Solution 5 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2
0=a11 + 4*a23 + 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
2*m3*n1*r4110
r28=---------------
2
b12
2 2
kap*m3 *r4110 - n1 *r4110
r29=---------------------------
2
b12
r210=0
r211=0
r212=0
r213=0
2
- n1 *r4110
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
- 2*a11*m3*r4110
r316=-------------------
2
b12
4*a23*m3*r4110 - 2*b12*n1*r4110
r317=---------------------------------
2
b12
r318=0
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
2*n1*r4110
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
- 2*a11*n1*r4110
r344=-------------------
2
b12
r345=0
r346=0
r347=0
4*a23*n1*r4110
r348=----------------
2
b12
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= - r4110
r428=0
r429=0
r430= - r4110
r431=0
r432=0
4*a23*r4110
r433=-------------
b12
2 2
a11 *r4110 + b12 *kap*r4110
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*r4110
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
2*a11*r4110
r453=-------------
b12
- 4*a11*a23*r4110
r454=--------------------
2
b12
r455=0
r456=0
r457=r4110
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
2 2 2
- 2*a11 *r4110 - 4*a23 *r4110 - b12 *kap*r4110
r464=-------------------------------------------------
2
b12
r465=0
r466=0
r467=0
r468=0
r469=0
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
- 4*a23*r4110
r486=----------------
b12
- 2*a11*r4110
r487=----------------
b12
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=2*r4110
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
2 2 2
- 2*a11 *r4110 - 8*a23 *r4110 - b12 *kap*r4110
r4114=-------------------------------------------------
2
b12
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
n3=0
n2=0
a33=0
a22=a11
a13=0
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, a23, a11, n1, m3, 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.
{a23,r4110,n1,b12}
Relevance for the application:
Modulo the following equation:
2 2 2
0=a11 + 4*a23 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2
HAM=u1 *a11 + u1*v2*b12 + u1*n1 + u2 *a11 + 2*u2*u3*a23 - u2*v1*b12 + v3*m3
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 2 2 2 2
FI=u1 *u3 *( - 2*a11 - 8*a23 - b12 *kap) + u1 *v1 *b12 + 4*u1*u2*u3*a23*n1
2 2 2
+ 2*u1*u2*v1*v2*b12 - 2*u1*u3 *v2*a11*b12 - 4*u1*u3 *v3*a23*b12
2 2
- 2*u1*u3 *a11*n1 + 2*u1*u3*v1*v3*b12
2 2 2 2 2 2 2 2 2 2
+ u2 *u3 *( - 2*a11 - 4*a23 - b12 *kap) + u2 *v2 *b12 - u2 *n1
3 2 2
- 4*u2*u3 *a11*a23 + 2*u2*u3 *v1*a11*b12 + 2*u2*u3*v2*v3*b12
4 2 2 3
+ 2*u2*u3*v3*b12*n1 + u3 *(a11 + b12 *kap) + 4*u3 *v1*a23*b12
2 2 2 2 2 2 2
- u3 *v1 *b12 - u3 *v2 *b12 + u3 *v2*(4*a23*m3 - 2*b12*n1)
2 2 2 2
- 2*u3 *v3*a11*m3 + u3 *(kap*m3 - n1 ) + 2*u3*v1*m3*n1
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*v2*b12 + u1*n1 + u2**2*a11 + 2*u2*u3*a23 - u2*v1*b12 + v3*m3$
FI=u1**2*u3**2*( - 2*a11**2 - 8*a23**2 - b12**2*kap) + u1**2*v1**2*b12**2 + 4*u1
*u2*u3*a23*n1 + 2*u1*u2*v1*v2*b12**2 - 2*u1*u3**2*v2*a11*b12 - 4*u1*u3**2*v3*a23
*b12 - 2*u1*u3**2*a11*n1 + 2*u1*u3*v1*v3*b12**2 + u2**2*u3**2*( - 2*a11**2 - 4*
a23**2 - b12**2*kap) + u2**2*v2**2*b12**2 - u2**2*n1**2 - 4*u2*u3**3*a11*a23 + 2
*u2*u3**2*v1*a11*b12 + 2*u2*u3*v2*v3*b12**2 + 2*u2*u3*v3*b12*n1 + u3**4*(a11**2
+ b12**2*kap) + 4*u3**3*v1*a23*b12 - u3**2*v1**2*b12**2 - u3**2*v2**2*b12**2 +
u3**2*v2*(4*a23*m3 - 2*b12*n1) - 2*u3**2*v3*a11*m3 + u3**2*(kap*m3**2 - n1**2) +
2*u3*v1*m3*n1$