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
3 2 2
r13=( - a11 *n3*r29 + a11 *a22*n3*r29 + 2*a11 *a33*n3*r29 - 2*a11*a22*a33*n3*r29
2 2 3 2
- a11*a33 *n3*r29 + a11*n1 *n3*r4114 - a11*n3 *r4114 + a22*a33 *n3*r29
1 2 1 3 1 2
- ---*a22*n1 *n3*r4114 + ---*a22*n3 *r4114 - ---*a33*n1 *n3*r4114
2 2 2
1 3 4 3 3 2
+ ---*a33*n3 *r4114)/(a11 - a11 *a22 - 3*a11 *a33 + 3*a11 *a22*a33
2
2 2 2 3 3
+ 3*a11 *a33 - 3*a11*a22*a33 - a11*a33 + a22*a33 )
3 2 2
r14=( - a11 *n2*r29 + a11 *a22*n2*r29 + 2*a11 *a33*n2*r29 - 2*a11*a22*a33*n2*r29
2 2 2 2
- a11*a33 *n2*r29 + a11*n1 *n2*r4114 - a11*n2*n3 *r4114 + a22*a33 *n2*r29
1 2 1 2 1 2
- ---*a22*n1 *n2*r4114 + ---*a22*n2*n3 *r4114 - ---*a33*n1 *n2*r4114
2 2 2
1 2 4 3 3 2
+ ---*a33*n2*n3 *r4114)/(a11 - a11 *a22 - 3*a11 *a33 + 3*a11 *a22*a33
2
2 2 2 3 3
+ 3*a11 *a33 - 3*a11*a22*a33 - a11*a33 + a22*a33 )
3 2 2
r15=( - a11 *n1*r29 + a11 *a22*n1*r29 + 2*a11 *a33*n1*r29 - 2*a11*a22*a33*n1*r29
2 3 2 2
- a11*a33 *n1*r29 + a11*n1 *r4114 - a11*n1*n3 *r4114 + a22*a33 *n1*r29
1 3 1 2 1 3
- ---*a22*n1 *r4114 + ---*a22*n1*n3 *r4114 - ---*a33*n1 *r4114
2 2 2
1 2 4 3 3 2
+ ---*a33*n1*n3 *r4114)/(a11 - a11 *a22 - 3*a11 *a33 + 3*a11 *a22*a33
2
2 2 2 3 3
+ 3*a11 *a33 - 3*a11*a22*a33 - a11*a33 + a22*a33 )
r20=0
r21=0
r22=0
r23=0
r24=0
r26=0
r27=0
r28=0
r210=0
r212=0
- 2*a11*n2*n3*r4114 + a22*n2*n3*r4114 + a33*n2*n3*r4114
r213=--------------------------------------------------------------------
3 2 2 2 2
a11 - a11 *a22 - 2*a11 *a33 + 2*a11*a22*a33 + a11*a33 - a22*a33
4 3 3 2 2
r214=(a11 *r29 - 2*a11 *a22*r29 - 2*a11 *a33*r29 + a11 *a22 *r29
2 2 2 2 2 2 2
+ 4*a11 *a22*a33*r29 + a11 *a33 *r29 - a11 *n2 *r4114 + a11 *n3 *r4114
2 2 2
- 2*a11*a22 *a33*r29 - 2*a11*a22*a33 *r29 + a11*a22*n1 *r4114
1 2 3 2 2
+ ---*a11*a22*n2 *r4114 - ---*a11*a22*n3 *r4114 - a11*a33*n1 *r4114
2 2
3 2 1 2 2 2
+ ---*a11*a33*n2 *r4114 - ---*a11*a33*n3 *r4114 + a22 *a33 *r29
2 2
1 2 2 1 2 2 1 2
- ---*a22 *n1 *r4114 + ---*a22 *n3 *r4114 - ---*a22*a33*n2 *r4114
2 2 2
1 2 1 2 2 1 2 2 4
+ ---*a22*a33*n3 *r4114 + ---*a33 *n1 *r4114 - ---*a33 *n2 *r4114)/(a11
2 2 2
3 3 2 2 2 2
- a11 *a22 - 3*a11 *a33 + 3*a11 *a22*a33 + 3*a11 *a33 - 3*a11*a22*a33
3 3
- a11*a33 + a22*a33 )
r215=0
r216=0
r217=0
- 2*a11*n1*n3*r4114 + a22*n1*n3*r4114 + a33*n1*n3*r4114
r218=--------------------------------------------------------------------
3 2 2 2 2
a11 - a11 *a22 - 2*a11 *a33 + 2*a11*a22*a33 + a11*a33 - a22*a33
- 2*a11*n1*n2*r4114 + a22*n1*n2*r4114 + a33*n1*n2*r4114
r219=--------------------------------------------------------------------
3 2 2 2 2
a11 - a11 *a22 - 2*a11 *a33 + 2*a11*a22*a33 + a11*a33 - a22*a33
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
r314=0
r315=0
- n3*r4102
