Solution 4 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
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Equations
The following unsolved equations remain:
2 2
0=a11 + b12 *kap
Expressions
The solution is given through the following expressions:
r10=0
- 2*m3*n2*n3*r4110
r11=---------------------
2
a11*b12
- 2*m3*n1*n3*r4110
r12=---------------------
2
a11*b12
2 3
- 2*kap*m3 *n3*r4110 + 2*n3 *r4110
r13=-------------------------------------
2
a11*b12
2
2*n2*n3 *r4110
r14=----------------
2
a11*b12
2
2*n1*n3 *r4110
r15=----------------
2
a11*b12
r20=0
r21=0
r22=0
r23=0
r24=0
2*a11*m3*n2*r4110 + 2*b12*n1*n3*r4110
r27=---------------------------------------
2
a11*b12
2*a11*m3*n1*r4110 - 2*b12*n2*n3*r4110
r28=---------------------------------------
2
a11*b12
2 2 2 2
kap*m3 *r4110 - n1 *r4110 - n2 *r4110 - n3 *r4110
r29=---------------------------------------------------
2
b12
- 2*n1*n3*r4110
r210=------------------
a11*b12
- 4*m3*n3*r4110
r211=------------------
2
b12
r212=0
2*n2*n3*r4110
r213=---------------
2
b12
2
- n1 *r4110
r214=--------------
2
b12
2*n2*n3*r4110
r215=---------------
a11*b12
- 4*m3*n3*r4110
r217=------------------
2
b12
2*n1*n3*r4110
r218=---------------
2
b12
2*n1*n2*r4110
r219=---------------
2
b12
2
- 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
2*n3*r4110
r310=------------
a11
r311=0
r313=0
r314=0
2*n3*r4110
r315=------------
a11
- 2*a11*m3*r4110
r316=-------------------
2
b12
- 2*n1*r4110
r317=---------------
b12
2*n2*r4110
r318=------------
b12
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
2*n1*r4110
r326=------------
b12
r327=0
- 4*n3*r4110
r328=---------------
b12
- 2*a11*n2*r4110
r329=-------------------
2
b12
r330=0
r331=0
r332=0
2 2
4*a11 *n3*r4110 + 2*b12 *kap*n3*r4110
r333=---------------------------------------
2
a11*b12
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- 2*n2*r4110
r341=---------------
b12
4*n3*r4110
r342=------------
b12
r343=0
- 2*a11*n1*r4110
r344=-------------------
2
b12
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
2 2
4*a11 *n3*r4110 + 2*b12 *kap*n3*r4110
r353=---------------------------------------
2
a11*b12
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
r433=0
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
r454=0
r455=0
r456=0
r457=r4110
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
2 2
- 2*a11 *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
r486=0
- 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*a11 *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
a33=0
a23=0
a22=a11
a13=0
2*n3*r4110
r312=------------
a11
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, n2, a11, n3, 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.
