Solution 23 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
2 2 3 3
2*a33*c12*n3*r26 - i*n2 *n3*r417 + n2 *n3*r420 + i*n3 *r417 - n3 *r420
r10=------------------------------------------------------------------------
2
2*a33 *c12
3 3 2 2
2*a33*c12*n2*r26 - i*n2 *r417 + n2 *r420 + i*n2*n3 *r417 - n2*n3 *r420
r11=------------------------------------------------------------------------
2
2*a33 *c12
3 3 2 2
2*i*a33*c12*n2*r26 + n2 *r417 + i*n2 *r420 - n2*n3 *r417 - i*n2*n3 *r420
r12=--------------------------------------------------------------------------
2
2*a33 *c12
r13=0
r14=0
r15=0
2 2
r20=(2*i*a33 *c12*m1*r26 - 2*a33 *c12*m2*r26 + 4*i*a33*b21*c12*n2*r26
2 2 2
- a33*m1*n2 *r417 - i*a33*m1*n2 *r420 - a33*m1*n3 *r417
2 2 2
- i*a33*m1*n3 *r420 + i*a33*m2*n2 *r417 - a33*m2*n2 *r420
2 2 2
- i*a33*m2*n3 *r417 + a33*m2*n3 *r420 - 2*b21*n2*n3 *r417
2 3 2 2
- 2*i*b21*n2*n3 *r420 + 2*i*c12*n2 *r448 + 2*i*c12*n2*n3 *r448)/(4*a33
*c12*n2)
r21=( - i*a33*m2*n3*r417 + a33*m2*n3*r420 - b21*n2*n3*r417 - i*b21*n2*n3*r420
2
+ 2*i*c12*n2*n3*r448)/(2*a33 *c12)
r22=( - a33*m1*n2*r417 - i*a33*m1*n2*r420 - i*a33*m2*n2*r417 + a33*m2*n2*r420
2 2 2 2
- 2*b21*n2 *r417 - 2*i*b21*n2 *r420 + 2*i*c12*n2 *r448)/(2*a33 *c12)
r23=( - i*a33*m1*n3*r417 + a33*m1*n3*r420 - i*b21*n2*n3*r417 + b21*n2*n3*r420
2
- 2*c12*n2*n3*r448)/(2*a33 *c12)
r24=( - i*a33*m1*n2*r417 + a33*m1*n2*r420 + a33*m2*n2*r417 + i*a33*m2*n2*r420
2 2 2 2
- 2*i*b21*n2 *r417 + 2*b21*n2 *r420 - 2*c12*n2 *r448)/(2*a33 *c12)
- i*n2*n3*r417 + n2*n3*r420
r27=------------------------------
2*a33*c12
n2*n3*r417 + i*n2*n3*r420
r28=---------------------------
2*a33*c12
- i*n2*n3*r417 + n2*n3*r420
r210=------------------------------
2*a33*c12
2 2
- 2*i*a33*c12*r26 + n3 *r417 + i*n3 *r420
r212=--------------------------------------------
2*a33*c12
r213=0
r214=0
n2*n3*r417 + i*n2*n3*r420
r215=---------------------------
2*a33*c12
2 2 2 2
2*i*a33*c12*r26 + 2*n2 *r417 + 2*i*n2 *r420 - n3 *r417 - i*n3 *r420
r216=---------------------------------------------------------------------
2*a33*c12
2 2
i*n2 *r417 - n2 *r420
r217=-----------------------
a33*c12
r218=0
r219=0
r220=0
r30=( - a33*m1*n3*r448 - i*a33*m2*n3*r448 + a33*n2*n3*r417 - i*a33*n2*n3*r420
2
- 2*b21*n2*n3*r448)/(2*a33 *n2)
2 2 2 2 2 2
r31=(a33 *m1*m2*r417 + i*a33 *m1*m2*r420 + i*a33 *m2 *r417 - a33 *m2 *r420
- i*a33*b21*m1*n2*r417 + a33*b21*m1*n2*r420 + 3*a33*b21*m2*n2*r417
+ 3*i*a33*b21*m2*n2*r420 - 2*a33*c12*m1*n2*r448 - 2*i*a33*c12*m2*n2*r448
2 2 2 2
+ 2*a33*c12*n2 *r417 - 2*i*a33*c12*n2 *r420 - 2*i*b21 *n2 *r417
2 2 2 2
+ 2*b21 *n2 *r420 - 4*b21*c12*n2 *r448)/(4*a33 *c12*n2)
n3*r417
