Solution 8 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2 2 2 2
0=a11 *n2 + a11 *n3 + b12 *kap*n3
Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
2
n3 *r312
r13=----------
2
b12
n2*n3*r312
r14=------------
2
b12
n1*n3*r312
r15=------------
2
b12
r20=0
r21=0
r22=0
r23=0
r24=0
n1*r312
r27=---------
b12
- n2*r312
r28=------------
b12
1 2 2 1 2 2 1 2 2
- ---*a11 *n1 *r312 - ---*a11 *n2 *r312 + ---*b12 *kap*n3 *r312
2 2 2
r29=------------------------------------------------------------------
2
a11*b12 *n3
- n1*r312
r210=------------
b12
r211=0
r212=0
2 2 2 2 2 2
a11 *n2 *r312 + a11 *n3 *r312 + b12 *kap*n3 *r312
r213=---------------------------------------------------
2
a11*b12 *n2
1 2 2 1 2 2 1 2 2
- ---*a11 *n1 *r312 + ---*a11 *n3 *r312 + ---*b12 *kap*n3 *r312
2 2 2
r214=------------------------------------------------------------------
2
a11*b12 *n3
n2*r312
r215=---------
b12
r217=0
a11*n1*r312
r218=-------------
2
b12
a11*n1*n2*r312
r219=----------------
2
b12 *n3
1 2 2 1 2 2 1 2 2
- ---*a11 *n2 *r312 + ---*a11 *n3 *r312 + ---*b12 *kap*n3 *r312
2 2 2
r220=------------------------------------------------------------------
2
a11*b12 *n3
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=r312
r311=0
r313=0
r314=0
r315=r312
r316=0
- a11*n1*r312
r317=----------------
b12*n3
2 2 2 2 2 2
a11 *n2 *r312 - a11 *n3 *r312 - b12 *kap*n3 *r312
r318=---------------------------------------------------
a11*b12*n2*n3
2 2
- a11 *r312 - b12 *kap*r312
r319=------------------------------
2
b12
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
a11*n1*r312
r326=-------------
b12*n3
r327=0
- 2*a11*r312
r328=---------------
b12
2 2 2 2 2 2
- a11 *n2 *r312 + a11 *n3 *r312 + b12 *kap*n3 *r312
r329=------------------------------------------------------
2
b12 *n2*n3
r330=0
r331=0
r332=0
3 4 2 1 4 2 1 2 2 2
r333=(---*a11 *n2 *r312 + ---*a11 *n3 *r312 + ---*a11 *b12 *kap*n2 *r312
2 2 2
2 2 2 1 4 2 2 2 2 2
+ a11 *b12 *kap*n3 *r312 + ---*b12 *kap *n3 *r312)/(a11 *b12 *n2 )
2
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
2 2 2 2 2 2
- a11 *n2 *r312 + a11 *n3 *r312 + b12 *kap*n3 *r312
r341=------------------------------------------------------
a11*b12*n2*n3
2*a11*r312
r342=------------
b12
r343=0
2
- a11 *n1*r312
r344=-----------------
2
b12 *n3
r345=0
r346=0
r347=0
2 2
a11 *n1*r312 + b12 *kap*n1*r312
r348=---------------------------------
2
b12 *n2
r349=0
r350=0
r351=0
r352=0
