
b12
r315=0
- 2*a22*m3*r4110
r316=-------------------
2
b12
r317=0
2*n2*r4110
r318=------------
b12
r319=0
r320=0
r321=0
r322=0
2*m3*r4110
r323=------------
b12
r324=0
r325=0
r326=0
r327=0
r328=0
- 2*a22*n2*r4110
r329=-------------------
2
b12
- 2*a22*m3*r4110
r330=-------------------
2
b12
r331=0
2*n2*r4110
r332=------------
b12
r333=0
- 2*a22*n2*r4110
r334=-------------------
2
b12
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- 2*n2*r4110
r341=---------------
b12
r342=0
- 2*a22*m3*r4110
r343=-------------------
2
b12
r344=0
r345=0
- 2*n2*r4110
r346=---------------
b12
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
- 2*a22*n2*r4110
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= - r4110
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
2*a22*r4110
r453=-------------
b12
r454=0
r455=0
r456=0
r457=0
r458=0
r459=0
r460= - r4110
r461=0
r462=0
r463=0
2
- a22 *r4110
r464=---------------
2
b12
r465=0
r466=0
2*a22*r4110
r467=-------------
b12
r468=0
2
- a22 *r4110
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*r4110
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=2*r4110
r495=0
- 2*a22*r4110
r496=----------------
b12
r497=0
r498=0
r499=0
r4100=0
- 2*a22*r4110
r4101=----------------
b12
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4118=0
2
- a22 *r4110
r4119=---------------
2
b12
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
n3=0
n1=0
1
a33=---*a22
2
a23=0
a13=0
1 2 1 2
---*a22 - ---*b12 *kap
2 2
a11=-------------------------
a22
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:
r4110, m3, a22, 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,n2,b12,{r4110,2*r4110,a22 *r4110,2*b12 *kap*r4110}}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 1 2 1 2 2 2
HAM=(u1 *(---*a22 - ---*b12 *kap) + u1*v2*a22*b12 + u2 *a22 - u2*v1*a22*b12
2 2
1 2 2
+ u2*a22*n2 + ---*u3 *a22 + v3*a22*m3)/a22
2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 2 2 2 2 2 2
FI= - u1 *u2 *a22 - 2*u1 *u2*a22*n2 + u1 *v1 *b12 + u1 *( - kap*m3 - n2 )
2 2
- 2*u1*u2 *v2*a22*b12 - 2*u1*u2*u3*v3*a22*b12 + 2*u1*u2*v1*v2*b12
2
- 2*u1*u2*v2*b12*n2 + 2*u1*u3*v1*v3*b12 - 2*u1*u3*v1*a22*m3
4 2 3 3
- 2*u1*u3*v3*b12*n2 - u2 *a22 + 2*u2 *v1*a22*b12 - 2*u2 *a22*n2
2 2 2 2 2 2 2 2 2 2
- u2 *u3 *a22 - u2 *v1 *b12 + 2*u2 *v1*b12*n2 - 2*u2 *v3*a22*m3 - u2 *n2
2 2
+ 2*u2*u3 *v1*a22*b12 - 2*u2*u3 *a22*n2 + 2*u2*v1*v3*b12*m3 - 2*u2*v3*m3*n2
2 2 2 2 2 2 2
- u3 *v1 *b12 + 2*u3 *v1*b12*n2 - 2*u3 *v3*a22*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*(1/2*a22**2 - 1/2*b12**2*kap) + u1*v2*a22*b12 + u2**2*a22**2 - u2*v1*
a22*b12 + u2*a22*n2 + 1/2*u3**2*a22**2 + v3*a22*m3)/a22$
FI= - u1**2*u2**2*a22**2 - 2*u1**2*u2*a22*n2 + u1**2*v1**2*b12**2 + u1**2*( -
kap*m3**2 - n2**2) - 2*u1*u2**2*v2*a22*b12 - 2*u1*u2*u3*v3*a22*b12 + 2*u1*u2*v1*
v2*b12**2 - 2*u1*u2*v2*b12*n2 + 2*u1*u3*v1*v3*b12**2 - 2*u1*u3*v1*a22*m3 - 2*u1*
u3*v3*b12*n2 - u2**4*a22**2 + 2*u2**3*v1*a22*b12 - 2*u2**3*a22*n2 - u2**2*u3**2*
a22**2 - u2**2*v1**2*b12**2 + 2*u2**2*v1*b12*n2 - 2*u2**2*v3*a22*m3 - u2**2*n2**
2 + 2*u2*u3**2*v1*a22*b12 - 2*u2*u3**2*a22*n2 + 2*u2*v1*v3*b12*m3 - 2*u2*v3*m3*
n2 - u3**2*v1**2*b12**2 + 2*u3**2*v1*b12*n2 - 2*u3**2*v3*a22*m3 - u3**2*n2**2 -
2*u3*v1*v2*b12*m3 + 2*u3*v2*m3*n2 - v2**2*m3**2 - v3**2*m3**2$