Solution 3 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
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Equations
The following unsolved equations remain:
2 2 2
0=4*a13 + 4*a23 + b12 *kap
Expressions
The solution is given through the following expressions:
r10=0
2
- 4*a13*m3*n1*n2*r4110 - 4*a23*m3*n2 *r4110
r11=----------------------------------------------
4
b12 *kap
2
- 4*a13*m3*n1 *r4110 - 4*a23*m3*n1*n2*r4110
r12=----------------------------------------------
4
b12 *kap
4 2 3 2 2 2 3
r13=( - 4*a13 *m3 *n1*r4110 - 4*a13 *a23*m3 *n2*r4110 + 2*a13 *b12 *n1 *r4110
2 2 3 1 4 3 3 4
+ 2*a23 *b12 *n1 *r4110 + ---*b12 *kap*n1 *r4110)/(a13 *b12 )
2
r14=0
r15=0
r20=0
r21=0
r22=0
r23=0
r24=0
2 2
r27=( - 8*a13*a23*m3*n1*r4110 + 4*a13*b12*n1 *r4110 - 8*a23 *m3*n2*r4110
2 4
+ 4*a23*b12*n1*n2*r4110 + 2*b12 *kap*m3*n2*r4110)/(b12 *kap)
2
r28=( - 8*a13 *m3*n1*r4110 - 8*a13*a23*m3*n2*r4110 - 4*a13*b12*n1*n2*r4110
2 2 4
- 4*a23*b12*n2 *r4110 + 2*b12 *kap*m3*n1*r4110)/(b12 *kap)
4 2 3 2 2 2
r29=(4*a13 *n1 *r4110 + 8*a13 *a23*n1*n2*r4110 + 4*a13 *a23 *n2 *r4110
2 2 2 2 2 2 2
+ a13 *b12 *kap *m3 *r4110 - 3*a13 *b12 *kap*n1 *r4110
2 2 2 2 2 2
- a13 *b12 *kap*n2 *r4110 - 2*a23 *b12 *kap*n1 *r4110
1 4 2 2 2 4
- ---*b12 *kap *n1 *r4110)/(a13 *b12 *kap)
2
2
- 4*a13*n1 *r4110 - 4*a23*n1*n2*r4110
r210=----------------------------------------
3
b12 *kap
r211=0
r212=0
r213=0
2 2 2 2
r214=(4*a13 *n1 *r4110 + 8*a13*a23*n1*n2*r4110 + 4*a23 *n2 *r4110
2 2 4
- b12 *kap*n1 *r4110)/(b12 *kap)
2
4*a13*n1*n2*r4110 + 4*a23*n2 *r4110
r215=-------------------------------------
3
b12 *kap
r217=0
r218=0
2*n1*n2*r4110
r219=---------------
2
b12
2 2 2 2
r220=(4*a13 *n1 *r4110 + 8*a13*a23*n1*n2*r4110 + 4*a23 *n2 *r4110
2 2 4
- b12 *kap*n2 *r4110)/(b12 *kap)
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
4*a13*n1*r4110 + 4*a23*n2*r4110
r310=---------------------------------
2
b12 *kap
r311=0
4*a13*n1*r4110 + 4*a23*n2*r4110
r312=---------------------------------
2
b12 *kap
r313=0
r314=0
4*a13*n1*r4110 + 4*a23*n2*r4110
r315=---------------------------------
2
b12 *kap
r316=0
2
r317=(8*a13 *n1*r4110 + 8*a13*a23*n2*r4110 + 4*a23*b12*kap*m3*r4110
2 3
- 2*b12 *kap*n1*r4110)/(b12 *kap)
2
r318=( - 8*a13*a23*n1*r4110 + 4*a13*b12*kap*m3*r4110 - 8*a23 *n2*r4110
2 3
+ 2*b12 *kap*n2*r4110)/(b12 *kap)
4 2 8 2
r319=( - ---*a13 *n1*r4110 - 4*a13*a23*n2*r4110 + ---*a23 *n1*r4110
3 3
2 2 2
+ ---*b12 *kap*n1*r4110)/(a13*b12 )
3
