Solution 1 to problem over
Expressions |
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 *r460
r20=----------
2
b12
r21=0
2
m3 *r460
r22=----------
2
b12
r23=0
r24=0
r27=0
2*m3*n1*r460
r28=--------------
2
b12
2 2
kap*m3 *r460 - n1 *r460
r29=-------------------------
2
b12
r210=0
r211=0
r212=0
r213=0
2
- n1 *r460
r214=-------------
2
b12
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
- a11*m3*r460
r316=----------------
2
b12
- 2*n1*r460
r317=--------------
b12
r318=0
r319=0
r320=0
r321=0
r322=0
- 2*m3*r460
r323=--------------
b12
r324=0
r325=0
2*n1*r460
r326=-----------
b12
- 2*a11*m3*r460
r327=------------------
2
b12
r328=0
r329=0
2 2
a11 *m3*r460 - b12 *kap*m3*r460
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
- a11*n1*r460
r344=----------------
2
b12
r345=0
r346=0
r347=0
r348=0
2 2
- a11 *n1*r460 - b12 *kap*n1*r460
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
r427= - r460
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
1 2
---*a11 *r460
4
r434=---------------
2
b12
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r446=2*r460
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
- a11*r460
r453=-------------
b12
r454=0
r455=0
r456=0
r457=r460
r458=0
r459=0
r461=0
r462=0
r463=0
1 2 1 2
---*a11 *r460 - ---*b12 *kap*r460
2 2
r464=-----------------------------------
2
b12
r465=0
r466=0
2 2
- a11 *r460 + b12 *kap*r460
r467=------------------------------
a11*b12
r468=0
1 4 1 2 2 1 4 2
---*a11 *r460 - ---*a11 *b12 *kap*r460 + ---*b12 *kap *r460
4 2 4
r469=-------------------------------------------------------------
2 2
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
- a11*r460
r487=-------------
b12
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=0
r495=0
2*a11*r460
r496=------------
b12
r497=0
r498=0
r499=0
r4100=0
2 2
a11 *r460 - b12 *kap*r460
r4101=---------------------------
a11*b12
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
r4119= - kap*r460
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:
r460, a11, m3, 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.
{n1,b12,r460,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$