r316=-------------
a11 - a22
r317=0
r318=0
n3*r4114
r319=-----------
a11 - a22
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
- n2*r4102
r326=-------------
a11 - a22
- n3*r4102
r327=-------------
a11 - a22
r328=0
n2*r4114
r329=-----------
a11 - a22
r330=0
- n2*r4102
r331=-------------
a11 - a22
r332=0
2 2
a11 *n3*r4114 - 2*a11*a22*n3*r4114 + a22 *n3*r4114
r333=--------------------------------------------------------------------
3 2 2 2 2
a11 - a11 *a22 - 2*a11 *a33 + 2*a11*a22*a33 + a11*a33 - a22*a33
a11*n2*r4114 - a22*n2*r4114
r334=-----------------------------
2 2
a11 - 2*a11*a33 + a33
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- n1*r4102
r341=-------------
a11 - a22
r342=0
- n3*r4102
r343=-------------
a11 - a22
n1*r4114
r344=-----------
a11 - a22
r345=0
- n1*r4102
r346=-------------
a11 - a22
- n2*r4102
r347=-------------
a11 - a22
r348=0
2 2
a11 *n1*r4114 - 2*a11*a22*n1*r4114 + a22 *n1*r4114
r349=--------------------------------------------------------------------
3 2 2 2 2
a11 - a11 *a22 - 2*a11 *a33 + 2*a11*a22*a33 + a11*a33 - a22*a33
r350=0
r351=0
- n1*r4102
r352=-------------
a11 - a22
- n3*r4114
r353=-------------
a11 - a33
- n2*r4114
r354=-------------
a11 - a33
- n1*r4114
r355=-------------
a11 - a33
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
r425=0
r426=0
1
r427= - ---*r446
2
r428=0
r429=0
1
r430= - ---*r446
2
a11*r4102 - a33*r4102
r431=-----------------------
a11 - a22
r432=0
r433=0
1 1
- ---*a22*r4114 + ---*a33*r4114
2 2
r434=----------------------------------
a11 - a22
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
r447=0
r448=0
r449=0
r450=0
r451=0
a11*r4102 - a33*r4102
r452=-----------------------
a11 - a22
r453=0
r454=0
1
r455= - ---*r446
2
r456=0
r458=0
r459=0
1
r460= - ---*r446
2
r461=r4102
r462=0
r463=0
r464=0
r465=0
r466=r4102
r467=0
r468=0
1 2 1 2 2
r469=(---*a11 *a22*r4114 - ---*a11 *a33*r4114 - a11*a22 *r4114
2 2
1 3 1 2 3
+ a11*a22*a33*r4114 + ---*a22 *r4114 - ---*a22 *a33*r4114)/(a11
2 2
2 2 2 2
- a11 *a22 - 2*a11 *a33 + 2*a11*a22*a33 + a11*a33 - a22*a33 )
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=r446
r484=0
r485=0
r486=0
r487=0
a11*r4102 - a33*r4102
r488=-----------------------
a11 - a22
r489=0
r490=0
r491=0
r492=0
r493=0
r494=r446
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4103=0
r4104=0
1
r4105= - ---*r446
2
r4106=0
1
r4107= - ---*r446
2
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4115=0
r4117=0
r4118=0
a11*r4114 - a22*r4114
r4119=-----------------------
a11 - a33
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
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, r4114, r4102, r446, n3, n2, n1, a33, a22, 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.
