Solution 3 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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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
r20=0
r21=0
r22=0
r23=0
r24=0
r26=0
r27=0
r28=0
r29=0
r210=0
r212=0
r213=0
r214=0
r215=0
r216=0
r217=0
r218=0
r219=0
1
- ---*m3*r483
2
r30=----------------
a11
1
- ---*m2*r483
2
r31=----------------
a11
1
- ---*m3*r483
2
r32=----------------
a11
1
- ---*m2*r483
2
r33=----------------
a11
1
- ---*m1*r483
2
r34=----------------
a11
r35=0
1
- ---*m1*r483
2
r36=----------------
a11
1
- ---*m3*r483
2
r37=----------------
a11
1
- ---*m2*r483
2
r38=----------------
a11
1
- ---*m1*r483
2
r39=----------------
a11
1
- ---*n3*r483
2
r310=----------------
a11
r311=0
1
- ---*n3*r483
2
r312=----------------
a11
r313=0
r314=0
1
- ---*n3*r483
2
r315=----------------
a11
r316=0
r317=0
r318=0
r319=0
1
- ---*n2*r483
2
r320=----------------
a11
r321=0
1
- ---*n2*r483
2
r322=----------------
a11
r323=0
r324=0
1
- ---*n2*r483
2
r325=----------------
a11
r326=0
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
1
- ---*n1*r483
2
r335=----------------
a11
r336=0
1
- ---*n1*r483
2
r337=----------------
a11
r338=0
r339=0
1
- ---*n1*r483
2
r340=----------------
a11
r341=0
r342=0
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
1
- ---*c33*r483
2
r40=-----------------
a11
- c23*r483
r41=-------------
a11
1 1
- ---*c22*r483 - ---*c33*r483
2 2
r42=--------------------------------
a11
- c23*r483
r43=-------------
a11
1
- ---*c22*r483
2
r44=-----------------
a11
- c13*r483
r45=-------------
a11
- c12*r483
r46=-------------
a11
- c13*r483
r47=-------------
a11
- c12*r483
r48=-------------
a11
1
- ---*c33*r483
2
r49=-----------------
a11
- c23*r483
r410=-------------
a11
1
- ---*c22*r483
2
r411=-----------------
a11
- c13*r483
r412=-------------
a11
- c12*r483
r413=-------------
a11
1
- ---*b33*r483
2
r415=-----------------
a11
1
- ---*b32*r483
2
r416=-----------------
a11
1
- ---*b33*r483
2
r417=-----------------
a11
1
- ---*b32*r483
2
r418=-----------------
a11
1
- ---*b31*r483
2
r419=-----------------
a11
r420=0
1
- ---*b31*r483
2
r421=-----------------
a11
1
- ---*b33*r483
2
r422=-----------------
a11
1
- ---*b32*r483
2
r423=-----------------
a11
1
- ---*b31*r483
2
r424=-----------------
a11
1
r425=---*r483
2
r426=0
r427=0
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
r434=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
r446=r483
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
r453=0
r454=0
1
r455= - ---*r483
2
r456=0
r458=0
r459=0
1
r460= - ---*r483
2
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
1
- ---*b11*r483
2
r474=-----------------
a11
r475=0
1
- ---*b11*r483
2
r476=-----------------
a11
r477=0
r478=0
1
- ---*b11*r483
2
r479=-----------------
a11
r480=0
r481=0
r482=0
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=r483
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
1
r4105= - ---*r483
2
r4106=0
1
r4107= - ---*r483
2
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=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:
r483, m3, c33, m2, m1, c22, n3, c23, c13, c12, b32,
b31, b33, b11, n2, n1, a11
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.
