Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=a11 + b12 *kap
Expressions
The solution is given through the following expressions:
r10=0
a11*m3*n2*n3*r494
r11=-------------------
4
b12 *kap
3 2 2
r12=( - 4*a11 *m3*n1*n3*r494 - 2*a11 *b12*n2*n3 *r494
2 3 2 6 2
- 3*a11*b12 *kap*m3*n1*n3*r494 - 2*b12 *kap*n2*n3 *r494)/(b12 *kap )
2 3
a11*kap*m3 *n3*r494 - a11*n3 *r494
r13=------------------------------------
4
b12 *kap
2 2 2
- 2*a11 *m3*n1*n3*r494 - a11*b12*n2*n3 *r494 - 2*b12 *kap*m3*n1*n3*r494
r14=--------------------------------------------------------------------------
5
b12 *kap
2
- a11*n1*n3 *r494
r15=--------------------
4
b12 *kap
r20=0
r21=0
r22=0
r23=0
r24=0
- a11*n1*n3*r494 + b12*kap*m3*n2*r494
r27=----------------------------------------
3
b12 *kap
a11*n2*n3*r494 + b12*kap*m3*n1*r494
r28=-------------------------------------
3
b12 *kap
1 2 1 2 1 2 1 2
---*kap*m3 *r494 - ---*n1 *r494 - ---*n2 *r494 - ---*n3 *r494
2 2 2 2
r29=---------------------------------------------------------------
2
b12
a11*n1*n3*r494
r210=----------------
3
b12 *kap
- 2*m3*n3*r494
r211=-----------------
2
b12
r212=0
n2*n3*r494
r213=------------
2
b12
1 2
- ---*n1 *r494
2
r214=-----------------
2
b12
- a11*n2*n3*r494
r215=-------------------
3
b12 *kap
- 2*m3*n3*r494
r217=-----------------
2
b12
n1*n3*r494
r218=------------
2
b12
n1*n2*r494
r219=------------
2
b12
1 2
- ---*n2 *r494
2
r220=-----------------
2
b12
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
- a11*n3*r494
r310=----------------
2
b12 *kap
r311=0
- a11*n3*r494
r312=----------------
2
b12 *kap
r313=0
r314=0
- a11*n3*r494
r315=----------------
2
b12 *kap
- a11*m3*r494
r316=----------------
2
b12
- n1*r494
r317=------------
b12
n2*r494
r318=---------
b12
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
n1*r494
r326=---------
b12
r327=0
- 2*n3*r494
r328=--------------
b12
- a11*n2*r494
r329=----------------
2
b12
r330=0
r331=0
r332=0
a11*n3*r494
r333=-------------
2
b12
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- n2*r494
r341=------------
b12
2*n3*r494
r342=-----------
b12
r343=0
- a11*n1*r494
r344=----------------
2
b12
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
a11*n3*r494
r353=-------------
2
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
1
r427= - ---*r494
2
r428=0
r429=0
1
r430= - ---*r494
2
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=r494
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
a11*r494
r453=----------
b12
r454=0
r455=0
r456=0
1
r457=---*r494
2
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
2 1 2
- a11 *r494 - ---*b12 *kap*r494
2
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=r494
r484=0
r485=0
r486=0
- a11*r494
r487=-------------
b12
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
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
1
r4110=---*r494
2
r4111=0
r4112=0
r4113=0
2 1 2
- a11 *r494 - ---*b12 *kap*r494
2
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
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:
r494, n3, m3, a11, n2, 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.
{r494,a11,b12,n2}
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 2 4 2 6 3 2 4 2
FI=u1 *u3 *( - 2*a11 *b12 *kap - b12 *kap ) + 2*u1 *u3*a11*b12 *kap *n3