1 4 2 1 4 2 1 2 2 2
r353=(---*a11 *n2 *r312 + ---*a11 *n3 *r312 - ---*a11 *b12 *kap*n2 *r312
2 2 2
2 2 2 1 4 2 2 2 2 2
+ a11 *b12 *kap*n3 *r312 + ---*b12 *kap *n3 *r312)/(a11 *b12 *n2 )
2
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
- ---*a11*r312
2
r427=-----------------
n3
r428=0
r429=0
1
- ---*a11*r312
2
r430=-----------------
n3
r431=0
r432=0
2 2
a11 *r312 + b12 *kap*r312
r433=---------------------------
b12*n2
1 4 2 2 1 4 2
- ---*a11 *n3*r312 - a11 *b12 *kap*n3*r312 - ---*b12 *kap *n3*r312
2 2
r434=---------------------------------------------------------------------
2 2
a11*b12 *n2
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
a11*r312
r446=----------
n3
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
2
a11 *r312
r453=-----------
b12*n3
3 2
- a11 *r312 - a11*b12 *kap*r312
r454=----------------------------------
2
b12 *n2
r455=0
r456=0
1
---*a11*r312
2
r457=--------------
n3
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
4 2 1 4 2 1 2 2 2
r464=( - a11 *n2 *r312 - ---*a11 *n3 *r312 - ---*a11 *b12 *kap*n2 *r312
2 2
2 2 2 1 4 2 2 2 2
- a11 *b12 *kap*n3 *r312 - ---*b12 *kap *n3 *r312)/(a11*b12 *n2 *n3)
2
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
a11*r312
r483=----------
n3
r484=0
r485=0
2 2
- a11 *r312 - b12 *kap*r312
r486=------------------------------
b12*n2
2
- a11 *r312
r487=--------------
b12*n3
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
a11*r312
r494=----------
n3
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
---*a11*r312
2
r4110=--------------
n3
r4111=0
r4112=0
r4113=0
4 2 4 2 1 2 2 2
r4114=( - a11 *n2 *r312 - a11 *n3 *r312 - ---*a11 *b12 *kap*n2 *r312
2
2 2 2 4 2 2 2 2
- 2*a11 *b12 *kap*n3 *r312 - b12 *kap *n3 *r312)/(a11*b12 *n2 *n3)
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m3=0
m2=0
m1=0
a33=0
1 2 1 2
---*a11 *n3 + ---*b12 *kap*n3
2 2
a23=-------------------------------
a11*n2
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:
r312, a11, n3, n1, n2, 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
{n1,n2,b12,n3,a11,r312,a11 + b12 *kap}
Relevance for the application:
Modulo the following equation:
2 2 2 2 2 2
0=a11 *n2 + a11 *n3 + b12 *kap*n3
the system of equations related to the Hamiltonian HAM:
2 2 2 2
HAM=(u1 *a11 *n2 + u1*v2*a11*b12*n2 + u1*a11*n1*n2 + u2 *a11 *n2
2 2 2
+ u2*u3*(a11 *n3 + b12 *kap*n3) - u2*v1*a11*b12*n2 + u2*a11*n2
+ u3*a11*n2*n3)/(a11*n2)