{{c33*r483,
3 3
576*a11 *c22*n2*r483 + 576*a11 *c33*n2*r483,
c23*r483,
c22*r483,
c13*r483,
c12*r483,
b33*r483,
b31*r483,
b32*r483,
r483},
a11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2
HAM=u1 *a11 + u1*v1*b11 + u1*n1 + u2 *a11 + u2*n2 + u3*v1*b31 + u3*v2*b32
2
+ u3*v3*b33 + u3*n3 + 2*v1*v2*c12 + 2*v1*v3*c13 + v1*m1 + v2 *c22
2
+ 2*v2*v3*c23 + v2*m2 + v3 *c33 + v3*m3
has apart from the Hamiltonian and Casimirs only the following first integral:
1 2 2 1 2 2
FI= - ---*u1 *v2 *a11 - ---*u1 *v3 *a11 + u1*u2*v1*v2*a11 + u1*u3*v1*v3*a11
2 2
1 3 1 2 1 2 1 2
- ---*u1*v1 *b11 - ---*u1*v1 *n1 - ---*u1*v1*v2 *b11 - ---*u1*v1*v3 *b11
2 2 2 2
1 2 1 2 1 2 2 1 2 2
- ---*u1*v2 *n1 - ---*u1*v3 *n1 - ---*u2 *v1 *a11 - ---*u2 *v3 *a11
2 2 2 2
1 2 1 2 1 2
+ u2*u3*v2*v3*a11 - ---*u2*v1 *n2 - ---*u2*v2 *n2 - ---*u2*v3 *n2
2 2 2
1 2 2 1 3 1 2 1 2
+ ---*u3 *v3 *a11 - ---*u3*v1 *b31 - ---*u3*v1 *v2*b32 - ---*u3*v1 *v3*b33
2 2 2 2
1 2 1 2 1 2 1 3
- ---*u3*v1 *n3 - ---*u3*v1*v2 *b31 - ---*u3*v1*v3 *b31 - ---*u3*v2 *b32
2 2 2 2
1 2 1 2 1 2 1 3
- ---*u3*v2 *v3*b33 - ---*u3*v2 *n3 - ---*u3*v2*v3 *b32 - ---*u3*v3 *b33
2 2 2 2
1 2 3 3 1 3 1 2 2
- ---*u3*v3 *n3 - v1 *v2*c12 - v1 *v3*c13 - ---*v1 *m1 - ---*v1 *v2 *c22
2 2 2
2 1 2 1 2 2 1 2
- v1 *v2*v3*c23 - ---*v1 *v2*m2 - ---*v1 *v3 *c33 - ---*v1 *v3*m3
2 2 2
3 2 1 2 2 3
- v1*v2 *c12 - v1*v2 *v3*c13 - ---*v1*v2 *m1 - v1*v2*v3 *c12 - v1*v3 *c13
2
1 2 1 4 3 1 3
- ---*v1*v3 *m1 - ---*v2 *c22 - v2 *v3*c23 - ---*v2 *m2
2 2 2
2 2 1 1 1 2 3
+ v2 *v3 *( - ---*c22 - ---*c33) - ---*v2 *v3*m3 - v2*v3 *c23
2 2 2
1 2 1 4 1 3
- ---*v2*v3 *m2 - ---*v3 *c33 - ---*v3 *m3
2 2 2
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*v1*b11 + u1*n1 + u2**2*a11 + u2*n2 + u3*v1*b31 + u3*v2*b32 +
u3*v3*b33 + u3*n3 + 2*v1*v2*c12 + 2*v1*v3*c13 + v1*m1 + v2**2*c22 + 2*v2*v3*c23
+ v2*m2 + v3**2*c33 + v3*m3$
FI= - 1/2*u1**2*v2**2*a11 - 1/2*u1**2*v3**2*a11 + u1*u2*v1*v2*a11 + u1*u3*v1*v3*
a11 - 1/2*u1*v1**3*b11 - 1/2*u1*v1**2*n1 - 1/2*u1*v1*v2**2*b11 - 1/2*u1*v1*v3**2
*b11 - 1/2*u1*v2**2*n1 - 1/2*u1*v3**2*n1 - 1/2*u2**2*v1**2*a11 - 1/2*u2**2*v3**2
*a11 + u2*u3*v2*v3*a11 - 1/2*u2*v1**2*n2 - 1/2*u2*v2**2*n2 - 1/2*u2*v3**2*n2 + 1
/2*u3**2*v3**2*a11 - 1/2*u3*v1**3*b31 - 1/2*u3*v1**2*v2*b32 - 1/2*u3*v1**2*v3*
b33 - 1/2*u3*v1**2*n3 - 1/2*u3*v1*v2**2*b31 - 1/2*u3*v1*v3**2*b31 - 1/2*u3*v2**3
*b32 - 1/2*u3*v2**2*v3*b33 - 1/2*u3*v2**2*n3 - 1/2*u3*v2*v3**2*b32 - 1/2*u3*v3**
3*b33 - 1/2*u3*v3**2*n3 - v1**3*v2*c12 - v1**3*v3*c13 - 1/2*v1**3*m1 - 1/2*v1**2
*v2**2*c22 - v1**2*v2*v3*c23 - 1/2*v1**2*v2*m2 - 1/2*v1**2*v3**2*c33 - 1/2*v1**2
*v3*m3 - v1*v2**3*c12 - v1*v2**2*v3*c13 - 1/2*v1*v2**2*m1 - v1*v2*v3**2*c12 - v1
*v3**3*c13 - 1/2*v1*v3**2*m1 - 1/2*v2**4*c22 - v2**3*v3*c23 - 1/2*v2**3*m2 + v2
**2*v3**2*( - 1/2*c22 - 1/2*c33) - 1/2*v2**2*v3*m3 - v2*v3**3*c23 - 1/2*v2*v3**2
*m2 - 1/2*v3**4*c33 - 1/2*v3**3*m3$