Solution 3 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
r11=0
r12=0
r13=0
r14=0
r15=0
2
m3 *r460
r20=----------
2
b12
r21=0
2
m3 *r460
r22=----------
2
b12
r23=0
r24=0
r27=0
2*kap*m3*n1*r460
r28=------------------
2
a11
2 2 2 5 2 2 3 2 2 2
a11 *kap *m3 *r460 - ---*a11 *kap*n1 *r460 + ---*b12 *kap *n1 *r460
2 2
r29=---------------------------------------------------------------------
4
a11
r210=0
r211=0
r212=0
r213=0
2
- kap*n1 *r460
r214=-----------------
2
a11
r215=0
r217=0
r218=0
r219=0
2
kap*m3 *r460
r220=--------------
2
b12
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
2*m3*r460
r314=-----------
b12
r315=0
- kap*m3*r460
r316=----------------
a11
- 2*b12*kap*n1*r460
r317=----------------------
2
a11
r318=0
r319=0
r320=0
r321=0
r322=0
- 2*m3*r460
r323=--------------
b12
r324=0
r325=0
2*b12*kap*n1*r460
r326=-------------------
2
a11
- 2*kap*m3*r460
r327=------------------
a11
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 2 2
a11 *kap*n1*r460 - 2*b12 *kap *n1*r460
r344=----------------------------------------
3
a11
r345=0
r346=0
r347=0
r348=0
- 2*kap*n1*r460
r349=------------------
a11
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
2
- b12 *kap*r460
r427=------------------
2
a11
r428=0
r429=0
2 2
a11 *r460 - b12 *kap*r460
r430=---------------------------
2
a11
r431=0
r432=0
r433=0
3 2 2 2 1 4 3
---*a11 *b12 *kap *r460 - ---*b12 *kap *r460
4 2
r434=----------------------------------------------
4
a11
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
2
2*b12 *kap*r460
r446=-----------------
2
a11
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
- b12*kap*r460
r453=-----------------
a11
r454=0
r455=0
r456=0
2
b12 *kap*r460
r457=---------------
2
a11
r458=0
r459=0
r461=0
r462=0
r463=0
r464=0
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
2 2
- 2*a11 *r460 + 2*b12 *kap*r460
r483=----------------------------------
2
a11
r484=0
r485=0
r486=0
- b12*kap*r460
r487=-----------------
a11
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
2 2
- 2*a11 *r460 + 2*b12 *kap*r460
r494=----------------------------------
2
a11
r495=0
2*b12*kap*r460
r496=----------------
a11
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
2 2
- a11 *r460 + b12 *kap*r460
r4110=------------------------------
2
a11
r4111=0
r4112=0
r4113=0
2 2 2
2*a11 *kap*r460 - 2*b12 *kap *r460
r4114=------------------------------------
2
a11
r4115=0
r4116=0
r4117=0
r4118=0
r4119= - kap*r460
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
n3=0
n2=0
1 2
---*b12 *kap
2
a33=--------------
a11
a23=0
a22=0
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:
r460, m3, a11, 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.
{a11,b12,n1,r460}
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 1 2 2
HAM=(u1 *a11 + u1*v2*a11*b12 + u1*a11*n1 - u2*v1*a11*b12 + ---*u3 *b12 *kap
2
+ v3*a11*m3)/a11
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 4 2 2 2 4 2 2 4 2
FI= - u1 *u2 *a11 *b12 *kap + u1 *u3 *(2*a11 *b12 *kap - 2*a11 *b12 *kap )
2 2 4 2 2 4 2 4 2
+ u1 *v1 *( - a11 *b12 + a11 *b12 *kap) + u1 *a11 *kap*m3
2 3 2 3 3
- 2*u1*u2 *a11 *b12 *kap*n1 + 2*u1*u2*u3*v3*a11 *b12 *kap
4 2 2 4 2 3 3
+ u1*u2*v1*v2*( - 2*a11 *b12 + 2*a11 *b12 *kap) - u1*u3 *v2*a11 *b12 *kap
