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
r11=0
r12=0
r13=0
r14=0
r15=0
2
- m3 *r453
r20=-------------
a11*b12
r21=0
2
- m3 *r453
r22=-------------
a11*b12
r23=0
r24=0
r27=0
- 2*m3*n1*r453
r28=-----------------
a11*b12
2 2
- kap*m3 *r453 + n1 *r453
r29=----------------------------
a11*b12
r210=0
r211=0
r212=0
r213=0
2
n1 *r453
r214=----------
a11*b12
r215=0
r217=0
r218=0
r219=0
2
- kap*m3 *r453
r220=-----------------
a11*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*r453
r314=--------------
a11
r315=0
m3*r453
r316=---------
b12
2*n1*r453
r317=-----------
a11
r318=0
r319=0
r320=0
r321=0
r322=0
2*m3*r453
r323=-----------
a11
r324=0
r325=0
- 2*n1*r453
r326=--------------
a11
2*m3*r453
r327=-----------
b12
r328=0
r329=0
2 2
- a11 *m3*r453 + b12 *kap*m3*r453
r330=------------------------------------
2
a11 *b12
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
n1*r453
r344=---------
b12
r345=0
r346=0
r347=0
r348=0
2 2
a11 *n1*r453 + b12 *kap*n1*r453
r349=---------------------------------
2
a11 *b12
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
b12*r453
r427=----------
a11
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
1
- ---*a11*r453
4
r434=-----------------
b12
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
- 2*b12*r453
r446=---------------
a11
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
r454=0
r455=0
r456=0
- b12*r453
r457=-------------
a11
r458=0
r459=0
- b12*r453
r460=-------------
a11
r461=0
r462=0
r463=0
1 2 1 2
- ---*a11 *r453 + ---*b12 *kap*r453
2 2
r464=--------------------------------------
a11*b12
r465=0
r466=0
2 2
a11 *r453 - b12 *kap*r453
r467=---------------------------
2
a11
r468=0
1 4 1 2 2 1 4 2
- ---*a11 *r453 + ---*a11 *b12 *kap*r453 - ---*b12 *kap *r453
4 2 4
r469=----------------------------------------------------------------
3
a11 *b12
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r479=0
r480=0
r482=0
r483=0
r484=0
r485=0
r486=0
r487=r453
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=0
r495=0
r496= - 2*r453
r497=0
r498=0
r499=0
r4100=0
2 2
- a11 *r453 + b12 *kap*r453
r4101=------------------------------
2
a11
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4118=0
b12*kap*r453
r4119=--------------
a11
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
n3=0
n2=0
1
a33=---*a11
2
a23=0
1 2 1 2
---*a11 - ---*b12 *kap
2 2
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:
r453, 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.
1 2 2 1 4 2
{{---*a11 *b12 *kap*r453 + ---*b12 *kap *r453,
8 8
r453,
1 2 2 1 4 2
---*a11 *b12 *kap*r453 - ---*b12 *kap *r453,
2 2
2 2 4 2
a11 *b12 *kap*r453 - b12 *kap *r453,
1*1*r453,
1*1*1*r453},
2 2
a11 + b12 *kap,
b12,
a11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 1 2 1 2
HAM=(u1 *a11 + u1*v2*a11*b12 + u1*a11*n1 + u2 *(---*a11 - ---*b12 *kap)
2 2
1 2 2
- u2*v1*a11*b12 + ---*u3 *a11 + v3*a11*m3)/a11