r32=---------
a33
n2*r417
r33=---------
a33
2 2 2 2 2 2
r34=(a33 *m1 *r417 + i*a33 *m1 *r420 + i*a33 *m1*m2*r417 - a33 *m1*m2*r420
+ 3*a33*b21*m1*n2*r417 + 3*i*a33*b21*m1*n2*r420 + i*a33*b21*m2*n2*r417
- a33*b21*m2*n2*r420 - 2*i*a33*c12*m1*n2*r448 + 2*a33*c12*m2*n2*r448
2 2 2 2
+ 2*i*a33*c12*n2 *r417 + 2*a33*c12*n2 *r420 + 2*b21 *n2 *r417
2 2 2 2
+ 2*i*b21 *n2 *r420 - 4*i*b21*c12*n2 *r448)/(4*a33 *c12*n2)
- i*n3*r417 + n3*r420
r35=------------------------
a33
n2*r420
r36=---------
a33
- i*n3*r420
r37=--------------
a33
n2*r417
r38=---------
a33
n2*r420
r39=---------
a33
r310=(a33*m1*n3*r417 + i*a33*m1*n3*r420 + i*a33*m2*n3*r417 - a33*m2*n3*r420
+ 2*b21*n2*n3*r417 + 2*i*b21*n2*n3*r420 + 4*i*c12*n2*n3*r448)/(4*a33*c12
*n2)
r311=
- i*a33*m2*r417 + a33*m2*r420 - b21*n2*r417 - i*b21*n2*r420 + 2*i*c12*n2*r448
--------------------------------------------------------------------------------
2*a33*c12
r312=0
r313=( - i*a33*m1*r417 + a33*m1*r420 + a33*m2*r417 + i*a33*m2*r420
- 2*i*b21*n2*r417 + 2*b21*n2*r420 - 4*c12*n2*r448)/(2*a33*c12)
r314=0
r315=0
- i*n3*r417 + n3*r420
r316=------------------------
c12
- i*n2*r417 + n2*r420
r317=------------------------
2*c12
n2*r417 + i*n2*r420
r318=---------------------
2*c12
r319=0
r320=(a33*m1*r417 + i*a33*m1*r420 + i*a33*m2*r417 - a33*m2*r420 + 2*b21*n2*r417
+ 2*i*b21*n2*r420)/(4*a33*c12)
n3*r448
r323=---------
a33
r325=
- a33*m1*r417 - i*a33*m1*r420 - b21*n2*r417 - i*b21*n2*r420 + 2*i*c12*n2*r448
--------------------------------------------------------------------------------
2*a33*c12
- i*n2*r417 + n2*r420
r326=------------------------
2*c12
- n3*r417 - i*n3*r420
r328=------------------------
2*c12
r329=0
r330=0
- n2*r417 - i*n2*r420
r332=------------------------
2*c12
r333=0
r334=0
r335=(i*a33*m1*r417 - a33*m1*r420 - a33*m2*r417 - i*a33*m2*r420
+ 2*i*b21*n2*r417 - 2*b21*n2*r420)/(4*a33*c12)
- n3*r448
r336=------------
a33
a33*m2*r417 + i*a33*m2*r420 - i*b21*n2*r417 + b21*n2*r420 - 2*c12*n2*r448
r337=---------------------------------------------------------------------------
2*a33*c12
r338=0
r339
a33*m1*r417 + i*a33*m1*r420 + b21*n2*r417 + i*b21*n2*r420 - 2*i*c12*n2*r448
=-----------------------------------------------------------------------------
2*a33*c12
a33*m2*r417 + i*a33*m2*r420 - i*b21*n2*r417 + b21*n2*r420 - 2*c12*n2*r448
r340=---------------------------------------------------------------------------
2*a33*c12
r341=0
n3*r417 + i*n3*r420
r342=---------------------
2*c12
r343=0
r344=0
r345=0
- i*n2*r417 + n2*r420
r347=------------------------
2*c12
r348=0
r349=0
r350=0
i*n2*r417 - n2*r420
r351=---------------------
2*c12