2 2
{{r312,b12 *kap*r312,a11*b12 *kap*r312},a11,n1,b12}
Relevance for the application:
Modulo the following equation:
2 2
0=a11 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2
HAM=u1 *a11 + u1*v2*b12 + u1*n1 + u2 *a11 - u2*v1*b12 + u2*n2 + u3*n3 + v3*m3
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 3 2 2 2 2
FI=u1 *u3 *( - 2*a11 - a11*b12 *kap) + u1 *u3*(4*a11 *n3 + 2*b12 *kap*n3)
2 2 2 2 2 2
+ u1 *v1 *a11*b12 - u1 *a11*n2 + 2*u1*u2*v1*v2*a11*b12
2 2 2 2
+ 2*u1*u2*a11*n1*n2 - 2*u1*u3 *v2*a11 *b12 - 2*u1*u3 *a11 *n1
2
+ 2*u1*u3*v1*v3*a11*b12 + 4*u1*u3*v2*a11*b12*n3 - 2*u1*u3*v3*a11*b12*n2
2
+ 2*u1*u3*a11*n1*n3 - 4*u1*v1*a11*m3*n3 + 2*u1*v3*b12*n2*n3 + 2*u1*n1*n3
2 2 3 2 2 2 2
+ u2 *u3 *( - 2*a11 - a11*b12 *kap) + u2 *u3*(4*a11 *n3 + 2*b12 *kap*n3)
2 2 2 2 2 2 2 2 2
+ u2 *v2 *a11*b12 - u2 *a11*n1 + 2*u2*u3 *v1*a11 *b12 - 2*u2*u3 *a11 *n2
2
- 4*u2*u3*v1*a11*b12*n3 + 2*u2*u3*v2*v3*a11*b12 + 2*u2*u3*v3*a11*b12*n1
2
+ 2*u2*u3*a11*n2*n3 - 4*u2*v2*a11*m3*n3 - 2*u2*v3*b12*n1*n3 + 2*u2*n2*n3
4 3 2 2 2 2 2
+ u3 *(a11 + a11*b12 *kap) - u3 *v1 *a11*b12 + 2*u3 *v1*a11*b12*n2
2 2 2 2 2 2
- u3 *v2 *a11*b12 - 2*u3 *v2*a11*b12*n1 - 2*u3 *v3*a11 *m3
2 2 2 2 2 2 2
+ u3 *(a11*kap*m3 - a11*n1 - a11*n2 - a11*n3 ) + 2*u3*v1 *b12 *n3
2 2
+ u3*v1*(2*a11*m3*n1 - 2*b12*n2*n3) + 2*u3*v2 *b12 *n3
2 2
+ u3*v2*(2*a11*m3*n2 + 2*b12*n1*n3) + 2*u3*v3 *b12 *n3
2 3
+ u3*( - 2*kap*m3 *n3 + 2*n3 ) - 2*v1*m3*n1*n3 - 2*v2*m3*n2*n3
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 - u2*v1*b12 + u2*n2 + u3*n3 + v3*
m3$
FI=u1**2*u3**2*( - 2*a11**3 - a11*b12**2*kap) + u1**2*u3*(4*a11**2*n3 + 2*b12**2
*kap*n3) + u1**2*v1**2*a11*b12**2 - u1**2*a11*n2**2 + 2*u1*u2*v1*v2*a11*b12**2 +
2*u1*u2*a11*n1*n2 - 2*u1*u3**2*v2*a11**2*b12 - 2*u1*u3**2*a11**2*n1 + 2*u1*u3*
v1*v3*a11*b12**2 + 4*u1*u3*v2*a11*b12*n3 - 2*u1*u3*v3*a11*b12*n2 + 2*u1*u3*a11*
n1*n3 - 4*u1*v1*a11*m3*n3 + 2*u1*v3*b12*n2*n3 + 2*u1*n1*n3**2 + u2**2*u3**2*( -
2*a11**3 - a11*b12**2*kap) + u2**2*u3*(4*a11**2*n3 + 2*b12**2*kap*n3) + u2**2*v2
**2*a11*b12**2 - u2**2*a11*n1**2 + 2*u2*u3**2*v1*a11**2*b12 - 2*u2*u3**2*a11**2*
n2 - 4*u2*u3*v1*a11*b12*n3 + 2*u2*u3*v2*v3*a11*b12**2 + 2*u2*u3*v3*a11*b12*n1 +
2*u2*u3*a11*n2*n3 - 4*u2*v2*a11*m3*n3 - 2*u2*v3*b12*n1*n3 + 2*u2*n2*n3**2 + u3**
4*(a11**3 + a11*b12**2*kap) - u3**2*v1**2*a11*b12**2 + 2*u3**2*v1*a11*b12*n2 -
u3**2*v2**2*a11*b12**2 - 2*u3**2*v2*a11*b12*n1 - 2*u3**2*v3*a11**2*m3 + u3**2*(
a11*kap*m3**2 - a11*n1**2 - a11*n2**2 - a11*n3**2) + 2*u3*v1**2*b12**2*n3 + u3*
v1*(2*a11*m3*n1 - 2*b12*n2*n3) + 2*u3*v2**2*b12**2*n3 + u3*v2*(2*a11*m3*n2 + 2*
b12*n1*n3) + 2*u3*v3**2*b12**2*n3 + u3*( - 2*kap*m3**2*n3 + 2*n3**3) - 2*v1*m3*
n1*n3 - 2*v2*m3*n2*n3$