{{a11*r4102 - a33*r4102,
1 1
---*a22*r4114 - ---*a33*r4114,
2 2
3 2
a11 *r4102 - a11 *a33*r4102,
1 3 1 3 2 2
---*a11 *a22*n1*n2*r4114 - ---*a11 *a33*n1*n2*r4114 - a11 *a22 *n1*n2*r4114
2 2
1 2 1 2 2
+ ---*a11 *a22*a33*n1*n2*r4114 + ---*a11 *a33 *n1*n2*r4114
2 2
1 3 1 2
+ ---*a11*a22 *n1*n2*r4114 + ---*a11*a22 *a33*n1*n2*r4114
2 2
2 1 3
- a11*a22*a33 *n1*n2*r4114 - ---*a22 *a33*n1*n2*r4114
2
1 2 2
+ ---*a22 *a33 *n1*n2*r4114,
2
r446,
r4102,
1 1 2
---*a11*a22*a33*r446 - ---*a11*a33 *r446,
2 2
r4114,
a11*r4114 - a22*r4114},
a11 - a33,
a22 - a33,
a11 - a22,
a33,
a22,
a11,
a11 + a22}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{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 *a11 + u1*n1 + u2 *a22 + u2*n2 + u3 *a33 + u3*n3
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 2 2 2 2 2 2 2
FI= - u1 *v2 - u1 *v3 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2 *v1 - u2 *v3
2 2 2 2
+ 2*u2*u3*v2*v3 - u3 *v1 - u3 *v2
which the program can not factorize further.
{HAM,FI} = 0
2 3 2 2 3 2
FI=u1 *v1*( - a11 *n1 + 3*a11 *a33*n1 - 3*a11*a33 *n1 + a33 *n1) + u1*u2 *v1*(
4 3 3 2 2 2
a11 - a11 *a22 - 3*a11 *a33 + 3*a11 *a22*a33 + 3*a11 *a33
2 3 3
- 3*a11*a22*a33 - a11*a33 + a22*a33 )
3 2 2 3
+ u1*u2*v1*( - a11 *n2 + 3*a11 *a33*n2 - 3*a11*a33 *n2 + a33 *n2)
3 2 2 3
+ u1*u2*v2*( - a11 *n1 + 3*a11 *a33*n1 - 3*a11*a33 *n1 + a33 *n1)
2 4 3 2 2 3 4
+ u1*u3 *v1*(a11 - 4*a11 *a33 + 6*a11 *a33 - 4*a11*a33 + a33 )
3 2 2 3
+ u1*u3*v1*( - a11 *n3 + 3*a11 *a33*n3 - 3*a11*a33 *n3 + a33 *n3)
3 2 2 3 3
+ u1*u3*v3*( - a11 *n1 + 3*a11 *a33*n1 - 3*a11*a33 *n1 + a33 *n1) + u2 *v2*(
4 3 3 2 2 2
a11 - a11 *a22 - 3*a11 *a33 + 3*a11 *a22*a33 + 3*a11 *a33
2 3 3 2 4 3
- 3*a11*a22*a33 - a11*a33 + a22*a33 ) + u2 *u3*v3*(a11 - a11 *a22
3 2 2 2 2 3
- 3*a11 *a33 + 3*a11 *a22*a33 + 3*a11 *a33 - 3*a11*a22*a33 - a11*a33
3
+ a22*a33 )
2 3 2 2 3
+ u2 *v2*( - a11 *n2 + 3*a11 *a33*n2 - 3*a11*a33 *n2 + a33 *n2)
2 4 3 2 2 3 4
+ u2*u3 *v2*(a11 - 4*a11 *a33 + 6*a11 *a33 - 4*a11*a33 + a33 )
3 2 2 3
+ u2*u3*v2*( - a11 *n3 + 3*a11 *a33*n3 - 3*a11*a33 *n3 + a33 *n3)
3 2 2 3
