Solution 1 to problem over
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Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r10=0
2*a11*m3*n2*n3*r460
r11=----------------------
2 2 4
a11 *b12 + b12 *kap
2*a11*m3*n1*n3*r460
r12=----------------------
2 2 4
a11 *b12 + b12 *kap
2 3
2*a11*kap*m3 *n3*r460 - 2*a11*n3 *r460
r13=----------------------------------------
2 2 4
a11 *b12 + b12 *kap
2
- 2*a11*n2*n3 *r460
r14=----------------------
2 2 4
a11 *b12 + b12 *kap
2
- 2*a11*n1*n3 *r460
r15=----------------------
2 2 4
a11 *b12 + b12 *kap
2
m3 *r460
r20=----------
2
b12
r21=0
2
m3 *r460
r22=----------
2
b12
r23=0
r24=0
4 3 2 2
r27=( - 2*a11 *m3*n2*r460 - 2*a11 *b12*n1*n3*r460 - 2*a11 *b12 *kap*m3*n2*r460
3 4 2 2 4 6 2
- 2*a11*b12 *kap*n1*n3*r460)/(a11 *b12 + 2*a11 *b12 *kap + b12 *kap )
3 2 2
r28=(2*a11 *n2*n3*r460 + 2*a11 *b12*kap*m3*n1*r460 + 2*a11*b12 *kap*n2*n3*r460
3 2 4 2 3 5 2
+ 2*b12 *kap *m3*n1*r460)/(a11 *b12 + 2*a11 *b12 *kap + b12 *kap )
12 2 12 2 10 2 2 2
r29=(a11 *n2 *r460 + a11 *n3 *r460 + a11 *b12 *kap *m3 *r460
10 2 2 10 2 2
- a11 *b12 *kap*n1 *r460 + 5*a11 *b12 *kap*n2 *r460
10 2 2 8 4 3 2
+ 5*a11 *b12 *kap*n3 *r460 + 5*a11 *b12 *kap *m3 *r460
8 4 2 2 8 4 2 2
- 5*a11 *b12 *kap *n1 *r460 + 10*a11 *b12 *kap *n2 *r460
8 4 2 2 6 6 4 2
+ 10*a11 *b12 *kap *n3 *r460 + 10*a11 *b12 *kap *m3 *r460
6 6 3 2 6 6 3 2
- 10*a11 *b12 *kap *n1 *r460 + 10*a11 *b12 *kap *n2 *r460
6 6 3 2 4 8 5 2
+ 10*a11 *b12 *kap *n3 *r460 + 10*a11 *b12 *kap *m3 *r460
4 8 4 2 4 8 4 2
- 10*a11 *b12 *kap *n1 *r460 + 5*a11 *b12 *kap *n2 *r460
4 8 4 2 2 10 6 2
+ 5*a11 *b12 *kap *n3 *r460 + 5*a11 *b12 *kap *m3 *r460
2 10 5 2 2 10 5 2
- 5*a11 *b12 *kap *n1 *r460 + a11 *b12 *kap *n2 *r460
2 10 5 2 12 7 2 12 6 2
+ a11 *b12 *kap *n3 *r460 + b12 *kap *m3 *r460 - b12 *kap *n1 *r460)/(
12 2 10 4 8 6 2 6 8 3
a11 *b12 + 6*a11 *b12 *kap + 15*a11 *b12 *kap + 20*a11 *b12 *kap
4 10 4 2 12 5 14 6
+ 15*a11 *b12 *kap + 6*a11 *b12 *kap + b12 *kap )
2 2
2*a11 *m3*n2*r460 + 2*a11*b12*n1*n3*r460 + 2*b12 *kap*m3*n2*r460
r210=------------------------------------------------------------------
2 2 4
a11 *b12 + b12 *kap
- 4*kap*m3*n3*r460
r211=---------------------
2 2
a11 + b12 *kap
r212=0
2*kap*n2*n3*r460
r213=------------------
2 2
a11 + b12 *kap
8 2 8 2 6 2 2
r214=(a11 *n2 *r460 + a11 *n3 *r460 - a11 *b12 *kap*n1 *r460
6 2 2 6 2 2
+ 4*a11 *b12 *kap*n2 *r460 + 4*a11 *b12 *kap*n3 *r460
4 4 2 2 4 4 2 2
