Solution 8 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 2 3
- a22 *n3*r29 + 2*a22*a33*n3*r29 - a33 *n3*r29 - n1 *n3*r434 + n3 *r434
r13=--------------------------------------------------------------------------
3 2 2 3
a22 - 3*a22 *a33 + 3*a22*a33 - a33
r14
2 2 2 2
- a22 *n2*r29 + 2*a22*a33*n2*r29 - a33 *n2*r29 - n1 *n2*r434 + n2*n3 *r434
=-----------------------------------------------------------------------------
3 2 2 3
a22 - 3*a22 *a33 + 3*a22*a33 - a33
2 2 3 2
- a22 *n1*r29 + 2*a22*a33*n1*r29 - a33 *n1*r29 - n1 *r434 + n1*n3 *r434
r15=--------------------------------------------------------------------------
3 2 2 3
a22 - 3*a22 *a33 + 3*a22*a33 - a33
r20=0
r21=0
r23=0
r24=0
r25=0
r26=0
r27=0
r28=0
r210=0
r212=0
2*n2*n3*r434
r213=-------------------------
2 2
a22 - 2*a22*a33 + a33
2 2
- n1 *r434 + n2 *r434
r214=-------------------------
2 2
a22 - 2*a22*a33 + a33
r215=0
r216=0
r217=0
2*n1*n3*r434
r218=-------------------------
2 2
a22 - 2*a22*a33 + a33
2*n1*n2*r434
r219=-------------------------
2 2
a22 - 2*a22*a33 + a33
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
- n3*r425
r310=------------
a22 - a33
n2*r425
r311=-----------
a22 - a33
- n3*r425
r312=------------
a22 - a33
r313=0
r314=0
- n3*r425
r315=------------
a22 - a33
r316=0
r317=0
r318=0
- 2*n3*r434
r319=--------------
a22 - a33
- n2*r425
r320=------------
a22 - a33
r323=0
- n2*r425
r325=------------
a22 - a33
r326=0
r328=0
- 2*n2*r434
r329=--------------
a22 - a33
r330=0
r332=0
r333=0
r334=0
- n1*r425
r335=------------
a22 - a33
r336=0
- n1*r425
r337=------------
a22 - a33
r338=0
n2*r425
r339=-----------
a22 - a33
- n1*r425
r340=------------
a22 - a33
r341=0
r342=0
r343=0
- 2*n1*r434
r344=--------------
a22 - a33
r345=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
r40=0
r41=0
r42=0
r43=0
r45=0
r46=0
r47=0
r48=0
r49=0
r410=0
r411=0
r412=0
r413=0
r414=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
r426=0
r427=r425
r428=0
r429=0
r430=r425
r431=0
r432=0
r433=0
r435=0
r439=0
r442=0
r444=0
r445=0
r448=0
r450=0
r451=0
r453=0
r454=0
r455=0
r458=0
r460=0
r461=0
r463=0
r464=0
r465=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
r483=0
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r493=0
r495=0
r496=0
r498=0
r499=0
r4100=0
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
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
c11=0
b33=0
b32=0
b31=0
b23=0
b22=0
b21=0
b13=0
b12=0
b11=0
a23=0
a13=0
a12=0
a11=a22
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:
r29, r425, r434, n3, n2, n1, a33, a22
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.
{n2,r434,a11,a22 - a33,a33,a22}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11 - a22,
a12,
a13,
a23,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
b33,
c11,
c12,
c13,
c22,
c23,
c33,
m1,
m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a22 + u1*n1 + u2 *a22 + u2*n2 + u3 *a33 + u3*n3
has apart from the Hamiltonian and Casimirs the following 3 first integrals:
FI=u1*u2*(2*a22*n1*n2 - 2*a33*n1*n2)
2 2 2
+ u1*u3 *( - 2*a22 *n1 + 4*a22*a33*n1 - 2*a33 *n1)
3 2
+ u1*u3*(2*a22*n1*n3 - 2*a33*n1*n3) + u1*( - n1 + n1*n3 )
2 2 2 2 2
+ u2 *( - a22*n1 + a22*n2 + a33*n1 - a33*n2 )
2 2 2
+ u2*u3 *( - 2*a22 *n2 + 4*a22*a33*n2 - 2*a33 *n2)
2 2
+ u2*u3*(2*a22*n2*n3 - 2*a33*n2*n3) + u2*( - n1 *n2 + n2*n3 )
4 3 2 2 3
+ u3 *(a22 - 3*a22 *a33 + 3*a22*a33 - a33 )
3 2 2 2 3
+ u3 *( - 2*a22 *n3 + 4*a22*a33*n3 - 2*a33 *n3) + u3*( - n1 *n3 + n3 )
which the program can not factorize further.