+ u2*u3*v3*( - a11 *n2 + 3*a11 *a33*n2 - 3*a11*a33 *n2 + a33 *n2)
3 4 3 2 2 3 4
+ u3 *v3*(a11 - 4*a11 *a33 + 6*a11 *a33 - 4*a11*a33 + a33 )
2 3 2 2 3
+ u3 *v3*( - a11 *n3 + 3*a11 *a33*n3 - 3*a11*a33 *n3 + a33 *n3)
= a product of the elements of: {a11 - a33,
a11 - a33,
a11 - a33,
u1*v1 + u2*v2 + u3*v3,
2 2
- u1*n1 + u2 *(a11 - a22) - u2*n2 + u3 *(a11 - a33) - u3*n3}
{HAM,FI} = 0
3 3 2 2
FI=u1 *( - 2*a11 *n1 + 2*a11 *a22*n1 + 4*a11 *a33*n1 - 4*a11*a22*a33*n1
2 2 2 2 4 3
- 2*a11*a33 *n1 + 2*a22*a33 *n1) + u1 *u2 *(2*a11 - 4*a11 *a22
3 2 2 2 2 2
- 4*a11 *a33 + 2*a11 *a22 + 8*a11 *a22*a33 + 2*a11 *a33
2 2 2 2 2 3
- 4*a11*a22 *a33 - 4*a11*a22*a33 + 2*a22 *a33 ) + u1 *u2*( - 2*a11 *n2
2 2 2
+ 2*a11 *a22*n2 + 4*a11 *a33*n2 - 4*a11*a22*a33*n2 - 2*a11*a33 *n2
2 2 2 4 3 3
+ 2*a22*a33 *n2) + u1 *u3 *(2*a11 - 2*a11 *a22 - 6*a11 *a33
2 2 2 2 3 3
+ 6*a11 *a22*a33 + 6*a11 *a33 - 6*a11*a22*a33 - 2*a11*a33 + 2*a22*a33
2 3 2 2
) + u1 *u3*( - 2*a11 *n3 + 2*a11 *a22*n3 + 4*a11 *a33*n3
2 2 2 3
- 4*a11*a22*a33*n3 - 2*a11*a33 *n3 + 2*a22*a33 *n3) + u1*u2 *(2*a11 *n1
2 2 2
- 4*a11 *a22*n1 - 2*a11 *a33*n1 + 2*a11*a22 *n1 + 4*a11*a22*a33*n1
2 2
- 2*a22 *a33*n1) + u1*u2*( - 4*a11 *n1*n2 + 2*a11*a22*n1*n2
2
+ 6*a11*a33*n1*n2 - 2*a22*a33*n1*n2 - 2*a33 *n1*n2)
2 3 2 2 3
+ u1*u3 *(2*a11 *n1 - 6*a11 *a33*n1 + 6*a11*a33 *n1 - 2*a33 *n1) + u1*u3*(
2
- 4*a11 *n1*n3 + 2*a11*a22*n1*n3 + 6*a11*a33*n1*n3 - 2*a22*a33*n1*n3
2
- 2*a33 *n1*n3) + u1
3 2 3 2 3 2
*(2*a11*n1 - 2*a11*n1*n3 - a22*n1 + a22*n1*n3 - a33*n1 + a33*n1*n3 ) +
4 3 3 2 2 2 2 2 3
u2 *(a11 *a22 - a11 *a33 - 2*a11 *a22 + a11 *a22*a33 + a11 *a33 + a11*a22
2 2 3 2 2 3
+ a11*a22 *a33 - 2*a11*a22*a33 - a22 *a33 + a22 *a33 ) + u2 *(
3 2 2 2
2*a11 *n2 - 4*a11 *a22*n2 - 2*a11 *a33*n2 + 2*a11*a22 *n2
2 2 3 2
+ 4*a11*a22*a33*n2 - 2*a22 *a33*n2) + u2 *u3*(2*a11 *n3 - 4*a11 *a22*n3
2 2 2 2
- 2*a11 *a33*n3 + 2*a11*a22 *n3 + 4*a11*a22*a33*n3 - 2*a22 *a33*n3) + u2
2 2 2 2 2 2 2
*( - 2*a11 *n2 + 2*a11 *n3 + 2*a11*a22*n1 + a11*a22*n2 - 3*a11*a22*n3
2 2 2 2 2 2 2
- 2*a11*a33*n1 + 3*a11*a33*n2 - a11*a33*n3 - a22 *n1 + a22 *n3