- 3*a11 *b12 *kap *n1 *r460 + 6*a11 *b12 *kap *n2 *r460
4 4 2 2 2 6 3 2
+ 6*a11 *b12 *kap *n3 *r460 - 3*a11 *b12 *kap *n1 *r460
2 6 3 2 2 6 3 2
+ 4*a11 *b12 *kap *n2 *r460 + 4*a11 *b12 *kap *n3 *r460
8 4 2 8 4 2 8 4 2
- b12 *kap *n1 *r460 + b12 *kap *n2 *r460 + b12 *kap *n3 *r460)/(
8 2 6 4 4 6 2 2 8 3
a11 *b12 + 4*a11 *b12 *kap + 6*a11 *b12 *kap + 4*a11 *b12 *kap
10 4
+ b12 *kap )
- 2*a11*n2*n3*r460
r215=---------------------
2 3
a11 *b12 + b12 *kap
2 2
2*a11 *m3*n3*r460 - 2*b12 *kap*m3*n3*r460
r217=-------------------------------------------
2 2 4
a11 *b12 + b12 *kap
2*kap*n1*n3*r460
r218=------------------
2 2
a11 + b12 *kap
2*kap*n1*n2*r460
r219=------------------
2 2
a11 + b12 *kap
8 2 8 2 8 2
r220=(a11 *kap*m3 *r460 + a11 *n2 *r460 - a11 *n3 *r460
6 2 2 2 6 2 2
+ 4*a11 *b12 *kap *m3 *r460 + 3*a11 *b12 *kap*n2 *r460
6 2 2 4 4 3 2
- 4*a11 *b12 *kap*n3 *r460 + 6*a11 *b12 *kap *m3 *r460
4 4 2 2 4 4 2 2
+ 3*a11 *b12 *kap *n2 *r460 - 6*a11 *b12 *kap *n3 *r460
2 6 4 2 2 6 3 2
+ 4*a11 *b12 *kap *m3 *r460 + a11 *b12 *kap *n2 *r460
2 6 3 2 8 5 2 8 4 2
- 4*a11 *b12 *kap *n3 *r460 + b12 *kap *m3 *r460 - b12 *kap *n3 *r460)/(
8 2 6 4 4 6 2 2 8 3
a11 *b12 + 4*a11 *b12 *kap + 6*a11 *b12 *kap + 4*a11 *b12 *kap
10 4
+ b12 *kap )
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
- 2*a11*n3*r460
r310=------------------
2 2
a11 + b12 *kap
r311=0
- 2*a11*n3*r460
r312=------------------
2 2
a11 + b12 *kap
r313=0
2*m3*r460
r314=-----------
b12
- 2*a11*n3*r460
r315=------------------
2 2
a11 + b12 *kap
- 2*a11*kap*m3*r460
r316=----------------------
2 2
a11 + b12 *kap
- 2*b12*kap*n1*r460
r317=----------------------
2 2
a11 + b12 *kap
2
- 2*a11 *n2*r460
r318=---------------------
2 3
a11 *b12 + b12 *kap
r319=0
r320=0
r321=0
r322=0
- 2*m3*r460
r323=--------------
b12
r324=0
r325=0
2*b12*kap*n1*r460
r326=-------------------
2 2
a11 + b12 *kap
r327=0
- 4*b12*kap*n3*r460
r328=----------------------
2 2
a11 + b12 *kap
- 2*a11*kap*n2*r460
r329=----------------------
2 2
a11 + b12 *kap
- 2*kap*m3*r460
r330=------------------
a11
r331=0
- 2*n2*r460
r332=--------------
b12
2 2
- 2*b12 *kap *n3*r460
r333=------------------------
3 2
a11 + a11*b12 *kap
- 2*kap*n2*r460
r334=------------------
a11
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
2
2*a11 *n2*r460
r341=---------------------
2 3
a11 *b12 + b12 *kap
2 2
- 2*a11 *n3*r460 + 2*b12 *kap*n3*r460
r342=----------------------------------------
2 3
a11 *b12 + b12 *kap