- n2*r417 - i*n2*r420
r352=------------------------
2*c12
r353=0
r354=0
r355=0
2 2 2 2
r40=( - i*a33 *m1 *r448 + 2*a33 *m1*m2*r448 + 2*i*a33 *m1*n2*r417
2 2 2 2
+ 2*a33 *m1*n2*r420 + i*a33 *m2 *r448 - 2*a33 *m2*n2*r417
2
+ 2*i*a33 *m2*n2*r420 - 4*i*a33*b21*m1*n2*r448 + 4*a33*b21*m2*n2*r448
2 2 2 2 2
+ 4*i*a33*b21*n2 *r417 + 4*a33*b21*n2 *r420 - 4*i*b21 *n2 *r448)/(8*a33
2
*n2 )
r41=0
i*a33*m1*r417 - a33*m2*r417 + 2*i*b21*n2*r417
r42=-----------------------------------------------
2*a33*n2
r43=0
r44=0
r45=0
r46=(a33*m1*r417 + i*a33*m1*r420 + i*a33*m2*r417 - a33*m2*r420 + 2*b21*n2*r417
+ 2*i*b21*n2*r420)/(2*a33*n2)
r47=0
r48=0
a33*m1*r420 + i*a33*m2*r420 + 2*b21*n2*r420
r49=---------------------------------------------
2*a33*n2
r410=0
r411=0
r412=0
r413=0
r415
- a33*m1*r448 - i*a33*m2*r448 + a33*n2*r417 - i*a33*n2*r420 - 2*b21*n2*r448
=------------------------------------------------------------------------------
2*a33*n2
r416=0
r418=0
r419=0
r421=0
r422=r417
r423=0
r424=0
r425=(a33*m1*r417 + i*a33*m1*r420 + i*a33*m2*r417 - a33*m2*r420 + 2*b21*n2*r417
+ 2*i*b21*n2*r420 + 2*i*c12*n2*r448)/(4*c12*n2)
r426=0
r427=0
r428=0
r429=0
- i*a33*r417 + a33*r420
r431=--------------------------
2*c12
r432=0
r433=0
r435=0
r439
i*a33*m1*r448 - a33*m2*r448 - i*a33*n2*r417 - a33*n2*r420 + 2*i*b21*n2*r448
=-----------------------------------------------------------------------------
2*a33*n2
r442=0
r444= - r420
r445=0
r450=0
r451=0
r454=0
r455=0
r458=0
- i*r448
r460=-----------
2
r461=0
r463=0
r464=0
r465=0
r467=0
r468=0
r469=0
r470=0
r471=
- i*a33*m1*r448 + a33*m2*r448 + i*a33*n2*r417 + a33*n2*r420 - 2*i*b21*n2*r448
--------------------------------------------------------------------------------
2*a33*n2
r472=0
r473=i*r417
r474=0
r475=0
r476=r417 + i*r420
r477=0
r478=i*r417 + r420
r479=r417 + i*r420
r480=0
r481= - r448
r482=0
r483= - i*r448
r484=0
r485=0
r486=0
a33*r417 + i*a33*r420
r487=-----------------------
2*c12
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
- i*r448
r4107=-----------
2
r4108=0
r4109=0
r4110= - i*r448
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
r4125=0
i*b21*n3
m3=----------
a33
n1=i*n2
c23=0
c22=2*i*c12
c13=0
c11=0
b33=i*b21
b32=0
b31=0
b23=0
b22=0
b13=0
b12= - b21
b11=0
a23=0
a22=0
a13=0
a12=0
a11=0
2
- a33*b21*m1 - i*a33*b21*m2 + 2*i*a33*c12*n2 - 2*b21 *n2
c33=-----------------------------------------------------------
2*a33*n2
- a33*r417 - i*a33*r420
r453=--------------------------
2*c12
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:
r417, m1, r26, r420, r448, m2, b21, n3, n2, c12, a33
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.
