Solution 7 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
m3*r455
r30=---------
a11
m2*r455
r31=---------
a11
m3*r455
r32=---------
a11
m2*r455
r33=---------
a11
m1*r455
r34=---------
a11
r35=0
m1*r455
r36=---------
a11
m3*r455
r37=---------
a11
m2*r455
r38=---------
a11
m1*r455
r39=---------
a11
n3*r455
r310=---------
a11
r311=0
n3*r455
r312=---------
a11
r313=0
r314=0
n3*r455
r315=---------
a11
r316=0
r317=0
r318=0
r319=0
n2*r455
r320=---------
a11
r321=0
n2*r455
r322=---------
a11
r323=0
r324=0
n2*r455
r325=---------
a11
r326=0
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
r336=0
n1*r455
r337=---------
a11
r338=0
r339=0
n1*r455
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
c33*r455
r40=----------
a11
2*c23*r455
r41=------------
a11
c22*r455 + c33*r455
r42=---------------------
a11
2*c23*r455
r43=------------
a11
c22*r455
r44=----------
a11
2*c13*r455
r45=------------
a11
2*c12*r455
r46=------------
a11
2*c13*r455
r47=------------
a11
2*c12*r455
r48=------------
a11
c33*r455
r49=----------
a11
2*c23*r455
r410=------------
a11
c22*r455
r411=----------
a11
2*c13*r455
r412=------------
a11
2*c12*r455
r413=------------
a11
- b22*r455 + b33*r455
r415=------------------------
a11
b32*r455
r416=----------
a11
- b22*r455 + b33*r455
r417=------------------------
a11
b32*r455
r418=----------
a11
b31*r455
r419=----------
a11
r420=0
b31*r455
r421=----------
a11
- b22*r455 + b33*r455
r422=------------------------
a11
b32*r455
r423=----------
a11
b31*r455
r424=----------
a11
- a11*r455 + a33*r455
r425=------------------------
a11
r426=0
a33*r455
r427=----------
a11
r428=0
r429=0
a33*r455
r430=----------
a11
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= - 2*r455
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
r453=0
r454=0
r456=0
r458=0
r459=0
r460=r455
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
- b22*r455
r474=-------------
a11
r475=0
- b22*r455
r476=-------------
a11
r477=0
r478=0
- b22*r455
r479=-------------
a11
r480=0
r481=0
r482=0
r483= - 2*r455
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494= - 2*r455
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=r455
r4106=0
r4107=r455
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
n1*r455
r335=---------
a11
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:
r455, m3, c33, m2, m1, c22, n3, c23, c13, c12, n2,
b32, b31, b33, b22, n1, a33, 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.
{{r335,
4 3
8*a11 *r335 - 8*a11 *a33*r335,
a11*b22*r335 - a11*b33*r335,
b31*r335,
b32*r335,
c12*r335,
c22*r335,
c33*r335,
c13*r335,
3 3
24*a11 *c22*r335 + 24*a11 *c33*r335,
c23*r335},
a11 - a33,
a33,
a11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a11 + u1*n1 + u2 *a11 + u2*v2*b22 + u2*n2 + u3 *a33 + u3*v1*b31
+ u3*v2*b32 + u3*v3*b33 + u3*n3 + 2*v1*v2*c12 + 2*v1*v3*c13 + v1*m1
2 2
+ v2 *c22 + 2*v2*v3*c23 + v2*m2 + v3 *c33 + v3*m3
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2
FI=u1 *v2 *a11 + u1 *v3 *a11 - 2*u1*u2*v1*v2*a11 - 2*u1*u3*v1*v3*a11
3 2 2 2 2
- u1*v1 *b22 + u1*v1 *n1 - u1*v1*v2 *b22 - u1*v1*v3 *b22 + u1*v2 *n1
2 2 2 2 2 2
+ u1*v3 *n1 + u2 *v1 *a11 + u2 *v3 *a11 - 2*u2*u3*v2*v3*a11 + u2*v1 *n2
2 2 2 2 2 2 2 2
+ u2*v2 *n2 + u2*v3 *n2 + u3 *v1 *a33 + u3 *v2 *a33 + u3 *v3 *( - a11 + a33)
3 2 2 2
+ u3*v1 *b31 + u3*v1 *v2*b32 + u3*v1 *v3*( - b22 + b33) + u3*v1 *n3
2 2 3 2
+ u3*v1*v2 *b31 + u3*v1*v3 *b31 + u3*v2 *b32 + u3*v2 *v3*( - b22 + b33)
2 2 3 2
+ u3*v2 *n3 + u3*v2*v3 *b32 + u3*v3 *( - b22 + b33) + u3*v3 *n3
3 3 3 2 2 2
+ 2*v1 *v2*c12 + 2*v1 *v3*c13 + v1 *m1 + v1 *v2 *c22 + 2*v1 *v2*v3*c23
2 2 2 2 3 2
+ v1 *v2*m2 + v1 *v3 *c33 + v1 *v3*m3 + 2*v1*v2 *c12 + 2*v1*v2 *v3*c13
2 2 3 2 4
+ v1*v2 *m1 + 2*v1*v2*v3 *c12 + 2*v1*v3 *c13 + v1*v3 *m1 + v2 *c22
3 3 2 2 2 3
+ 2*v2 *v3*c23 + v2 *m2 + v2 *v3 *(c22 + c33) + v2 *v3*m3 + 2*v2*v3 *c23
2 4 3
+ v2*v3 *m2 + v3 *c33 + v3 *m3
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*n1 + u2**2*a11 + u2*v2*b22 + u2*n2 + u3**2*a33 + 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=u1**2*v2**2*a11 + u1**2*v3**2*a11 - 2*u1*u2*v1*v2*a11 - 2*u1*u3*v1*v3*a11 -
u1*v1**3*b22 + u1*v1**2*n1 - u1*v1*v2**2*b22 - u1*v1*v3**2*b22 + u1*v2**2*n1 +
u1*v3**2*n1 + u2**2*v1**2*a11 + u2**2*v3**2*a11 - 2*u2*u3*v2*v3*a11 + u2*v1**2*
n2 + u2*v2**2*n2 + u2*v3**2*n2 + u3**2*v1**2*a33 + u3**2*v2**2*a33 + u3**2*v3**2
*( - a11 + a33) + u3*v1**3*b31 + u3*v1**2*v2*b32 + u3*v1**2*v3*( - b22 + b33) +
u3*v1**2*n3 + u3*v1*v2**2*b31 + u3*v1*v3**2*b31 + u3*v2**3*b32 + u3*v2**2*v3*( -
b22 + b33) + u3*v2**2*n3 + u3*v2*v3**2*b32 + u3*v3**3*( - b22 + b33) + u3*v3**2
*n3 + 2*v1**3*v2*c12 + 2*v1**3*v3*c13 + v1**3*m1 + v1**2*v2**2*c22 + 2*v1**2*v2*
v3*c23 + v1**2*v2*m2 + v1**2*v3**2*c33 + v1**2*v3*m3 + 2*v1*v2**3*c12 + 2*v1*v2
**2*v3*c13 + v1*v2**2*m1 + 2*v1*v2*v3**2*c12 + 2*v1*v3**3*c13 + v1*v3**2*m1 + v2
**4*c22 + 2*v2**3*v3*c23 + v2**3*m2 + v2**2*v3**2*(c22 + c33) + v2**2*v3*m3 + 2*
v2*v3**3*c23 + v2*v3**2*m2 + v3**4*c33 + v3**3*m3$