r343=0
- 2*a11*kap*n1*r460
r344=----------------------
2 2
a11 + b12 *kap
2*n3*r460
r345=-----------
b12
2*n2*r460
r346=-----------
b12
r347=0
r348=0
- 2*kap*n1*r460
r349=------------------
a11
r350=0
r351=0
r352=0
2*a11*kap*n3*r460
r353=-------------------
2 2
a11 + 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
2
- b12 *kap*r460
r427=------------------
2 2
a11 + b12 *kap
r428=0
r429=0
2
a11 *r460
r430=-----------------
2 2
a11 + b12 *kap
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
2
2*b12 *kap*r460
r446=-----------------
2 2
a11 + b12 *kap
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
2*a11*b12*kap*r460
r453=--------------------
2 2
a11 + b12 *kap
r454=0
r455=0
r456=0
2
b12 *kap*r460
r457=-----------------
2 2
a11 + b12 *kap
r458=0
r459=0
r461=0
r462=0
r463=0
2 2
b12 *kap *r460
r464=-----------------
2 2
a11 + b12 *kap
r465=0
r466=0
2*b12*kap*r460
r467=----------------
a11
r468=0
2 2
b12 *kap *r460
r469=----------------
2
a11
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r479=0
r480=0
r482=0
2
- 2*a11 *r460
r483=-----------------
2 2
a11 + b12 *kap
r484=0
r485=0
r486=0
- 2*a11*b12*kap*r460
r487=-----------------------
2 2
a11 + b12 *kap
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
2
- 2*a11 *r460
r494=-----------------
2 2
a11 + b12 *kap
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
- 2*b12*kap*r460
r4101=-------------------
a11
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
2
- a11 *r460
r4110=-----------------
2 2
a11 + b12 *kap
r4111=0
r4112=0
r4113=0
2
- a11 *kap*r460
r4114=------------------
2 2
a11 + b12 *kap
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
a33=0
a23=0
2
- b12 *kap
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:
r460, n2, m3, n1, a11, n3, 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
{a22,a11 + b12 *kap,b12,r460,a11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2
HAM=(u1 *a11 + u1*v2*a11*b12 + u1*a11*n1 - u2 *b12 *kap - u2*v1*a11*b12
+ u2*a11*n2 + u3*a11*n3 + v3*a11*m3)/a11
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 4 2 2 4 2 2 2 4 2
FI=u1 *u2 *( - a11 *b12 *kap - a11 *b12 *kap ) - u1 *u3 *a11 *b12 *kap
2 3 2 2 2 4 2 2 4 2
+ 2*u1 *u3*a11 *b12 *kap*n3 - u1 *v1 *a11 *b12 + u1 *(a11 *kap*m3
4 2 4 2 2 2 2 2 2 2 2
+ a11 *n2 - a11 *n3 + a11 *b12 *kap *m3 - a11 *b12 *kap*n3 )
2 3 3 5 2
+ u1*u2 *v2*( - 2*a11 *b12 *kap - 2*a11*b12 *kap )
2 3 2 4 2