{HAM,FI} = 0
2 2 2
FI=u1*v1 *( - a22 *n1 + 2*a22*a33*n1 - a33 *n1)
2 2
+ u1*v1*v2*(a22 *n2 - 2*a22*a33*n2 + a33 *n2)
2 2 2
+ u1*v2 *( - a22 *n1 + 2*a22*a33*n1 - a33 *n1)
2 2 2
+ u1*v3 *( - a22 *n1 + 2*a22*a33*n1 - a33 *n1)
2 2 2
+ u2*v1 *( - a22 *n2 + 2*a22*a33*n2 - a33 *n2)
2 2 2
+ u2*v3 *( - a22 *n2 + 2*a22*a33*n2 - a33 *n2)
2 2 3 2 2 3
+ u3 *v1 *(a22 - 3*a22 *a33 + 3*a22*a33 - a33 )
2 2 3 2 2 3
+ u3 *v2 *(a22 - 3*a22 *a33 + 3*a22*a33 - a33 )
2 2 3 2 2 3
+ u3 *v3 *(a22 - 3*a22 *a33 + 3*a22*a33 - a33 )
2 2 2
+ u3*v1 *( - a22 *n3 + 2*a22*a33*n3 - a33 *n3)
2 2 2
+ u3*v2 *( - a22 *n3 + 2*a22*a33*n3 - a33 *n3)
2 2
+ u3*v2*v3*(a22 *n2 - 2*a22*a33*n2 + a33 *n2)
2 2 2
+ u3*v3 *( - a22 *n3 + 2*a22*a33*n3 - a33 *n3)
= a product of the elements of: {a22 - a33,
a22 - a33,
2 2 2 2 2
- u1*v1 *n1 + u1*v1*v2*n2 - u1*v2 *n1 - u1*v3 *n1 - u2*v1 *n2 - u2*v3 *n2
2 2 2 2 2 2
+ u3 *v1 *(a22 - a33) + u3 *v2 *(a22 - a33) + u3 *v3 *(a22 - a33)
2 2 2
- u3*v1 *n3 - u3*v2 *n3 + u3*v2*v3*n2 - u3*v3 *n3}
{HAM,FI} = {2,
a22 - a33,
a22 - a33,
u1*v1 + u2*v2 + u3*v3,
n2,
1 1
u1*v3*a22 - u3*v1*a33 - ---*v1*n3 + ---*v3*n1}
2 2
2 2
FI=u1*( - a22 *n1 + 2*a22*a33*n1 - a33 *n1)
2 2
+ u2*( - a22 *n2 + 2*a22*a33*n2 - a33 *n2)
2 3 2 2 3
+ u3 *(a22 - 3*a22 *a33 + 3*a22*a33 - a33 )
2 2
+ u3*( - a22 *n3 + 2*a22*a33*n3 - a33 *n3)
= a product of the elements of: {a22 - a33,
a22 - a33,
2
- u1*n1 - u2*n2 + u3 *(a22 - a33) - u3*n3}
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a22 + u1*n1 + u2**2*a22 + u2*n2 + u3**2*a33 + u3*n3$
FI=u1*u2*(2*a22*n1*n2 - 2*a33*n1*n2) + u1*u3**2*( - 2*a22**2*n1 + 4*a22*a33*n1 -
2*a33**2*n1) + u1*u3*(2*a22*n1*n3 - 2*a33*n1*n3) + u1*( - n1**3 + n1*n3**2) +
u2**2*( - a22*n1**2 + a22*n2**2 + a33*n1**2 - a33*n2**2) + u2*u3**2*( - 2*a22**2
*n2 + 4*a22*a33*n2 - 2*a33**2*n2) + u2*u3*(2*a22*n2*n3 - 2*a33*n2*n3) + u2*( -
n1**2*n2 + n2*n3**2) + u3**4*(a22**3 - 3*a22**2*a33 + 3*a22*a33**2 - a33**3) +
u3**3*( - 2*a22**2*n3 + 4*a22*a33*n3 - 2*a33**2*n3) + u3*( - n1**2*n3 + n3**3)$
FI=u1*v1**2*( - a22**2*n1 + 2*a22*a33*n1 - a33**2*n1) + u1*v1*v2*(a22**2*n2 - 2*
a22*a33*n2 + a33**2*n2) + u1*v2**2*( - a22**2*n1 + 2*a22*a33*n1 - a33**2*n1) +
u1*v3**2*( - a22**2*n1 + 2*a22*a33*n1 - a33**2*n1) + u2*v1**2*( - a22**2*n2 + 2*
a22*a33*n2 - a33**2*n2) + u2*v3**2*( - a22**2*n2 + 2*a22*a33*n2 - a33**2*n2) +
u3**2*v1**2*(a22**3 - 3*a22**2*a33 + 3*a22*a33**2 - a33**3) + u3**2*v2**2*(a22**
3 - 3*a22**2*a33 + 3*a22*a33**2 - a33**3) + u3**2*v3**2*(a22**3 - 3*a22**2*a33 +
3*a22*a33**2 - a33**3) + u3*v1**2*( - a22**2*n3 + 2*a22*a33*n3 - a33**2*n3) +
u3*v2**2*( - a22**2*n3 + 2*a22*a33*n3 - a33**2*n3) + u3*v2*v3*(a22**2*n2 - 2*a22
*a33*n2 + a33**2*n2) + u3*v3**2*( - a22**2*n3 + 2*a22*a33*n3 - a33**2*n3)$
FI=u1*( - a22**2*n1 + 2*a22*a33*n1 - a33**2*n1) + u2*( - a22**2*n2 + 2*a22*a33*
n2 - a33**2*n2) + u3**2*(a22**3 - 3*a22**2*a33 + 3*a22*a33**2 - a33**3) + u3*( -
a22**2*n3 + 2*a22*a33*n3 - a33**2*n3)$