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 5 2 5 2 3 2 2 3 2 2
FI=u1 *u3 *( - 2*a11 *n2 - 2*a11 *n3 - a11 *b12 *kap*n2 - 4*a11 *b12 *kap*n3
4 2 2 2 4 2 4 3
- 2*a11*b12 *kap *n3 ) + u1 *u3*(a11 *n2 *n3 + a11 *n3
2 2 2 2 2 3 4 2 3
- a11 *b12 *kap*n2 *n3 + 2*a11 *b12 *kap*n3 + b12 *kap *n3 )
2 2 3 2 2
+ u1 *v1 *a11 *b12 *n2
2 3 4 3 2 2 2 2 2
+ u1 *( - a11 *n2 + a11 *n2 *n3 + a11*b12 *kap*n2 *n3 )
4 2 2
+ u1*u2*u3*(2*a11 *n1*n2*n3 + 2*a11 *b12 *kap*n1*n2*n3)
3 2 2 3 3
+ 2*u1*u2*v1*v2*a11 *b12 *n2 + 2*u1*u2*a11 *n1*n2
2 4 2
- 2*u1*u3 *v2*a11 *b12*n2
2 4 2 3
+ u1*u3 *v3*( - 2*a11 *b12*n2*n3 - 2*a11 *b12 *kap*n2*n3)
2 4 2 3 2 2
- 2*u1*u3 *a11 *n1*n2 + 2*u1*u3*v1*v3*a11 *b12 *n2
3 2
+ 4*u1*u3*v2*a11 *b12*n2 *n3
3 3 3 2 3 2
+ u1*u3*v3*( - 2*a11 *b12*n2 + 2*a11 *b12*n2*n3 + 2*a11*b12 *kap*n2*n3 )
3 2 2 3 2 2 2
+ 2*u1*u3*a11 *n1*n2 *n3 + 2*u1*v3*a11 *b12*n2 *n3 + 2*u1*a11 *n1*n2 *n3 +
2 2 5 2 5 2 3 2 2 3 2 2
u2 *u3 *( - 2*a11 *n2 - a11 *n3 - a11 *b12 *kap*n2 - 2*a11 *b12 *kap*n3
4 2 2 2 4 2 4 3
- a11*b12 *kap *n3 ) + u2 *u3*(3*a11 *n2 *n3 + a11 *n3
2 2 2 2 2 3 4 2 3
+ a11 *b12 *kap*n2 *n3 + 2*a11 *b12 *kap*n3 + b12 *kap *n3 )
2 2 3 2 2
+ u2 *v2 *a11 *b12 *n2
2 3 2 2 3 2 2 2 2 2
+ u2 *( - a11 *n1 *n2 + a11 *n2 *n3 + a11*b12 *kap*n2 *n3 )
3 5 3 2
+ u2*u3 *( - 2*a11 *n2*n3 - 2*a11 *b12 *kap*n2*n3)
2 4 2
+ 2*u2*u3 *v1*a11 *b12*n2
2 4 3 4 2 2 2 2
+ u2*u3 *( - 2*a11 *n2 + 2*a11 *n2*n3 + 2*a11 *b12 *kap*n2*n3 )
3 2 3 2 2
- 4*u2*u3*v1*a11 *b12*n2 *n3 + 2*u2*u3*v2*v3*a11 *b12 *n2
3 2
+ 2*u2*u3*v3*a11 *b12*n1*n2
3 3 3 3 2 3
+ u2*u3*(2*a11 *n2 *n3 + 2*a11 *n2*n3 + 2*a11*b12 *kap*n2*n3 )
2 2 2 3 2
- 2*u2*v3*a11 *b12*n1*n2 *n3 + 2*u2*a11 *n2 *n3
4 5 2 3 2 2 4 2 2
+ u3 *( - a11 *n3 - 2*a11 *b12 *kap*n3 - a11*b12 *kap *n3 )
3 4 2 3
+ u3 *v1*(2*a11 *b12*n2*n3 + 2*a11 *b12 *kap*n2*n3)
3 4 2 2 2 2 2 2 3 2 2
+ u3 *( - 2*a11 *n2 *n3 - 2*a11 *b12 *kap*n2 *n3) - u3 *v1 *a11 *b12 *n2
2 3 3 3 2 3 2
+ u3 *v1*(2*a11 *b12*n2 - 2*a11 *b12*n2*n3 - 2*a11*b12 *kap*n2*n3 )
2 2 3 2 2 2 3 2
- u3 *v2 *a11 *b12 *n2 - 2*u3 *v2*a11 *b12*n1*n2
2 3 2 2 3 4 2 2 2
+ u3 *( - a11 *n1 *n2 - a11 *n2 + a11*b12 *kap*n2 *n3 )
2 2 2 2 2 3
+ 2*u3*v1 *a11 *b12 *n2 *n3 - 2*u3*v1*a11 *b12*n2 *n3
2 2 2 2 2 2
+ 2*u3*v2 *a11 *b12 *n2 *n3 + 2*u3*v2*a11 *b12*n1*n2 *n3
2 2 2 2 2 2 3
+ 2*u3*v3 *a11 *b12 *n2 *n3 + 2*u3*a11 *n2 *n3
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a11**2*n2 + u1*v2*a11*b12*n2 + u1*a11*n1*n2 + u2**2*a11**2*n2 + u2*u3