+ u1*u2 *( - 2*a11 *b12 *kap*n1 - 2*a11*b12 *kap *n1)
4 2 4 2 3
- 2*u1*u2*v1*v2*a11 *b12 + u1*u2*v2*(2*a11 *b12*n2 + 2*a11 *b12 *kap*n2)
4 2 3
+ u1*u2*v3*(2*a11 *b12*n3 + 2*a11 *b12 *kap*n3)
2 2 2 3 3
+ 2*u1*u2*a11 *b12 *kap*n1*n2 - 2*u1*u3 *v2*a11 *b12 *kap
2 3 2 4 2
- 2*u1*u3 *a11 *b12 *kap*n1 - 2*u1*u3*v1*v3*a11 *b12
4 2 3 4
+ u1*u3*v2*( - 2*a11 *b12*n3 + 2*a11 *b12 *kap*n3) + 2*u1*u3*v3*a11 *b12*n2
2 2 4 2 2
+ 2*u1*u3*a11 *b12 *kap*n1*n3 + u1*v1*(2*a11 *m3*n3 - 2*a11 *b12 *kap*m3*n3)
3 3 2
- 2*u1*v3*a11 *b12*n2*n3 - 2*u1*a11 *n1*n3
4 2 4 2 6 3
+ u2 *(a11 *b12 *kap + b12 *kap )
3 3 3 5 2
+ u2 *v1*(2*a11 *b12 *kap + 2*a11*b12 *kap )
3 3 2 4 2 2 2 2 4 2
+ u2 *( - 2*a11 *b12 *kap*n2 - 2*a11*b12 *kap *n2) + u2 *u3 *a11 *b12 *kap
2 4 2 2 2 4 2 2 4
- 2*u2 *u3*a11*b12 *kap *n3 + u2 *v1 *(a11 *b12 + a11 *b12 *kap)
2 4 2 3 2 2 2 4
+ u2 *v1*( - 2*a11 *b12*n2 - 2*a11 *b12 *kap*n2) + u2 *v2 *a11 *b12 *kap
2 3 2 4 2 2 4 2
+ u2 *v3*( - 2*a11 *b12 *kap*m3 - 2*a11*b12 *kap *m3) + u2 *(a11 *n2
4 2 2 2 2 2 2 2 2 2 2
+ a11 *n3 - a11 *b12 *kap*n1 + a11 *b12 *kap*n2 + a11 *b12 *kap*n3 )
2 3 3 2 3 2
+ 2*u2*u3 *v1*a11 *b12 *kap - 2*u2*u3 *a11 *b12 *kap*n2
2 3 2 4
- 4*u2*u3*v1*a11 *b12 *kap*n3 + 2*u2*u3*v2*v3*a11 *b12 *kap
2 3 2 2
+ 2*u2*u3*v3*a11 *b12 *kap*n1 + 2*u2*u3*a11 *b12 *kap*n2*n3
4 2 3
+ u2*v1*v3*( - 2*a11 *b12*m3 - 2*a11 *b12 *kap*m3)
2 2
- 4*u2*v2*a11 *b12 *kap*m3*n3
4 3 2 2
+ u2*v3*(2*a11 *m3*n2 + 2*a11 *b12*n1*n3 + 2*a11 *b12 *kap*m3*n2)
3 2 2 2 4 2 2 4
- 2*u2*a11 *n2*n3 + u3 *v1 *a11 *b12 - 2*u3 *v1*a11 *b12*n2
2 2 2 4 2 2 3
- u3 *v2 *a11 *b12 *kap - 2*u3 *v2*a11 *b12 *kap*n1
2 3 2
- 2*u3 *v3*a11 *b12 *kap*m3
2 4 2 4 2 2 2 2 2 2 2 2
+ u3 *(a11 *n2 + a11 *n3 + a11 *b12 *kap *m3 - a11 *b12 *kap*n1 )
2 3 2 4 2 3
- 2*u3*v1 *a11 *b12 *n3 + u3*v1*v2*(2*a11 *b12*m3 + 2*a11 *b12 *kap*m3)
3 2 2 2 3 2
+ u3*v1*(2*a11 *b12*n2*n3 + 2*a11 *b12 *kap*m3*n1) - 2*u3*v2 *a11 *b12 *n3
4 3 2 3 2
+ u3*v2*( - 2*a11 *m3*n2 - 2*a11 *b12*n1*n3) - 2*u3*v3 *a11 *b12 *n3
3 2 3 3 3
+ u3*(2*a11 *kap*m3 *n3 - 2*a11 *n3 ) + 2*v1*a11 *m3*n1*n3
2 4 2 2 2 2 3
+ v2 *(a11 *m3 + a11 *b12 *kap*m3 ) + 2*v2*a11 *m3*n2*n3
2 4 2 2 2 2
+ v3 *(a11 *m3 + a11 *b12 *kap*m3 )
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a11**2 + u1*v2*a11*b12 + u1*a11*n1 - u2**2*b12**2*kap - u2*v1*a11*b12
+ u2*a11*n2 + u3*a11*n3 + v3*a11*m3)/a11$
FI=u1**2*u2**2*( - a11**4*b12**2*kap - a11**2*b12**4*kap**2) - u1**2*u3**2*a11**