{n3,n2,a33}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a33*b21*m1 + i*a33*b21*m2 - 2*i*a33*c12*n2 + 2*a33*c33*n2 + 2*b21**2*n2,
a11,
a12,
a13,
a22,
a23,
b11,
b12 + b21,
b13,
b22,
b23,
b31,
b32,
- i*b21 + b33,
c11,
c13,
- 2*i*c12 + c22,
c23,
n1 - i*n2,
a33*m3 - i*b21*n3}$
The system of equations related to the Hamiltonian HAM:
2
HAM= - u1*v2*b21 + i*u1*n2 + u2*v1*b21 + u2*n2 + u3 *a33 + i*u3*v3*b21 + u3*n3
2
+ 2*v1*v2*c12 + v1*m1 + 2*i*v2 *c12 + v2*m2
2
2 - a33*b21*m1 - i*a33*b21*m2 + 2*i*a33*c12*n2 - 2*b21 *n2
+ v3 *-----------------------------------------------------------
2*a33*n2
i*v3*b21*n3
+ -------------
a33
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 2 2 2 2 2 2 2 2 2
FI= - 8*i*u1 *v1 *a33 *n2 - 4*i*u1 *v2 *a33 *n2 - 8*i*u1*u3*v1*v3*a33 *n2
2 2 2 3 3
- 8*u1*u3*v2*v3*a33 *n2 - 8*u1*v1 *a33*n2 - 8*i*u1*v1*v2*a33*n2
2 3
- 8*u1*v2 *a33*n2
2 2 2 2
+ u1*v2*v3 *( - 4*i*a33 *m1*n2 + 4*a33 *m2*n2 - 8*i*a33*b21*n2 )
2 2 2 2 2 2 2
- 8*u1*v2*v3*a33*n2 *n3 - 4*i*u2 *v1 *a33 *n2 + 8*u2*u3*v1*v3*a33 *n2
2 3
+ 8*i*u2*v1 *a33*n2
2 2 2 2
+ u2*v1*v3 *(4*i*a33 *m1*n2 - 4*a33 *m2*n2 + 8*i*a33*b21*n2 )
2 2 2 2 2 3
+ 8*u2*v1*v3*a33*n2 *n3 + 4*i*u3 *v3 *a33 *n2 - 16*u3*v1*v3*a33*n2
3
+ 8*i*u3*v2*v3*a33*n2
3 2 2 2
+ u3*v3 *( - 4*a33 *m1*n2 - 4*i*a33 *m2*n2 - 8*a33*b21*n2 )
2 2 4
+ 8*i*u3*v3 *a33*n2 *n3 - 8*v1*v2*n2
2 2 2 3 3
+ v1*v3 *( - 4*i*a33*m1*n2 + 4*a33*m2*n2 - 8*i*b21*n2 ) - 8*v1*v3*n2 *n3
2 4 2 2 2 3
+ 8*i*v2 *n2 + v2*v3 *( - 4*a33*m1*n2 - 4*i*a33*m2*n2 - 8*b21*n2 )
3 4 2 2 2 2 2
+ 8*i*v2*v3*n2 *n3 + v3 *( - i*a33 *m1 + 2*a33 *m1*m2 + i*a33 *m2
2 2
- 4*i*a33*b21*m1*n2 + 4*a33*b21*m2*n2 - 4*i*b21 *n2 )
3 2
+ v3 *( - 4*a33*m1*n2*n3 - 4*i*a33*m2*n2*n3 - 8*b21*n2 *n3)
2 4 2 2
+ v3 *(4*i*n2 + 4*i*n2 *n3 )
Stop of a subroutine.
Number of garbage collections exceeds max_gc_fac.
The factorization crashed!
which the program can not factorize further.