2 2 2 2 2 2
- a22*a33*n2 + a22*a33*n3 + a33 *n1 - a33 *n2 )
2 3 2 2 3
+ u2*u3 *(2*a11 *n2 - 6*a11 *a33*n2 + 6*a11*a33 *n2 - 2*a33 *n2) + u2*u3*(
2
- 4*a11 *n2*n3 + 2*a11*a22*n2*n3 + 6*a11*a33*n2*n3 - 2*a22*a33*n2*n3
2 2 2 2
- 2*a33 *n2*n3) + u2*(2*a11*n1 *n2 - 2*a11*n2*n3 - a22*n1 *n2
2 2 2 4 3 3
+ a22*n2*n3 - a33*n1 *n2 + a33*n2*n3 ) + u3 *( - a11 *a22 + a11 *a33
2 2 2 2 3 3
+ 3*a11 *a22*a33 - 3*a11 *a33 - 3*a11*a22*a33 + 3*a11*a33 + a22*a33
4 3 3 2 2 3
- a33 ) + u3 *(2*a11 *n3 - 6*a11 *a33*n3 + 6*a11*a33 *n3 - 2*a33 *n3) +
2 3 2 3 2 3
u3*(2*a11*n1 *n3 - 2*a11*n3 - a22*n1 *n3 + a22*n3 - a33*n1 *n3 + a33*n3 )
= a product of the elements of: {2,
3 3 2 2
u1 *( - a11 *n1 + a11 *a22*n1 + 2*a11 *a33*n1 - 2*a11*a22*a33*n1
2 2 2 2 4 3 3
- a11*a33 *n1 + a22*a33 *n1) + u1 *u2 *(a11 - 2*a11 *a22 - 2*a11 *a33
2 2 2 2 2 2
+ a11 *a22 + 4*a11 *a22*a33 + a11 *a33 - 2*a11*a22 *a33
2 2 2 2 3 2
- 2*a11*a22*a33 + a22 *a33 ) + u1 *u2*( - a11 *n2 + a11 *a22*n2
2 2 2 2 2
+ 2*a11 *a33*n2 - 2*a11*a22*a33*n2 - a11*a33 *n2 + a22*a33 *n2) + u1 *u3
4 3 3 2 2 2
*(a11 - a11 *a22 - 3*a11 *a33 + 3*a11 *a22*a33 + 3*a11 *a33
2 3 3 2 3 2
- 3*a11*a22*a33 - a11*a33 + a22*a33 ) + u1 *u3*( - a11 *n3 + a11 *a22*n3
2 2 2 2
+ 2*a11 *a33*n3 - 2*a11*a22*a33*n3 - a11*a33 *n3 + a22*a33 *n3) + u1*u2 *
3 2 2 2
(a11 *n1 - 2*a11 *a22*n1 - a11 *a33*n1 + a11*a22 *n1 + 2*a11*a22*a33*n1
2 2
- a22 *a33*n1) + u1*u2*( - 2*a11 *n1*n2 + a11*a22*n1*n2 + 3*a11*a33*n1*n2
2
- a22*a33*n1*n2 - a33 *n1*n2)
2 3 2 2 3
+ u1*u3 *(a11 *n1 - 3*a11 *a33*n1 + 3*a11*a33 *n1 - a33 *n1) + u1*u3*(
2
- 2*a11 *n1*n3 + a11*a22*n1*n3 + 3*a11*a33*n1*n3 - a22*a33*n1*n3
2
- a33 *n1*n3) + u1
3 2 3 2 3 2
2*a11*n1 - 2*a11*n1*n3 - a22*n1 + a22*n1*n3 - a33*n1 + a33*n1*n3
*------------------------------------------------------------------------ +
2
4 3 3 2 2 2 2 2 3
u2 *(a11 *a22 - a11 *a33 - 2*a11 *a22 + a11 *a22*a33 + a11 *a33 + a11*a22
2 2 3 2 2 3
+ a11*a22 *a33 - 2*a11*a22*a33 - a22 *a33 + a22 *a33 )/2 + u2 *(
3 2 2 2
a11 *n2 - 2*a11 *a22*n2 - a11 *a33*n2 + a11*a22 *n2 + 2*a11*a22*a33*n2
2 2 3 2 2