2 2 6 2 2 4 2 2 6 2
+ u1 *v1 *b12 *kap - u1 *b12 *kap *n2 + 2*u1*u2*v1*v2*b12 *kap
4 2 2 5 2
+ 2*u1*u2*b12 *kap *n1*n2 - 2*u1*u3 *v2*a11*b12 *kap
2 4 2 6 2
- 2*u1*u3 *a11*b12 *kap *n1 + 2*u1*u3*v1*v3*b12 *kap
5 2 5 2
+ 4*u1*u3*v2*b12 *kap *n3 - 2*u1*u3*v3*b12 *kap *n2
4 2 4 2
+ 2*u1*u3*b12 *kap *n1*n3 - 4*u1*v1*b12 *kap *m3*n3
3 2 2
- 2*u1*v3*a11*b12 *kap*n2*n3 - 2*u1*a11*b12 *kap*n1*n3
2 2 2 4 2 6 3 2 4 2
+ u2 *u3 *( - 2*a11 *b12 *kap - b12 *kap ) + 2*u2 *u3*a11*b12 *kap *n3
2 2 6 2 2 4 2 2 2 5 2
+ u2 *v2 *b12 *kap - u2 *b12 *kap *n1 + 2*u2*u3 *v1*a11*b12 *kap
2 4 2 5 2
- 2*u2*u3 *a11*b12 *kap *n2 - 4*u2*u3*v1*b12 *kap *n3
6 2 5 2
+ 2*u2*u3*v2*v3*b12 *kap + 2*u2*u3*v3*b12 *kap *n1
4 2 4 2
+ 2*u2*u3*b12 *kap *n2*n3 - 4*u2*v2*b12 *kap *m3*n3
3
+ 2*u2*v3*a11*b12 *kap*n1*n3 + u2
2 2 2 3 2
*( - 4*a11 *b12*kap*m3*n1*n3 - 2*a11*b12 *kap*n2*n3 - 4*b12 *kap *m3*n1*n3)
2 2 6 2 2 5 2 2 2 6 2
- u3 *v1 *b12 *kap + 2*u3 *v1*b12 *kap *n2 - u3 *v2 *b12 *kap
2 5 2 2 4 2
- 2*u3 *v2*b12 *kap *n1 - 2*u3 *v3*a11*b12 *kap *m3
2 4 3 2 4 2 2 4 2 2 4 2 2
+ u3 *(b12 *kap *m3 - b12 *kap *n1 - b12 *kap *n2 - b12 *kap *n3 )
2 4
- 2*u3*v1 *a11*b12 *kap*n3
3 4 2
+ u3*v1*(2*a11*b12 *kap*n2*n3 + 2*b12 *kap *m3*n1)
2 4
- 2*u3*v2 *a11*b12 *kap*n3
3 4 2
+ u3*v2*( - 2*a11*b12 *kap*n1*n3 + 2*b12 *kap *m3*n2)
2 4
- 2*u3*v3 *a11*b12 *kap*n3
2 2 2 2 3 3
+ u3*(2*a11*b12 *kap *m3 *n3 - 2*a11*b12 *kap*n3 ) + v1*( - 8*a11 *m3*n1*n3
2 2 2 3 2
- 4*a11 *b12*n2*n3 - 6*a11*b12 *kap*m3*n1*n3 - 4*b12 *kap*n2*n3 )
2
+ 2*v2*a11*b12 *kap*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**2*b12**4*kap**2 - b12**6*kap**3) + 2*u1**2*u3*a11*b12
**4*kap**2*n3 + u1**2*v1**2*b12**6*kap**2 - u1**2*b12**4*kap**2*n2**2 + 2*u1*u2*
v1*v2*b12**6*kap**2 + 2*u1*u2*b12**4*kap**2*n1*n2 - 2*u1*u3**2*v2*a11*b12**5*kap
**2 - 2*u1*u3**2*a11*b12**4*kap**2*n1 + 2*u1*u3*v1*v3*b12**6*kap**2 + 4*u1*u3*v2
*b12**5*kap**2*n3 - 2*u1*u3*v3*b12**5*kap**2*n2 + 2*u1*u3*b12**4*kap**2*n1*n3 -
4*u1*v1*b12**4*kap**2*m3*n3 - 2*u1*v3*a11*b12**3*kap*n2*n3 - 2*u1*a11*b12**2*kap
*n1*n3**2 + u2**2*u3**2*( - 2*a11**2*b12**4*kap**2 - b12**6*kap**3) + 2*u2**2*u3
*a11*b12**4*kap**2*n3 + u2**2*v2**2*b12**6*kap**2 - u2**2*b12**4*kap**2*n1**2 +
2*u2*u3**2*v1*a11*b12**5*kap**2 - 2*u2*u3**2*a11*b12**4*kap**2*n2 - 4*u2*u3*v1*
b12**5*kap**2*n3 + 2*u2*u3*v2*v3*b12**6*kap**2 + 2*u2*u3*v3*b12**5*kap**2*n1 + 2
*u2*u3*b12**4*kap**2*n2*n3 - 4*u2*v2*b12**4*kap**2*m3*n3 + 2*u2*v3*a11*b12**3*
kap*n1*n3 + u2*( - 4*a11**2*b12*kap*m3*n1*n3 - 2*a11*b12**2*kap*n2*n3**2 - 4*b12
**3*kap**2*m3*n1*n3) - u3**2*v1**2*b12**6*kap**2 + 2*u3**2*v1*b12**5*kap**2*n2 -
u3**2*v2**2*b12**6*kap**2 - 2*u3**2*v2*b12**5*kap**2*n1 - 2*u3**2*v3*a11*b12**4
*kap**2*m3 + u3**2*(b12**4*kap**3*m3**2 - b12**4*kap**2*n1**2 - b12**4*kap**2*n2
**2 - b12**4*kap**2*n3**2) - 2*u3*v1**2*a11*b12**4*kap*n3 + u3*v1*(2*a11*b12**3*
kap*n2*n3 + 2*b12**4*kap**2*m3*n1) - 2*u3*v2**2*a11*b12**4*kap*n3 + u3*v2*( - 2*
a11*b12**3*kap*n1*n3 + 2*b12**4*kap**2*m3*n2) - 2*u3*v3**2*a11*b12**4*kap*n3 +
u3*(2*a11*b12**2*kap**2*m3**2*n3 - 2*a11*b12**2*kap*n3**3) + v1*( - 8*a11**3*m3*
n1*n3 - 4*a11**2*b12*n2*n3**2 - 6*a11*b12**2*kap*m3*n1*n3 - 4*b12**3*kap*n2*n3**
2) + 2*v2*a11*b12**2*kap*m3*n2*n3$