{HAM,FI} = { - 16*i,
u1*v1 + u2*v2 + u3*v3,
n2,
a33,
2 i*u1*v2*v3*a33*b21*n2 u1*v2*a33*n2*n3
u1*u3*v2*a33 *n2 + ----------------------- + -----------------
2 2
2
- u1*v3*a33*n2 2 i*u2*v1*v3*a33*b21*n2
+ ------------------ + u2*u3*v1*a33 *n2 + -----------------------
2 2
2
u2*v1*a33*n2*n3 - u3*v1 *a33*b21*n2
+ ----------------- + ----------------------
2 2
2
i*u3*v1*v2*a33*b21*n2 - u3*v1*a33*n2 2
+ ----------------------- + ------------------ - i*u3*v2*a33*n2
2 2
2
- v1 *v3*a33*c12*n2 + v1*v2*v3
2
- a33*b21*m1 - i*a33*b21*m2 - 2*i*a33*c12*n2 - 2*b21 *n2
*-----------------------------------------------------------
2
2
i*v1*v2*b21*n2*n3 - a33*m2*n2 + i*b21*n2
+ ------------------- + v1*v3*--------------------------
2 2
2 2 3
v2*v3*b21*n2 - i*v2*n2 *n3 i*v3*n2
+ --------------- + ---------------- + ----------}
2 2 2
2 2 2 2 2 2 2 2
FI= - 2*i*u1 *v1*a33 *n2 - 2*u1 *v2*a33 *n2 + 2*u1*u2*v1*a33 *n2
2 3 2 3 2
+ 2*i*u1*u3 *v2*a33 *n2 + 2*i*u1*u3*v2*a33 *n2*n3 + 4*i*u1*v1 *a33 *c12*n2
2 2 2 2 2
+ 4*u1*v1 *v2*a33 *c12*n2 + u1*v1 *(2*i*a33 *m2*n2 + 2*a33*b21*n2 )
2 2 2 2
+ 4*i*u1*v1*v2 *a33 *c12*n2 + u1*v1*v2*(2*i*a33 *m1*n2 + 2*i*a33*b21*n2 )
3 2 2 2
- 4*u1*v1*a33*n2 + u1*v2 *(2*i*a33 *m2*n2 + 2*a33*b21*n2 )
2 2 3 2
+ 2*u1*v2*v3 *a33 *c12*n2 + u1*v2*(4*i*a33*n2 - 2*i*a33*n2*n3 )
2 2 2 2
+ u1*v3 *( - a33 *m1*n2 - i*a33 *m2*n2 - 2*a33*b21*n2 )
2 2 2 2 2 3
+ 2*i*u1*v3*a33*n2 *n3 - 2*i*u2 *v1*a33 *n2 - 2*i*u2*u3 *v1*a33 *n2
2 2 2 3 2
- 2*i*u2*u3*v1*a33 *n2*n3 + 2*u2*u3*v3*a33 *n2 - 4*u2*v1 *a33 *c12*n2
2 2 2 2 2
+ u2*v1 *( - 2*i*a33 *m1*n2 - 2*i*a33*b21*n2 ) - 2*u2*v1*v3 *a33 *c12*n2
2
+ 2*i*u2*v1*a33*n2*n3
2 2 2 2 2
+ u2*v3 *(i*a33 *m1*n2 - a33 *m2*n2 + 2*i*a33*b21*n2 ) + 2*u2*v3*a33*n2 *n3
3 3 2 2 2 2 2 2
+ 2*u3 *v3*a33 *n2 + 2*i*u3 *v1*a33 *n2 + 2*u3 *v2*a33 *n2
2 2 3 3 2 2 2
+ u3 *v3 *(i*a33 *m1 - a33 *m2 + 2*i*a33 *b21*n2) + 4*u3 *v3*a33 *n2*n3
2
+ 4*u3*v1*v2*v3*a33 *c12*n2
2 2 2
+ u3*v1*v3*(2*a33 *m1*n2 + 2*i*a33 *m2*n2 + 4*a33*b21*n2 )
2 2 2
+ 2*i*u3*v1*a33*n2 *n3 + u3*v2*v3*(2*a33 *m2*n2 - 2*i*a33*b21*n2 )
2 3 2
+ 2*u3*v2*a33*n2 *n3 - 2*i*u3*v3 *a33 *c12*n2
2 2 2 3 2
+ u3*v3 *(i*a33 *m1*n3 - a33 *m2*n3 + 2*i*a33*b21*n2*n3) + 4*v1 *a33*c12*n2