2 3 2 4 2
+ u1*u3 *(a11 *b12 *kap*n1 - 2*a11*b12 *kap *n1)
4 2 2 4 2 2 4 2
+ u1*u3*v1*v3*( - 2*a11 *b12 + 2*a11 *b12 *kap) + u2 *v1 *a11 *b12
2 2 2 4 2 2 2 2 2 3 3
+ u2 *v2 *a11 *b12 *kap - u2 *a11 *b12 *kap*n1 - u2*u3 *v1*a11 *b12 *kap
2 4 3 2
+ 2*u2*u3*v2*v3*a11 *b12 *kap - 2*u2*u3*v2*a11 *b12 *kap*m3
2 3 4
+ 2*u2*u3*v3*a11 *b12 *kap*n1 - 2*u2*v1*v3*a11 *b12*m3
4 3 2 4 2 1 6 3
+ u3 *(---*a11 *b12 *kap - ---*b12 *kap )
4 2
2 2 4 2 2 4 2 2 2 4
+ u3 *v1 *(a11 *b12 - a11 *b12 *kap) - u3 *v2 *a11 *b12 *kap
2 2 3 2 3 2
- 2*u3 *v2*a11 *b12 *kap*n1 - u3 *v3*a11 *b12 *kap*m3
2 2 2 2 2 5 2 2 2 3 4 2 2
+ u3 *(a11 *b12 *kap *m3 - ---*a11 *b12 *kap*n1 + ---*b12 *kap *n1 )
2 2
4 2 2 2 4 2
+ 2*u3*v1*v2*a11 *b12*m3 + 2*u3*v1*a11 *b12 *kap*m3*n1 + v2 *a11 *m3
2 4 2
+ v3 *a11 *m3
1
= a product of the elements of: { - ---,
4
2 2 4 2 2 2 4 2 2 4 2
4*u1 *u2 *a11 *b12 *kap + u1 *u3 *( - 8*a11 *b12 *kap + 8*a11 *b12 *kap )
2 2 4 2 2 4 2 4 2
+ u1 *v1 *(4*a11 *b12 - 4*a11 *b12 *kap) - 4*u1 *a11 *kap*m3
2 3 2 3 3
+ 8*u1*u2 *a11 *b12 *kap*n1 - 8*u1*u2*u3*v3*a11 *b12 *kap
4 2 2 4 2 3 3
+ u1*u2*v1*v2*(8*a11 *b12 - 8*a11 *b12 *kap) + 4*u1*u3 *v2*a11 *b12 *kap
2 3 2 4 2
+ u1*u3 *( - 4*a11 *b12 *kap*n1 + 8*a11*b12 *kap *n1)
4 2 2 4 2 2 4 2
+ u1*u3*v1*v3*(8*a11 *b12 - 8*a11 *b12 *kap) - 4*u2 *v1 *a11 *b12
2 2 2 4 2 2 2 2
- 4*u2 *v2 *a11 *b12 *kap + 4*u2 *a11 *b12 *kap*n1
2 3 3 2 4
+ 4*u2*u3 *v1*a11 *b12 *kap - 8*u2*u3*v2*v3*a11 *b12 *kap
3 2 2 3
+ 8*u2*u3*v2*a11 *b12 *kap*m3 - 8*u2*u3*v3*a11 *b12 *kap*n1
4 4 2 4 2 6 3
+ 8*u2*v1*v3*a11 *b12*m3 + u3 *( - 3*a11 *b12 *kap + 2*b12 *kap )
2 2 4 2 2 4 2 2 2 4
+ u3 *v1 *( - 4*a11 *b12 + 4*a11 *b12 *kap) + 4*u3 *v2 *a11 *b12 *kap
2 2 3 2 3 2
+ 8*u3 *v2*a11 *b12 *kap*n1 + 4*u3 *v3*a11 *b12 *kap*m3
2 2 2 2 2 2 2 2 4 2 2
+ u3 *( - 4*a11 *b12 *kap *m3 + 10*a11 *b12 *kap*n1 - 6*b12 *kap *n1 )
4 2 2 2 4 2
- 8*u3*v1*v2*a11 *b12*m3 - 8*u3*v1*a11 *b12 *kap*m3*n1 - 4*v2 *a11 *m3
2 4 2
- 4*v3 *a11 *m3 }
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a11**2 + u1*v2*a11*b12 + u1*a11*n1 - u2*v1*a11*b12 + 1/2*u3**2*b12**2
*kap + v3*a11*m3)/a11$
FI= - u1**2*u2**2*a11**4*b12**2*kap + u1**2*u3**2*(2*a11**4*b12**2*kap - 2*a11**
2*b12**4*kap**2) + u1**2*v1**2*( - a11**4*b12**2 + a11**2*b12**4*kap) + u1**2*
a11**4*kap*m3**2 - 2*u1*u2**2*a11**3*b12**2*kap*n1 + 2*u1*u2*u3*v3*a11**3*b12**3
*kap + u1*u2*v1*v2*( - 2*a11**4*b12**2 + 2*a11**2*b12**4*kap) - u1*u3**2*v2*a11
**3*b12**3*kap + u1*u3**2*(a11**3*b12**2*kap*n1 - 2*a11*b12**4*kap**2*n1) + u1*
u3*v1*v3*( - 2*a11**4*b12**2 + 2*a11**2*b12**4*kap) + u2**2*v1**2*a11**4*b12**2
+ u2**2*v2**2*a11**2*b12**4*kap - u2**2*a11**2*b12**2*kap*n1**2 - u2*u3**2*v1*
a11**3*b12**3*kap + 2*u2*u3*v2*v3*a11**2*b12**4*kap - 2*u2*u3*v2*a11**3*b12**2*
kap*m3 + 2*u2*u3*v3*a11**2*b12**3*kap*n1 - 2*u2*v1*v3*a11**4*b12*m3 + u3**4*(3/4
*a11**2*b12**4*kap**2 - 1/2*b12**6*kap**3) + u3**2*v1**2*(a11**4*b12**2 - a11**2
*b12**4*kap) - u3**2*v2**2*a11**2*b12**4*kap - 2*u3**2*v2*a11**2*b12**3*kap*n1 -
u3**2*v3*a11**3*b12**2*kap*m3 + u3**2*(a11**2*b12**2*kap**2*m3**2 - 5/2*a11**2*
b12**2*kap*n1**2 + 3/2*b12**4*kap**2*n1**2) + 2*u3*v1*v2*a11**4*b12*m3 + 2*u3*v1
*a11**2*b12**2*kap*m3*n1 + v2**2*a11**4*m3**2 + v3**2*a11**4*m3**2$