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
2 2
- 8*a13 *n1*r4110 - 8*a13*a23*n2*r4110 + 2*b12 *kap*n1*r4110
r326=---------------------------------------------------------------
3
b12 *kap
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
3 2 2
r333=(8*a13 *n1*r4110 + 8*a13 *a23*n2*r4110 + 8*a13*a23 *n1*r4110
2 3 2
- 2*a13*b12 *kap*n1*r4110 + 8*a23 *n2*r4110 + 2*a23*b12 *kap*n2*r4110)/(
4
b12 *kap)
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
2 2
8*a13*a23*n1*r4110 + 8*a23 *n2*r4110 - 2*b12 *kap*n2*r4110
r341=------------------------------------------------------------
3
b12 *kap
r342=0
r343=0
r344=0
r345=0
r346=0
r347=0
4*a13*n2*r4110 + 4*a23*n1*r4110
r348=---------------------------------
2
b12
r349=0
r350=0
r351=0
r352=0
3 2 2
r353=(8*a13 *n1*r4110 + 8*a13 *a23*n2*r4110 + 8*a13*a23 *n1*r4110
2 3 2
+ 2*a13*b12 *kap*n1*r4110 + 8*a23 *n2*r4110 - 2*a23*b12 *kap*n2*r4110)/(
4
b12 *kap)
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
- 4*a13*r4110
r432=----------------
b12
4*a23*r4110
r433=-------------
b12
2 2
- 4*a13 *r4110 - 4*a23 *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
4*a13*r4110
r451=-------------
b12
r452=0
r453=0
r454=0
r455=0
r456=0
r457=r4110
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
2 2 2
- 8*a13 *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
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=2*r4110
r495=0
r496=0
r497=0
r498=0
8*a13*a23*r4110
r499=-----------------
2
b12
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
- 4*a13 *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
a33=0
a22=0
a11=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, n2, n1, a23, a13, 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.
{{8*r4110,
2
b12 *kap*r4110,
2
4*a13 *r4110,
2*r4110,
r4110,
2 2
4*a13 *r4110 + b12 *kap*r4110,
4*r4110},
b12,
a13}
Relevance for the application:
Modulo the following equation:
2 2 2
0=4*a13 + 4*a23 + b12 *kap
the system of equations related to the Hamiltonian HAM:
HAM=2*u1*u3*a13 + u1*v2*b12 + u1*n1 + 2*u2*u3*a23 - u2*v1*b12 + u2*n2 + v3*m3
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 5 2 3 2 2 3 4 2 2
FI=u1 *u3 *( - 4*a13 *b12 *kap - 8*a13 *a23 *b12 *kap - a13 *b12 *kap ) + u1 *u3
6 5 4 2 4 2
*(8*a13 *n1 + 8*a13 *a23*n2 + 8*a13 *a23 *n1 + 2*a13 *b12 *kap*n1
3 3 3 2 2 2 3 4
+ 8*a13 *a23 *n2 - 2*a13 *a23*b12 *kap*n2) + u1 *v1 *a13 *b12 *kap
2 5 2 4 3 2 2 3 2 2
+ u1 *(4*a13 *n1 + 8*a13 *a23*n1*n2 + 4*a13 *a23 *n2 - a13 *b12 *kap*n2 )