2 2 2 2
+ v1 *v3 *(2*a33 *c12*m1 + 2*i*a33 *c12*m2 + 4*a33*b21*c12*n2)
2 2 2
- 4*i*v1 *v3*a33*c12*n2*n3 + 4*v1*v2 *a33*c12*n2
2 2 2
+ v1*v2*v3 *(2*i*a33 *c12*m1 - 2*a33 *c12*m2 + 4*i*a33*b21*c12*n2)
+ 4*v1*v2*v3*a33*c12*n2*n3
2 2 3 2 2 2
+ v1*v2*(2*a33*m1*n2 + 2*i*a33*m2*n2 + 4*b21*n2 ) + v1*v3 *(i*a33 *m1
2 2
- a33 *m1*m2 + 3*i*a33*b21*m1*n2 - a33*b21*m2*n2 + 2*a33*c12*n2
2 2 2
+ 2*i*b21 *n2 ) + v1*v3*(2*a33*m1*n2*n3 + 2*b21*n2 *n3)
4 2 2
+ v1*(2*i*n2 - 2*i*n2 *n3 )
2 2 2 3 2
+ v2 *( - 2*i*a33*m1*n2 + 2*a33*m2*n2 - 4*i*b21*n2 ) + v2*v3 *(
2 2 2
i*a33 *m1*m2 - a33 *m2 + a33*b21*m1*n2 + 3*i*a33*b21*m2*n2
2 2 2 2
- 2*i*a33*c12*n2 + 2*b21 *n2 ) + v2*v3*(2*a33*m2*n2*n3 - 2*i*b21*n2 *n3)
4 2 2
+ v2*(2*n2 - 2*n2 *n3 )
4 2 2
+ v3 *(a33 *c12*m1 + i*a33 *c12*m2 + 2*a33*b21*c12*n2)
3 2
- 2*i*v3 *a33*c12*n2*n3 + v3
2 2 2 2 2
*( - i*a33*m1*n2 - i*a33*m1*n3 - a33*m2*n2 + a33*m2*n3 - 2*i*b21*n2*n3 )
3 3
+ v3*(2*n2 *n3 - 2*n2*n3 )
= a product of the elements of: {i,
2 2 2 2 2 2 2 2
- 2*u1 *v1*a33 *n2 + 2*i*u1 *v2*a33 *n2 - 2*i*u1*u2*v1*a33 *n2
2 3 2 3 2
+ 2*u1*u3 *v2*a33 *n2 + 2*u1*u3*v2*a33 *n2*n3 + 4*u1*v1 *a33 *c12*n2
2 2 2 2 2
- 4*i*u1*v1 *v2*a33 *c12*n2 + u1*v1 *(2*a33 *m2*n2 - 2*i*a33*b21*n2 )
2 2 2 2
+ 4*u1*v1*v2 *a33 *c12*n2 + u1*v1*v2*(2*a33 *m1*n2 + 2*a33*b21*n2 )
3 2 2 2
+ 4*i*u1*v1*a33*n2 + u1*v2 *(2*a33 *m2*n2 - 2*i*a33*b21*n2 )
2 2 3 2
- 2*i*u1*v2*v3 *a33 *c12*n2 + u1*v2*(4*a33*n2 - 2*a33*n2*n3 )
2 2 2 2 2
+ u1*v3 *(i*a33 *m1*n2 - a33 *m2*n2 + 2*i*a33*b21*n2 ) + 2*u1*v3*a33*n2 *n3
2 2 2 2 3 2
- 2*u2 *v1*a33 *n2 - 2*u2*u3 *v1*a33 *n2 - 2*u2*u3*v1*a33 *n2*n3
2 2 3 2
- 2*i*u2*u3*v3*a33 *n2 + 4*i*u2*v1 *a33 *c12*n2
2 2 2 2 2
+ u2*v1 *( - 2*a33 *m1*n2 - 2*a33*b21*n2 ) + 2*i*u2*v1*v3 *a33 *c12*n2
2 2 2 2 2
+ 2*u2*v1*a33*n2*n3 + u2*v3 *(a33 *m1*n2 + i*a33 *m2*n2 + 2*a33*b21*n2 )
2 3 3 2 2 2
- 2*i*u2*v3*a33*n2 *n3 - 2*i*u3 *v3*a33 *n2 + 2*u3 *v1*a33 *n2
2 2 2 2 2 3 3 2
- 2*i*u3 *v2*a33 *n2 + u3 *v3 *(a33 *m1 + i*a33 *m2 + 2*a33 *b21*n2)
2 2 2
- 4*i*u3 *v3*a33 *n2*n3 - 4*i*u3*v1*v2*v3*a33 *c12*n2
2 2 2
+ u3*v1*v3*( - 2*i*a33 *m1*n2 + 2*a33 *m2*n2 - 4*i*a33*b21*n2 )
2 2 2