- a22 *a33*n2) + u2 *u3*(a11 *n3 - 2*a11 *a22*n3 - a11 *a33*n3
2 2 2 2 2
+ a11*a22 *n3 + 2*a11*a22*a33*n3 - a22 *a33*n3) + u2 *( - 2*a11 *n2
2 2 2 2 2
+ 2*a11 *n3 + 2*a11*a22*n1 + a11*a22*n2 - 3*a11*a22*n3
2 2 2 2 2 2 2
- 2*a11*a33*n1 + 3*a11*a33*n2 - a11*a33*n3 - a22 *n1 + a22 *n3
2 2 2 2 2 2
- a22*a33*n2 + a22*a33*n3 + a33 *n1 - a33 *n2 )/2
2 3 2 2 3
+ u2*u3 *(a11 *n2 - 3*a11 *a33*n2 + 3*a11*a33 *n2 - a33 *n2) + u2*u3*(
2
- 2*a11 *n2*n3 + a11*a22*n2*n3 + 3*a11*a33*n2*n3 - a22*a33*n2*n3
2 2 2 2 2
- a33 *n2*n3) + u2*(2*a11*n1 *n2 - 2*a11*n2*n3 - a22*n1 *n2 + a22*n2*n3
2 2 4 3 3
- a33*n1 *n2 + a33*n2*n3 )/2 + u3 *( - a11 *a22 + a11 *a33
2 2 2 2 3 3
+ 3*a11 *a22*a33 - 3*a11 *a33 - 3*a11*a22*a33 + 3*a11*a33 + a22*a33
4 3 3 2 2 3
- a33 )/2 + u3 *(a11 *n3 - 3*a11 *a33*n3 + 3*a11*a33 *n3 - a33 *n3) + u3
2 3 2 3 2 3
2*a11*n1 *n3 - 2*a11*n3 - a22*n1 *n3 + a22*n3 - a33*n1 *n3 + a33*n3
*------------------------------------------------------------------------}
2
{HAM,FI} = 0
3 2 2 2
FI=u1*( - a11 *n1 + a11 *a22*n1 + 2*a11 *a33*n1 - 2*a11*a22*a33*n1 - a11*a33 *n1
2 2 4 3 3 2 2
+ a22*a33 *n1) + u2 *(a11 - 2*a11 *a22 - 2*a11 *a33 + a11 *a22
2 2 2 2 2
+ 4*a11 *a22*a33 + a11 *a33 - 2*a11*a22 *a33 - 2*a11*a22*a33
2 2 3 2 2
+ a22 *a33 ) + u2*( - a11 *n2 + a11 *a22*n2 + 2*a11 *a33*n2
2 2 2 4 3
- 2*a11*a22*a33*n2 - a11*a33 *n2 + a22*a33 *n2) + u3 *(a11 - a11 *a22
3 2 2 2 2 3
- 3*a11 *a33 + 3*a11 *a22*a33 + 3*a11 *a33 - 3*a11*a22*a33 - a11*a33
3 3 2 2
+ a22*a33 ) + u3*( - a11 *n3 + a11 *a22*n3 + 2*a11 *a33*n3
2 2
- 2*a11*a22*a33*n3 - a11*a33 *n3 + a22*a33 *n3)
= a product of the elements of: {a11 - a33,
a11 - a33,
a11 - a22,
2 2
- u1*n1 + u2 *(a11 - a22) - u2*n2 + u3 *(a11 - a33) - u3*n3}
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*n1 + u2**2*a22 + u2*n2 + u3**2*a33 + u3*n3$
FI= - u1**2*v2**2 - u1**2*v3**2 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2**2*v1**2 -
u2**2*v3**2 + 2*u2*u3*v2*v3 - u3**2*v1**2 - u3**2*v2**2$
FI=u1**2*v1*( - a11**3*n1 + 3*a11**2*a33*n1 - 3*a11*a33**2*n1 + a33**3*n1) + u1*
u2**2*v1*(a11**4 - a11**3*a22 - 3*a11**3*a33 + 3*a11**2*a22*a33 + 3*a11**2*a33**