4*b12**2*kap + 2*u1**2*u3*a11**3*b12**2*kap*n3 - u1**2*v1**2*a11**4*b12**2 + u1
**2*(a11**4*kap*m3**2 + a11**4*n2**2 - a11**4*n3**2 + a11**2*b12**2*kap**2*m3**2
- a11**2*b12**2*kap*n3**2) + u1*u2**2*v2*( - 2*a11**3*b12**3*kap - 2*a11*b12**5
*kap**2) + u1*u2**2*( - 2*a11**3*b12**2*kap*n1 - 2*a11*b12**4*kap**2*n1) - 2*u1*
u2*v1*v2*a11**4*b12**2 + u1*u2*v2*(2*a11**4*b12*n2 + 2*a11**2*b12**3*kap*n2) +
u1*u2*v3*(2*a11**4*b12*n3 + 2*a11**2*b12**3*kap*n3) + 2*u1*u2*a11**2*b12**2*kap*
n1*n2 - 2*u1*u3**2*v2*a11**3*b12**3*kap - 2*u1*u3**2*a11**3*b12**2*kap*n1 - 2*u1
*u3*v1*v3*a11**4*b12**2 + u1*u3*v2*( - 2*a11**4*b12*n3 + 2*a11**2*b12**3*kap*n3)
+ 2*u1*u3*v3*a11**4*b12*n2 + 2*u1*u3*a11**2*b12**2*kap*n1*n3 + u1*v1*(2*a11**4*
m3*n3 - 2*a11**2*b12**2*kap*m3*n3) - 2*u1*v3*a11**3*b12*n2*n3 - 2*u1*a11**3*n1*
n3**2 + u2**4*(a11**2*b12**4*kap**2 + b12**6*kap**3) + u2**3*v1*(2*a11**3*b12**3
*kap + 2*a11*b12**5*kap**2) + u2**3*( - 2*a11**3*b12**2*kap*n2 - 2*a11*b12**4*
kap**2*n2) + u2**2*u3**2*a11**2*b12**4*kap**2 - 2*u2**2*u3*a11*b12**4*kap**2*n3
+ u2**2*v1**2*(a11**4*b12**2 + a11**2*b12**4*kap) + u2**2*v1*( - 2*a11**4*b12*n2
- 2*a11**2*b12**3*kap*n2) + u2**2*v2**2*a11**2*b12**4*kap + u2**2*v3*( - 2*a11
**3*b12**2*kap*m3 - 2*a11*b12**4*kap**2*m3) + u2**2*(a11**4*n2**2 + a11**4*n3**2
- a11**2*b12**2*kap*n1**2 + a11**2*b12**2*kap*n2**2 + a11**2*b12**2*kap*n3**2)
+ 2*u2*u3**2*v1*a11**3*b12**3*kap - 2*u2*u3**2*a11**3*b12**2*kap*n2 - 4*u2*u3*v1
*a11**2*b12**3*kap*n3 + 2*u2*u3*v2*v3*a11**2*b12**4*kap + 2*u2*u3*v3*a11**2*b12
**3*kap*n1 + 2*u2*u3*a11**2*b12**2*kap*n2*n3 + u2*v1*v3*( - 2*a11**4*b12*m3 - 2*
a11**2*b12**3*kap*m3) - 4*u2*v2*a11**2*b12**2*kap*m3*n3 + u2*v3*(2*a11**4*m3*n2
+ 2*a11**3*b12*n1*n3 + 2*a11**2*b12**2*kap*m3*n2) - 2*u2*a11**3*n2*n3**2 + u3**2
*v1**2*a11**4*b12**2 - 2*u3**2*v1*a11**4*b12*n2 - u3**2*v2**2*a11**2*b12**4*kap
- 2*u3**2*v2*a11**2*b12**3*kap*n1 - 2*u3**2*v3*a11**3*b12**2*kap*m3 + u3**2*(a11
**4*n2**2 + a11**4*n3**2 + a11**2*b12**2*kap**2*m3**2 - a11**2*b12**2*kap*n1**2)
- 2*u3*v1**2*a11**3*b12**2*n3 + u3*v1*v2*(2*a11**4*b12*m3 + 2*a11**2*b12**3*kap
*m3) + u3*v1*(2*a11**3*b12*n2*n3 + 2*a11**2*b12**2*kap*m3*n1) - 2*u3*v2**2*a11**
3*b12**2*n3 + u3*v2*( - 2*a11**4*m3*n2 - 2*a11**3*b12*n1*n3) - 2*u3*v3**2*a11**3
*b12**2*n3 + u3*(2*a11**3*kap*m3**2*n3 - 2*a11**3*n3**3) + 2*v1*a11**3*m3*n1*n3
+ v2**2*(a11**4*m3**2 + a11**2*b12**2*kap*m3**2) + 2*v2*a11**3*m3*n2*n3 + v3**2*
(a11**4*m3**2 + a11**2*b12**2*kap*m3**2)$