2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 2 2 2
FI=u1 *u2 *a11 *b12 *kap - u1 *a11 *kap*m3
2 3 3
+ u1*u2 *v2*( - a11 *b12 + a11*b12 *kap)
2 3 2 3
+ u1*u2 *(a11 *n1 + a11*b12 *kap*n1) - 2*u1*u2*u3*v3*a11 *b12
2 3 2 3
+ u1*u3 *v2*a11 *b12 + u1*u3 *a11 *n1
4 1 4 1 2 2 1 4 2
+ u2 *( - ---*a11 + ---*a11 *b12 *kap - ---*b12 *kap )
4 2 4
3 3 3
+ u2 *v1*(a11 *b12 - a11*b12 *kap)
2 2 1 4 1 2 2 2 2 2 2
+ u2 *u3 *( - ---*a11 + ---*a11 *b12 *kap) - u2 *v1 *a11 *b12
2 2
2 2 2 2 2 3 2 2 2 2
- u2 *v2 *a11 *b12 + u2 *v3*( - a11 *m3 + a11*b12 *kap*m3) + u2 *a11 *n1
2 3 2 2 3
+ u2*u3 *v1*a11 *b12 - 2*u2*u3*v2*v3*a11 *b12 + 2*u2*u3*v2*a11 *m3
2 2 1 4 4
- 2*u2*u3*v3*a11 *b12*n1 + 2*u2*v1*v3*a11 *b12*m3 - ---*u3 *a11
4
2 2 2 2 2 2 2 3
+ u3 *v2 *a11 *b12 + 2*u3 *v2*a11 *b12*n1 + u3 *v3*a11 *m3
2 2 2 2 2 2
+ u3 *( - a11 *kap*m3 + a11 *n1 ) - 2*u3*v1*v2*a11 *b12*m3
2 2 2 2 2 2 2
- 2*u3*v1*a11 *m3*n1 - v2 *a11 *m3 - v3 *a11 *m3
1
= a product of the elements of: { - ---,
4
2 2 2 2 2 2 2
- 4*u1 *u2 *a11 *b12 *kap + 4*u1 *a11 *kap*m3
2 3 3
+ u1*u2 *v2*(4*a11 *b12 - 4*a11*b12 *kap)
2 3 2 3
+ u1*u2 *( - 4*a11 *n1 - 4*a11*b12 *kap*n1) + 8*u1*u2*u3*v3*a11 *b12
2 3 2 3
- 4*u1*u3 *v2*a11 *b12 - 4*u1*u3 *a11 *n1
4 4 2 2 4 2
+ u2 *(a11 - 2*a11 *b12 *kap + b12 *kap )
3 3 3
+ u2 *v1*( - 4*a11 *b12 + 4*a11*b12 *kap)
2 2 4 2 2 2 2 2 2
+ u2 *u3 *(2*a11 - 2*a11 *b12 *kap) + 4*u2 *v1 *a11 *b12
2 2 2 2 2 3 2
+ 4*u2 *v2 *a11 *b12 + u2 *v3*(4*a11 *m3 - 4*a11*b12 *kap*m3)
2 2 2 2 3 2 2
- 4*u2 *a11 *n1 - 4*u2*u3 *v1*a11 *b12 + 8*u2*u3*v2*v3*a11 *b12
3 2 2
- 8*u2*u3*v2*a11 *m3 + 8*u2*u3*v3*a11 *b12*n1 - 8*u2*v1*v3*a11 *b12*m3
4 4 2 2 2 2 2 2 2 3
+ u3 *a11 - 4*u3 *v2 *a11 *b12 - 8*u3 *v2*a11 *b12*n1 - 4*u3 *v3*a11 *m3
2 2 2 2 2 2
+ u3 *(4*a11 *kap*m3 - 4*a11 *n1 ) + 8*u3*v1*v2*a11 *b12*m3
2 2 2 2 2 2 2
+ 8*u3*v1*a11 *m3*n1 + 4*v2 *a11 *m3 + 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**2*(1/2*a11**2 - 1/2*b12**2*
kap) - u2*v1*a11*b12 + 1/2*u3**2*a11**2 + v3*a11*m3)/a11$
FI=u1**2*u2**2*a11**2*b12**2*kap - u1**2*a11**2*kap*m3**2 + u1*u2**2*v2*( - a11
**3*b12 + a11*b12**3*kap) + u1*u2**2*(a11**3*n1 + a11*b12**2*kap*n1) - 2*u1*u2*
u3*v3*a11**3*b12 + u1*u3**2*v2*a11**3*b12 + u1*u3**2*a11**3*n1 + u2**4*( - 1/4*
a11**4 + 1/2*a11**2*b12**2*kap - 1/4*b12**4*kap**2) + u2**3*v1*(a11**3*b12 - a11
*b12**3*kap) + u2**2*u3**2*( - 1/2*a11**4 + 1/2*a11**2*b12**2*kap) - u2**2*v1**2
*a11**2*b12**2 - u2**2*v2**2*a11**2*b12**2 + u2**2*v3*( - a11**3*m3 + a11*b12**2
*kap*m3) + u2**2*a11**2*n1**2 + u2*u3**2*v1*a11**3*b12 - 2*u2*u3*v2*v3*a11**2*
b12**2 + 2*u2*u3*v2*a11**3*m3 - 2*u2*u3*v3*a11**2*b12*n1 + 2*u2*v1*v3*a11**2*b12
*m3 - 1/4*u3**4*a11**4 + u3**2*v2**2*a11**2*b12**2 + 2*u3**2*v2*a11**2*b12*n1 +
u3**2*v3*a11**3*m3 + u3**2*( - a11**2*kap*m3**2 + a11**2*n1**2) - 2*u3*v1*v2*a11
**2*b12*m3 - 2*u3*v1*a11**2*m3*n1 - v2**2*a11**2*m3**2 - v3**2*a11**2*m3**2$