2 - 3*a11*a22*a33**2 - a11*a33**3 + a22*a33**3) + u1*u2*v1*( - a11**3*n2 + 3*a11
**2*a33*n2 - 3*a11*a33**2*n2 + a33**3*n2) + u1*u2*v2*( - a11**3*n1 + 3*a11**2*
a33*n1 - 3*a11*a33**2*n1 + a33**3*n1) + u1*u3**2*v1*(a11**4 - 4*a11**3*a33 + 6*
a11**2*a33**2 - 4*a11*a33**3 + a33**4) + u1*u3*v1*( - a11**3*n3 + 3*a11**2*a33*
n3 - 3*a11*a33**2*n3 + a33**3*n3) + u1*u3*v3*( - a11**3*n1 + 3*a11**2*a33*n1 - 3
*a11*a33**2*n1 + a33**3*n1) + u2**3*v2*(a11**4 - a11**3*a22 - 3*a11**3*a33 + 3*
a11**2*a22*a33 + 3*a11**2*a33**2 - 3*a11*a22*a33**2 - a11*a33**3 + a22*a33**3) +
u2**2*u3*v3*(a11**4 - a11**3*a22 - 3*a11**3*a33 + 3*a11**2*a22*a33 + 3*a11**2*
a33**2 - 3*a11*a22*a33**2 - a11*a33**3 + a22*a33**3) + u2**2*v2*( - a11**3*n2 +
3*a11**2*a33*n2 - 3*a11*a33**2*n2 + a33**3*n2) + u2*u3**2*v2*(a11**4 - 4*a11**3*
a33 + 6*a11**2*a33**2 - 4*a11*a33**3 + a33**4) + u2*u3*v2*( - a11**3*n3 + 3*a11
**2*a33*n3 - 3*a11*a33**2*n3 + a33**3*n3) + u2*u3*v3*( - a11**3*n2 + 3*a11**2*
a33*n2 - 3*a11*a33**2*n2 + a33**3*n2) + u3**3*v3*(a11**4 - 4*a11**3*a33 + 6*a11
**2*a33**2 - 4*a11*a33**3 + a33**4) + u3**2*v3*( - a11**3*n3 + 3*a11**2*a33*n3 -
3*a11*a33**2*n3 + a33**3*n3)$
FI=u1**3*( - 2*a11**3*n1 + 2*a11**2*a22*n1 + 4*a11**2*a33*n1 - 4*a11*a22*a33*n1
- 2*a11*a33**2*n1 + 2*a22*a33**2*n1) + u1**2*u2**2*(2*a11**4 - 4*a11**3*a22 - 4*
a11**3*a33 + 2*a11**2*a22**2 + 8*a11**2*a22*a33 + 2*a11**2*a33**2 - 4*a11*a22**2
*a33 - 4*a11*a22*a33**2 + 2*a22**2*a33**2) + u1**2*u2*( - 2*a11**3*n2 + 2*a11**2
*a22*n2 + 4*a11**2*a33*n2 - 4*a11*a22*a33*n2 - 2*a11*a33**2*n2 + 2*a22*a33**2*n2
) + u1**2*u3**2*(2*a11**4 - 2*a11**3*a22 - 6*a11**3*a33 + 6*a11**2*a22*a33 + 6*
a11**2*a33**2 - 6*a11*a22*a33**2 - 2*a11*a33**3 + 2*a22*a33**3) + u1**2*u3*( - 2
*a11**3*n3 + 2*a11**2*a22*n3 + 4*a11**2*a33*n3 - 4*a11*a22*a33*n3 - 2*a11*a33**2
*n3 + 2*a22*a33**2*n3) + u1*u2**2*(2*a11**3*n1 - 4*a11**2*a22*n1 - 2*a11**2*a33*
n1 + 2*a11*a22**2*n1 + 4*a11*a22*a33*n1 - 2*a22**2*a33*n1) + u1*u2*( - 4*a11**2*
n1*n2 + 2*a11*a22*n1*n2 + 6*a11*a33*n1*n2 - 2*a22*a33*n1*n2 - 2*a33**2*n1*n2) +