+ 2*u3*v1*a33*n2 *n3 + u3*v2*v3*( - 2*i*a33 *m2*n2 - 2*a33*b21*n2 )
2 3 2
- 2*i*u3*v2*a33*n2 *n3 - 2*u3*v3 *a33 *c12*n2
2 2 2 3 2
+ u3*v3 *(a33 *m1*n3 + i*a33 *m2*n3 + 2*a33*b21*n2*n3) - 4*i*v1 *a33*c12*n2
2 2 2 2
+ v1 *v3 *( - 2*i*a33 *c12*m1 + 2*a33 *c12*m2 - 4*i*a33*b21*c12*n2)
2 2 2
- 4*v1 *v3*a33*c12*n2*n3 - 4*i*v1*v2 *a33*c12*n2
2 2 2
+ v1*v2*v3 *(2*a33 *c12*m1 + 2*i*a33 *c12*m2 + 4*a33*b21*c12*n2)
- 4*i*v1*v2*v3*a33*c12*n2*n3
2 2 3 2 2 2
+ v1*v2*( - 2*i*a33*m1*n2 + 2*a33*m2*n2 - 4*i*b21*n2 ) + v1*v3 *(a33 *m1
2 2
+ i*a33 *m1*m2 + 3*a33*b21*m1*n2 + i*a33*b21*m2*n2 - 2*i*a33*c12*n2
2 2 2
+ 2*b21 *n2 ) + v1*v3*( - 2*i*a33*m1*n2*n3 - 2*i*b21*n2 *n3)
4 2 2
+ v1*(2*n2 - 2*n2 *n3 )
2 2 2 3 2 2
+ v2 *( - 2*a33*m1*n2 - 2*i*a33*m2*n2 - 4*b21*n2 ) + v2*v3 *(a33 *m1*m2
2 2 2
+ i*a33 *m2 - i*a33*b21*m1*n2 + 3*a33*b21*m2*n2 - 2*a33*c12*n2
2 2 2
- 2*i*b21 *n2 ) + v2*v3*( - 2*i*a33*m2*n2*n3 - 2*b21*n2 *n3)
4 2 2
+ v2*( - 2*i*n2 + 2*i*n2 *n3 )
4 2 2
+ v3 *( - i*a33 *c12*m1 + a33 *c12*m2 - 2*i*a33*b21*c12*n2)
3 2
- 2*v3 *a33*c12*n2*n3 + v3
2 2 2 2 2
*( - a33*m1*n2 - a33*m1*n3 + i*a33*m2*n2 - i*a33*m2*n3 - 2*b21*n2*n3 )
3 3
+ v3*( - 2*i*n2 *n3 + 2*i*n2*n3 )}
{HAM,FI} = {16*i,
u1*v1 + u2*v2 + u3*v3,
n2,
a33,
2
- u1*v3*a33*b21*n2 - u2*u3*a33 *n2 - i*u2*v3*a33*b21*n2
--------------------- + ------------------ + -----------------------
8 4 8
- u2*a33*n2*n3 2 u3*v1*a33*b21*n2
+ ----------------- + u3*v1*v2*a33 *c12 + ------------------
8 8
2 2
2*a33 *m2 - 3*i*a33*b21*n2 u3*a33*n2
+ u3*v2*---------------------------- + ------------
8 8
2
- v1 *v3*a33*b21*c12 i*v1*v2*v3*a33*b21*c12
+ ----------------------- + ------------------------
2 2
v1*v2*a33*c12*n3
+ ------------------
2
2
- a33*b21*m2 - 2*a33*c12*n2 + i*b21 *n2
+ v1*v3*------------------------------------------
8
2
a33*b21*m1 + 2*i*a33*b21*m2 + 2*i*a33*c12*n2 + 3*b21 *n2
+ v2*v3*----------------------------------------------------------
8
2
a33*m2*n3 - 2*i*b21*n2*n3 i*v3*b21*n2
+ v2*--------------------------- + --------------}
8 8
2 2
FI=2*i*u1*v2*a33*n2 - 2*i*u2*v1*a33*n2 + 2*u3*v3*a33*n2 + 2*i*v1*n2 + 2*v2*n2
2
+ v3 *(i*a33*m1 - a33*m2 + 2*i*b21*n2) + 2*v3*n2*n3