2 4 2
+ 8*u1*u2*u3 *a13 *a23*b12 *kap
4 2 3 2
+ u1*u2*u3*(4*a13 *b12 *kap*n2 + 4*a13 *a23*b12 *kap*n1)
3 4 3 2
+ 2*u1*u2*v1*v2*a13 *b12 *kap + 2*u1*u2*a13 *b12 *kap*n1*n2
2 3 3 3 4
- 4*u1*u3 *v3*a13 *a23*b12 *kap + 2*u1*u3*v1*v3*a13 *b12 *kap
4 3 2 3 3
+ u1*u3*v3*(8*a13 *a23*b12*n1 + 8*a13 *a23 *b12*n2 - 2*a13 *b12 *kap*n2)
4 3 2
+ u1*v3*(4*a13 *b12*n1*n2 + 4*a13 *a23*b12*n2 )
2 2 5 2 3 2 2 3 4 2 2
+ u2 *u3 *( - 8*a13 *b12 *kap - 4*a13 *a23 *b12 *kap - a13 *b12 *kap ) + u2
6 5 4 2 4 2
*u3*(8*a13 *n1 + 8*a13 *a23*n2 + 8*a13 *a23 *n1 - 2*a13 *b12 *kap*n1
3 3 3 2 2 2 3 4
+ 8*a13 *a23 *n2 + 2*a13 *a23*b12 *kap*n2) + u2 *v2 *a13 *b12 *kap
2 5 2 4 3 2 2 3 2 2
+ u2 *(4*a13 *n1 + 8*a13 *a23*n1*n2 + 4*a13 *a23 *n2 - a13 *b12 *kap*n1 )
2 4 3 3 4
+ 4*u2*u3 *v3*a13 *b12 *kap + 2*u2*u3*v2*v3*a13 *b12 *kap
5 4 3 3
+ u2*u3*v3*( - 8*a13 *b12*n1 - 8*a13 *a23*b12*n2 + 2*a13 *b12 *kap*n1)
4 2 3
+ u2*v3*( - 4*a13 *b12*n1 - 4*a13 *a23*b12*n1*n2)
4 5 2 3 2 2
+ u3 *( - 4*a13 *b12 *kap - 4*a13 *a23 *b12 *kap)
3 3 3 3 4 3 3
+ 4*u3 *v1*a13 *a23*b12 *kap - 4*u3 *v2*a13 *b12 *kap + u3 *(
4 4 2 3 2
- ---*a13 *b12 *kap*n1 - 4*a13 *a23*b12 *kap*n2
3
8 2 2 2 2 2 4 2
+ ---*a13 *a23 *b12 *kap*n1 + ---*a13 *b12 *kap *n1)
3 3
2 2 3 4 2 4 4 2
- u3 *v1 *a13 *b12 *kap + u3 *v1*( - 8*a13 *a23*b12*n1 + 4*a13 *b12 *kap*m3
3 2 3 3 2 2 3 4 2
- 8*a13 *a23 *b12*n2 + 2*a13 *b12 *kap*n2) - u3 *v2 *a13 *b12 *kap + u3
5 4 3 2
*v2*(8*a13 *b12*n1 + 8*a13 *a23*b12*n2 + 4*a13 *a23*b12 *kap*m3
3 3 2 5 2 4
- 2*a13 *b12 *kap*n1) + u3 *(4*a13 *n1 + 8*a13 *a23*n1*n2
3 2 2 3 2 2 2 3 2 2
+ 4*a13 *a23 *n2 + a13 *b12 *kap *m3 - 3*a13 *b12 *kap*n1
3 2 2 2 2 2 1 4 2 2
- a13 *b12 *kap*n2 - 2*a13*a23 *b12 *kap*n1 - ---*a13*b12 *kap *n1 )
2
2 4 2 3 2 5
+ u3*v1 *(4*a13 *b12 *n1 + 4*a13 *a23*b12 *n2) + u3*v1*( - 8*a13 *m3*n1
4 4 3 2
- 8*a13 *a23*m3*n2 - 4*a13 *b12*n1*n2 - 4*a13 *a23*b12*n2
3 2 2 4 2 3 2
+ 2*a13 *b12 *kap*m3*n1) + u3*v2 *(4*a13 *b12 *n1 + 4*a13 *a23*b12 *n2) +
4 4 2 3 2
u3*v2*( - 8*a13 *a23*m3*n1 + 4*a13 *b12*n1 - 8*a13 *a23 *m3*n2
3 3 2
+ 4*a13 *a23*b12*n1*n2 + 2*a13 *b12 *kap*m3*n2)
2 4 2 3 2 4 2
+ u3*v3 *(4*a13 *b12 *n1 + 4*a13 *a23*b12 *n2) + u3*( - 4*a13 *kap*m3 *n1
3 2 2 2 3 2 2 3
- 4*a13 *a23*kap*m3 *n2 + 2*a13 *b12 *kap*n1 + 2*a23 *b12 *kap*n1
1 4 2 3 4 2 3
+ ---*b12 *kap *n1 ) + v1*( - 4*a13 *m3*n1 - 4*a13 *a23*m3*n1*n2)
2
4 3 2
+ v2*( - 4*a13 *m3*n1*n2 - 4*a13 *a23*m3*n2 )
Stop of a subroutine.
Number of garbage collections exceeds max_gc_fac.