*(a11**2*n3 + b12**2*kap*n3) - u2*v1*a11*b12*n2 + u2*a11*n2**2 + u3*a11*n2*n3)/(
a11*n2)$
FI=u1**2*u3**2*( - 2*a11**5*n2**2 - 2*a11**5*n3**2 - a11**3*b12**2*kap*n2**2 - 4
*a11**3*b12**2*kap*n3**2 - 2*a11*b12**4*kap**2*n3**2) + u1**2*u3*(a11**4*n2**2*
n3 + a11**4*n3**3 - a11**2*b12**2*kap*n2**2*n3 + 2*a11**2*b12**2*kap*n3**3 + b12
**4*kap**2*n3**3) + u1**2*v1**2*a11**3*b12**2*n2**2 + u1**2*( - a11**3*n2**4 +
a11**3*n2**2*n3**2 + a11*b12**2*kap*n2**2*n3**2) + u1*u2*u3*(2*a11**4*n1*n2*n3 +
2*a11**2*b12**2*kap*n1*n2*n3) + 2*u1*u2*v1*v2*a11**3*b12**2*n2**2 + 2*u1*u2*a11
**3*n1*n2**3 - 2*u1*u3**2*v2*a11**4*b12*n2**2 + u1*u3**2*v3*( - 2*a11**4*b12*n2*
n3 - 2*a11**2*b12**3*kap*n2*n3) - 2*u1*u3**2*a11**4*n1*n2**2 + 2*u1*u3*v1*v3*a11
**3*b12**2*n2**2 + 4*u1*u3*v2*a11**3*b12*n2**2*n3 + u1*u3*v3*( - 2*a11**3*b12*n2
**3 + 2*a11**3*b12*n2*n3**2 + 2*a11*b12**3*kap*n2*n3**2) + 2*u1*u3*a11**3*n1*n2
**2*n3 + 2*u1*v3*a11**2*b12*n2**3*n3 + 2*u1*a11**2*n1*n2**2*n3**2 + u2**2*u3**2*
( - 2*a11**5*n2**2 - a11**5*n3**2 - a11**3*b12**2*kap*n2**2 - 2*a11**3*b12**2*
kap*n3**2 - a11*b12**4*kap**2*n3**2) + u2**2*u3*(3*a11**4*n2**2*n3 + a11**4*n3**
3 + a11**2*b12**2*kap*n2**2*n3 + 2*a11**2*b12**2*kap*n3**3 + b12**4*kap**2*n3**3
) + u2**2*v2**2*a11**3*b12**2*n2**2 + u2**2*( - a11**3*n1**2*n2**2 + a11**3*n2**
2*n3**2 + a11*b12**2*kap*n2**2*n3**2) + u2*u3**3*( - 2*a11**5*n2*n3 - 2*a11**3*
b12**2*kap*n2*n3) + 2*u2*u3**2*v1*a11**4*b12*n2**2 + u2*u3**2*( - 2*a11**4*n2**3
+ 2*a11**4*n2*n3**2 + 2*a11**2*b12**2*kap*n2*n3**2) - 4*u2*u3*v1*a11**3*b12*n2
**2*n3 + 2*u2*u3*v2*v3*a11**3*b12**2*n2**2 + 2*u2*u3*v3*a11**3*b12*n1*n2**2 + u2
*u3*(2*a11**3*n2**3*n3 + 2*a11**3*n2*n3**3 + 2*a11*b12**2*kap*n2*n3**3) - 2*u2*
v3*a11**2*b12*n1*n2**2*n3 + 2*u2*a11**2*n2**3*n3**2 + u3**4*( - a11**5*n3**2 - 2
*a11**3*b12**2*kap*n3**2 - a11*b12**4*kap**2*n3**2) + u3**3*v1*(2*a11**4*b12*n2*
n3 + 2*a11**2*b12**3*kap*n2*n3) + u3**3*( - 2*a11**4*n2**2*n3 - 2*a11**2*b12**2*
kap*n2**2*n3) - u3**2*v1**2*a11**3*b12**2*n2**2 + u3**2*v1*(2*a11**3*b12*n2**3 -
2*a11**3*b12*n2*n3**2 - 2*a11*b12**3*kap*n2*n3**2) - u3**2*v2**2*a11**3*b12**2*
n2**2 - 2*u3**2*v2*a11**3*b12*n1*n2**2 + u3**2*( - a11**3*n1**2*n2**2 - a11**3*
n2**4 + a11*b12**2*kap*n2**2*n3**2) + 2*u3*v1**2*a11**2*b12**2*n2**2*n3 - 2*u3*
v1*a11**2*b12*n2**3*n3 + 2*u3*v2**2*a11**2*b12**2*n2**2*n3 + 2*u3*v2*a11**2*b12*
n1*n2**2*n3 + 2*u3*v3**2*a11**2*b12**2*n2**2*n3 + 2*u3*a11**2*n2**2*n3**3$