Solution 3 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=a22 + b12 *kap
Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
r13=0
r14=0
r15=0
r20=0
r21=0
r22=0
r23=0
r24=0
r27=0
r28=0
2
- n1 *r460
r29=-------------
2
b12
r210=0
r211=0
r212=0
r213=0
2
- n1 *r460
r214=-------------
2
b12
r215=0
r217=0
r218=0
r219=0
r220=0
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
r314=0
r315=0
r316=0
- 2*n1*r460
r317=--------------
b12
r318=0
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
2*n1*r460
r326=-----------
b12
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
r343=0
- a22*n1*r460
r344=----------------
2
b12
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
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= - r460
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
1 2
---*a22 *r460
4
r434=---------------
2
b12
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r446=2*r460
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
- a22*r460
r453=-------------
b12
r454=0
r455=0
r456=0
r457=r460
r458=0
r459=0
r461=0
r462=0
r463=0
r464= - kap*r460
r465=0
r466=0
- 2*a22*r460
r467=---------------
b12
r468=0
2
a22 *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=0
r484=0
r485=0
r486=0
- a22*r460
r487=-------------
b12
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=0
r495=0
2*a22*r460
r496=------------
b12
r497=0
r498=0
r499=0
r4100=0
2*a22*r460
r4101=------------
b12
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4118=0
2
a22 *r460
r4119=-----------
2
b12
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m3=0
m2=0
m1=0
n3=0
n2=0
1
a33=---*a22
2
a23=0
a13=0
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:
r460, a22, n1, 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.
{r460,a22,b12,n1}
Relevance for the application:
Modulo the following equation:
2 2
0=a22 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2 1 2
HAM=u1 *a22 + u1*v2*b12 + u1*n1 + u2 *a22 - u2*v1*b12 + ---*u3 *a22
2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2
FI=u1 *u2 *a22 + 2*u1*u2 *v2*a22*b12 + 2*u1*u2*u3*v3*a22*b12
2 2 4 2 3
- u1*u3 *v2*a22*b12 - u1*u3 *a22*n1 + u2 *a22 - 2*u2 *v1*a22*b12
2 2 2 2 2 2 2 2 2 2 2
- u2 *u3 *b12 *kap + u2 *v1 *b12 + u2 *v2 *b12 - u2 *n1
2 2 1 4 2
- u2*u3 *v1*a22*b12 + 2*u2*u3*v2*v3*b12 + 2*u2*u3*v3*b12*n1 + ---*u3 *a22
4
2 2 2 2 2 2
- u3 *v2 *b12 - 2*u3 *v2*b12*n1 - u3 *n1
1
= a product of the elements of: {---,
4
2 2 2 2
4*u1 *u2 *a22 + 8*u1*u2 *v2*a22*b12 + 8*u1*u2*u3*v3*a22*b12
2 2 4 2 3
- 4*u1*u3 *v2*a22*b12 - 4*u1*u3 *a22*n1 + 4*u2 *a22 - 8*u2 *v1*a22*b12
2 2 2 2 2 2 2 2 2 2 2
- 4*u2 *u3 *b12 *kap + 4*u2 *v1 *b12 + 4*u2 *v2 *b12 - 4*u2 *n1
2 2 4 2
- 4*u2*u3 *v1*a22*b12 + 8*u2*u3*v2*v3*b12 + 8*u2*u3*v3*b12*n1 + u3 *a22
2 2 2 2 2 2
- 4*u3 *v2 *b12 - 8*u3 *v2*b12*n1 - 4*u3 *n1 }
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a22 + u1*v2*b12 + u1*n1 + u2**2*a22 - u2*v1*b12 + 1/2*u3**2*a22$
FI=u1**2*u2**2*a22**2 + 2*u1*u2**2*v2*a22*b12 + 2*u1*u2*u3*v3*a22*b12 - u1*u3**2
*v2*a22*b12 - u1*u3**2*a22*n1 + u2**4*a22**2 - 2*u2**3*v1*a22*b12 - u2**2*u3**2*
b12**2*kap + u2**2*v1**2*b12**2 + u2**2*v2**2*b12**2 - u2**2*n1**2 - u2*u3**2*v1
*a22*b12 + 2*u2*u3*v2*v3*b12**2 + 2*u2*u3*v3*b12*n1 + 1/4*u3**4*a22**2 - u3**2*
v2**2*b12**2 - 2*u3**2*v2*b12*n1 - u3**2*n1**2$