Solution 2 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
2 2 3
- a11 *n3*r29 - n1 *n3*r434 + n3 *r434
r13=-----------------------------------------
3
a11
2 2 2
- a11 *n2*r29 - n1 *n2*r434 + n2*n3 *r434
r14=--------------------------------------------
3
a11
2 3 2
- a11 *n1*r29 - n1 *r434 + n1*n3 *r434
r15=-----------------------------------------
3
a11
r20=0
r21=0
r22=0
r23=0
r24=0
r26=0
r27=0
r28=0
r210=0
r212=0
2*n2*n3*r434
r213=--------------
2
a11
2 2
- n1 *r434 + n2 *r434
r214=------------------------
2
a11
r215=0
r216=0
r217=0
2*n1*n3*r434
r218=--------------
2
a11
2*n1*n2*r434
r219=--------------
2
a11
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
n3*r320
r310=---------
n2
r311=0
n3*r320
r312=---------
n2
r313=0
r314=0
n3*r320
r315=---------
n2
- n3*r488
r316=------------
a11
r317=0
r318=0
- 2*n3*r434
r319=--------------
a11
r321=0
r322=r320
r323=0
r324=0
r325=r320
- n2*r488
r326=------------
a11
- n3*r488
r327=------------
a11
r328=0
- 2*n2*r434
r329=--------------
a11
r330=0
- n2*r488
r331=------------
a11
r332=0
r333=0
r334=0
n1*r320
r335=---------
n2
r336=0
n1*r320
r337=---------
n2
r338=0
r339=0
n1*r320
r340=---------
n2
- n1*r488
r341=------------
a11
r342=0
- n3*r488
r343=------------
a11
- 2*n1*r434
r344=--------------
a11
r345=0
- n1*r488
r346=------------
a11
- n2*r488
r347=------------
a11
r348=0
r349=0
r350=0
r351=0
- n1*r488
r352=------------
a11
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
r422=0
r423=0
r424=0
- a11*r320
r425=-------------
n2
r426=0
1
- a11*r320 - ---*n2*r483
2
r427=---------------------------
n2
r428=0
r429=0
1
- a11*r320 - ---*n2*r483
2
r430=---------------------------
n2
r431=r488
r432=0
r433=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=r488
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
r474=0
r475=0
r476=0
r477=0
r478=0
r479=0
r480=0
r481=0
r482=0
r484=0
r485=0
r486=0
r487=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
m3=0
m2=0
m1=0
c33=0
c23=0
c22=0
c13=0
c12=0
b33=0
b32=0
b31=0
b11=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:
r434, r29, r320, r483, r488, n3, 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.
{r488,n2,a11,n1}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2
HAM=u1 *a11 + u1*n1 + u2 *a11 + u2*n2 + u3*n3
has apart from the Hamiltonian and Casimirs the following 5 first integrals:
2 2
FI= - u1 *v1*n1 - u1*u2*v1*n2 - u1*u2*v2*n1 + u1*u3 *v1*a11 - u1*u3*v1*n3
2 2
- u1*u3*v3*n1 - u2 *v2*n2 + u2*u3 *v2*a11 - u2*u3*v2*n3 - u2*u3*v3*n2
3 2
+ u3 *v3*a11 - u3 *v3*n3
{HAM,FI} = 0
1 2 2 1 2 2 1 2 2
FI= - ---*u1 *v2 - ---*u1 *v3 + u1*u2*v1*v2 + u1*u3*v1*v3 - ---*u2 *v1
2 2 2
1 2 2 1 2 2 1 2 2
- ---*u2 *v3 + u2*u3*v2*v3 - ---*u3 *v1 - ---*u3 *v2
2 2 2
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1*v1 *n1 + u1*v2 *n1 + u1*v3 *n1 + u2*v1 *n2 + u2*v2 *n2 + u2*v3 *n2
2 2 2 2 2 2 2 2
- u3 *v1 *a11 - u3 *v2 *a11 - u3 *v3 *a11 + u3*v1 *n3 + u3*v2 *n3
2
+ u3*v3 *n3
{HAM,FI} = 0
2
FI= - u1*n1 - u2*n2 + u3 *a11 - u3*n3
{HAM,FI} = 0
2 2
FI=2*u1*u2*a11*n1*n2 - 2*u1*u3 *a11 *n1 + 2*u1*u3*a11*n1*n3
3 2 2 2 2 2 2
+ u1*( - n1 + n1*n3 ) + u2 *( - a11*n1 + a11*n2 ) - 2*u2*u3 *a11 *n2
2 2 4 3 3 2
+ 2*u2*u3*a11*n2*n3 + u2*( - n1 *n2 + n2*n3 ) + u3 *a11 - 2*u3 *a11 *n3
2 3
+ u3*( - n1 *n3 + n3 )
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*n1 + u2**2*a11 + u2*n2 + u3*n3$
FI= - u1**2*v1*n1 - u1*u2*v1*n2 - u1*u2*v2*n1 + u1*u3**2*v1*a11 - u1*u3*v1*n3 -
u1*u3*v3*n1 - u2**2*v2*n2 + u2*u3**2*v2*a11 - u2*u3*v2*n3 - u2*u3*v3*n2 + u3**3*
v3*a11 - u3**2*v3*n3$
FI= - 1/2*u1**2*v2**2 - 1/2*u1**2*v3**2 + u1*u2*v1*v2 + u1*u3*v1*v3 - 1/2*u2**2*
v1**2 - 1/2*u2**2*v3**2 + u2*u3*v2*v3 - 1/2*u3**2*v1**2 - 1/2*u3**2*v2**2$
FI=u1*v1**2*n1 + u1*v2**2*n1 + u1*v3**2*n1 + u2*v1**2*n2 + u2*v2**2*n2 + u2*v3**
2*n2 - u3**2*v1**2*a11 - u3**2*v2**2*a11 - u3**2*v3**2*a11 + u3*v1**2*n3 + u3*v2
**2*n3 + u3*v3**2*n3$
FI= - u1*n1 - u2*n2 + u3**2*a11 - u3*n3$
FI=2*u1*u2*a11*n1*n2 - 2*u1*u3**2*a11**2*n1 + 2*u1*u3*a11*n1*n3 + u1*( - n1**3 +
n1*n3**2) + u2**2*( - a11*n1**2 + a11*n2**2) - 2*u2*u3**2*a11**2*n2 + 2*u2*u3*
a11*n2*n3 + u2*( - n1**2*n2 + n2*n3**2) + u3**4*a11**3 - 2*u3**3*a11**2*n3 + u3*
( - n1**2*n3 + n3**3)$