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u1*u3*a13 + u1*v2*b12 + u1*n1 + 2*u2*u3*a23 - u2*v1*b12 + u2*n2 + v3*m3$
FI=u1**2*u3**2*( - 24*a13**5*b12**2*kap - 48*a13**3*a23**2*b12**2*kap - 6*a13**3
*b12**4*kap**2) + u1**2*u3*(48*a13**6*n1 + 48*a13**5*a23*n2 + 48*a13**4*a23**2*
n1 + 12*a13**4*b12**2*kap*n1 + 48*a13**3*a23**3*n2 - 12*a13**3*a23*b12**2*kap*n2
) + 6*u1**2*v1**2*a13**3*b12**4*kap + u1**2*(24*a13**5*n1**2 + 48*a13**4*a23*n1*
n2 + 24*a13**3*a23**2*n2**2 - 6*a13**3*b12**2*kap*n2**2) + 48*u1*u2*u3**2*a13**4
*a23*b12**2*kap + u1*u2*u3*(24*a13**4*b12**2*kap*n2 + 24*a13**3*a23*b12**2*kap*
n1) + 12*u1*u2*v1*v2*a13**3*b12**4*kap + 12*u1*u2*a13**3*b12**2*kap*n1*n2 - 24*
u1*u3**2*v3*a13**3*a23*b12**3*kap + 12*u1*u3*v1*v3*a13**3*b12**4*kap + u1*u3*v3*
(48*a13**4*a23*b12*n1 + 48*a13**3*a23**2*b12*n2 - 12*a13**3*b12**3*kap*n2) + u1*
v3*(24*a13**4*b12*n1*n2 + 24*a13**3*a23*b12*n2**2) + u2**2*u3**2*( - 48*a13**5*
b12**2*kap - 24*a13**3*a23**2*b12**2*kap - 6*a13**3*b12**4*kap**2) + u2**2*u3*(
48*a13**6*n1 + 48*a13**5*a23*n2 + 48*a13**4*a23**2*n1 - 12*a13**4*b12**2*kap*n1
+ 48*a13**3*a23**3*n2 + 12*a13**3*a23*b12**2*kap*n2) + 6*u2**2*v2**2*a13**3*b12
**4*kap + u2**2*(24*a13**5*n1**2 + 48*a13**4*a23*n1*n2 + 24*a13**3*a23**2*n2**2
- 6*a13**3*b12**2*kap*n1**2) + 24*u2*u3**2*v3*a13**4*b12**3*kap + 12*u2*u3*v2*v3
*a13**3*b12**4*kap + u2*u3*v3*( - 48*a13**5*b12*n1 - 48*a13**4*a23*b12*n2 + 12*
a13**3*b12**3*kap*n1) + u2*v3*( - 24*a13**4*b12*n1**2 - 24*a13**3*a23*b12*n1*n2)
+ u3**4*( - 24*a13**5*b12**2*kap - 24*a13**3*a23**2*b12**2*kap) + 24*u3**3*v1*
a13**3*a23*b12**3*kap - 24*u3**3*v2*a13**4*b12**3*kap + u3**3*( - 8*a13**4*b12**
2*kap*n1 - 24*a13**3*a23*b12**2*kap*n2 + 16*a13**2*a23**2*b12**2*kap*n1 + 4*a13
**2*b12**4*kap**2*n1) - 6*u3**2*v1**2*a13**3*b12**4*kap + u3**2*v1*( - 48*a13**4
*a23*b12*n1 + 24*a13**4*b12**2*kap*m3 - 48*a13**3*a23**2*b12*n2 + 12*a13**3*b12
**3*kap*n2) - 6*u3**2*v2**2*a13**3*b12**4*kap + u3**2*v2*(48*a13**5*b12*n1 + 48*
a13**4*a23*b12*n2 + 24*a13**3*a23*b12**2*kap*m3 - 12*a13**3*b12**3*kap*n1) + u3
**2*(24*a13**5*n1**2 + 48*a13**4*a23*n1*n2 + 24*a13**3*a23**2*n2**2 + 6*a13**3*
b12**2*kap**2*m3**2 - 18*a13**3*b12**2*kap*n1**2 - 6*a13**3*b12**2*kap*n2**2 -
12*a13*a23**2*b12**2*kap*n1**2 - 3*a13*b12**4*kap**2*n1**2) + u3*v1**2*(24*a13**
4*b12**2*n1 + 24*a13**3*a23*b12**2*n2) + u3*v1*( - 48*a13**5*m3*n1 - 48*a13**4*
a23*m3*n2 - 24*a13**4*b12*n1*n2 - 24*a13**3*a23*b12*n2**2 + 12*a13**3*b12**2*kap
*m3*n1) + u3*v2**2*(24*a13**4*b12**2*n1 + 24*a13**3*a23*b12**2*n2) + u3*v2*( -
48*a13**4*a23*m3*n1 + 24*a13**4*b12*n1**2 - 48*a13**3*a23**2*m3*n2 + 24*a13**3*
a23*b12*n1*n2 + 12*a13**3*b12**2*kap*m3*n2) + u3*v3**2*(24*a13**4*b12**2*n1 + 24
*a13**3*a23*b12**2*n2) + u3*( - 24*a13**4*kap*m3**2*n1 - 24*a13**3*a23*kap*m3**2
*n2 + 12*a13**2*b12**2*kap*n1**3 + 12*a23**2*b12**2*kap*n1**3 + 3*b12**4*kap**2*
n1**3) + v1*( - 24*a13**4*m3*n1**2 - 24*a13**3*a23*m3*n1*n2) + v2*( - 24*a13**4*
m3*n1*n2 - 24*a13**3*a23*m3*n2**2)$