u1*u3**2*(2*a11**3*n1 - 6*a11**2*a33*n1 + 6*a11*a33**2*n1 - 2*a33**3*n1) + u1*u3
*( - 4*a11**2*n1*n3 + 2*a11*a22*n1*n3 + 6*a11*a33*n1*n3 - 2*a22*a33*n1*n3 - 2*
a33**2*n1*n3) + u1*(2*a11*n1**3 - 2*a11*n1*n3**2 - a22*n1**3 + a22*n1*n3**2 -
a33*n1**3 + a33*n1*n3**2) + u2**4*(a11**3*a22 - a11**3*a33 - 2*a11**2*a22**2 +
a11**2*a22*a33 + a11**2*a33**2 + a11*a22**3 + a11*a22**2*a33 - 2*a11*a22*a33**2
- a22**3*a33 + a22**2*a33**2) + u2**3*(2*a11**3*n2 - 4*a11**2*a22*n2 - 2*a11**2*
a33*n2 + 2*a11*a22**2*n2 + 4*a11*a22*a33*n2 - 2*a22**2*a33*n2) + u2**2*u3*(2*a11
**3*n3 - 4*a11**2*a22*n3 - 2*a11**2*a33*n3 + 2*a11*a22**2*n3 + 4*a11*a22*a33*n3
- 2*a22**2*a33*n3) + u2**2*( - 2*a11**2*n2**2 + 2*a11**2*n3**2 + 2*a11*a22*n1**2
+ a11*a22*n2**2 - 3*a11*a22*n3**2 - 2*a11*a33*n1**2 + 3*a11*a33*n2**2 - a11*a33
*n3**2 - a22**2*n1**2 + a22**2*n3**2 - a22*a33*n2**2 + a22*a33*n3**2 + a33**2*n1
**2 - a33**2*n2**2) + u2*u3**2*(2*a11**3*n2 - 6*a11**2*a33*n2 + 6*a11*a33**2*n2
- 2*a33**3*n2) + u2*u3*( - 4*a11**2*n2*n3 + 2*a11*a22*n2*n3 + 6*a11*a33*n2*n3 -
2*a22*a33*n2*n3 - 2*a33**2*n2*n3) + u2*(2*a11*n1**2*n2 - 2*a11*n2*n3**2 - a22*n1
**2*n2 + a22*n2*n3**2 - a33*n1**2*n2 + a33*n2*n3**2) + u3**4*( - a11**3*a22 +
a11**3*a33 + 3*a11**2*a22*a33 - 3*a11**2*a33**2 - 3*a11*a22*a33**2 + 3*a11*a33**
3 + a22*a33**3 - a33**4) + u3**3*(2*a11**3*n3 - 6*a11**2*a33*n3 + 6*a11*a33**2*
n3 - 2*a33**3*n3) + u3*(2*a11*n1**2*n3 - 2*a11*n3**3 - a22*n1**2*n3 + a22*n3**3
- a33*n1**2*n3 + a33*n3**3)$
FI=u1*( - a11**3*n1 + a11**2*a22*n1 + 2*a11**2*a33*n1 - 2*a11*a22*a33*n1 - a11*
a33**2*n1 + a22*a33**2*n1) + u2**2*(a11**4 - 2*a11**3*a22 - 2*a11**3*a33 + a11**
2*a22**2 + 4*a11**2*a22*a33 + a11**2*a33**2 - 2*a11*a22**2*a33 - 2*a11*a22*a33**
2 + a22**2*a33**2) + u2*( - a11**3*n2 + a11**2*a22*n2 + 2*a11**2*a33*n2 - 2*a11*
a22*a33*n2 - a11*a33**2*n2 + a22*a33**2*n2) + u3**2*(a11**4 - a11**3*a22 - 3*a11
**3*a33 + 3*a11**2*a22*a33 + 3*a11**2*a33**2 - 3*a11*a22*a33**2 - a11*a33**3 +
a22*a33**3) + u3*( - a11**3*n3 + a11**2*a22*n3 + 2*a11**2*a33*n3 - 2*a11*a22*a33
*n3 - a11*a33**2*n3 + a22*a33**2*n3)$