= a product of the elements of: {i,
2 2
2*u1*v2*a33*n2 - 2*u2*v1*a33*n2 - 2*i*u3*v3*a33*n2 + 2*v1*n2 - 2*i*v2*n2
2
+ v3 *(a33*m1 + i*a33*m2 + 2*b21*n2) - 2*i*v3*n2*n3}
{HAM,FI} = 0
2 2 2 2 2 2 2 2
FI= - 2*u1 *v1*a33 *n2 + 2*i*u1 *v2*a33 *n2 - 2*i*u1*u2*v1*a33 *n2
2 3 2 3 2
+ 2*u1*u3 *v2*a33 *n2 + 2*u1*u3*v2*a33 *n2*n3 + 4*u1*v1 *a33 *c12*n2
2 2 2 2 2
+ 4*i*u1*v1 *v2*a33 *c12*n2 + u1*v1 *(2*a33 *m2*n2 - 2*i*a33*b21*n2 )
2 2 2 2
+ 4*u1*v1*v2 *a33 *c12*n2 + u1*v1*v2*(2*a33 *m1*n2 + 2*a33*b21*n2 )
3 3 2
+ 4*i*u1*v1*a33*n2 + 4*i*u1*v2 *a33 *c12*n2
2 2 2 2 2
+ u1*v2 *(2*a33 *m2*n2 - 2*i*a33*b21*n2 ) + 2*i*u1*v2*v3 *a33 *c12*n2
3 2
+ u1*v2*(4*a33*n2 - 2*a33*n2*n3 )
2 2 2 2 2
+ u1*v3 *(i*a33 *m1*n2 - a33 *m2*n2 + 2*i*a33*b21*n2 ) + 2*u1*v3*a33*n2 *n3
2 2 2 2 3 2
- 2*u2 *v1*a33 *n2 - 2*u2*u3 *v1*a33 *n2 - 2*u2*u3*v1*a33 *n2*n3
2 2 2 2 2
- 2*i*u2*u3*v3*a33 *n2 + u2*v1 *( - 2*a33 *m1*n2 - 2*a33*b21*n2 )
2 2 2
- 2*i*u2*v1*v3 *a33 *c12*n2 + 2*u2*v1*a33*n2*n3
2 2 2 2 2
+ u2*v3 *(a33 *m1*n2 + i*a33 *m2*n2 + 2*a33*b21*n2 ) - 2*i*u2*v3*a33*n2 *n3
3 3 2 2 2 2 2 2
- 2*i*u3 *v3*a33 *n2 + 2*u3 *v1*a33 *n2 - 2*i*u3 *v2*a33 *n2
2 2 3 3 2 2 2
+ u3 *v3 *(a33 *m1 + i*a33 *m2 + 2*a33 *b21*n2) - 4*i*u3 *v3*a33 *n2*n3
2 2
+ 4*u3*v1 *v3*a33 *c12*n2
2 2 2
+ u3*v1*v3*( - 2*i*a33 *m1*n2 + 2*a33 *m2*n2 - 4*i*a33*b21*n2 )
2 2 2
+ 2*u3*v1*a33*n2 *n3 + 4*u3*v2 *v3*a33 *c12*n2
2 2 2
+ u3*v2*v3*( - 2*i*a33 *m2*n2 - 2*a33*b21*n2 ) - 2*i*u3*v2*a33*n2 *n3
3 2
+ 2*u3*v3 *a33 *c12*n2
2 2 2
+ u3*v3 *(a33 *m1*n3 + i*a33 *m2*n3 + 2*a33*b21*n2*n3)
2 2
+ 4*v1 *v2*a33*c12*n2
2 2 2
+ v1*v2*v3 *(2*a33 *c12*m1 + 2*i*a33 *c12*m2 + 4*a33*b21*c12*n2)
- 4*i*v1*v2*v3*a33*c12*n2*n3
2 2 3 2 2 2
+ v1*v2*( - 2*i*a33*m1*n2 + 2*a33*m2*n2 - 4*i*b21*n2 ) + v1*v3 *(a33 *m1
2 2
+ i*a33 *m1*m2 + 3*a33*b21*m1*n2 + i*a33*b21*m2*n2 + 2*i*a33*c12*n2
2 2 2
+ 2*b21 *n2 ) + v1*v3*( - 2*i*a33*m1*n2*n3 - 2*i*b21*n2 *n3)
4 2 2 3 2
+ v1*(2*n2 - 2*n2 *n3 ) + 4*v2 *a33*c12*n2
2 2 2 2
+ v2 *v3 *(2*i*a33 *c12*m1 - 2*a33 *c12*m2 + 4*i*a33*b21*c12*n2)
2
+ 4*v2 *v3*a33*c12*n2*n3
2 2 2 3 2 2
+ v2 *( - 2*a33*m1*n2 - 2*i*a33*m2*n2 - 4*b21*n2 ) + v2*v3 *(a33 *m1*m2
2 2 2
+ i*a33 *m2 - i*a33*b21*m1*n2 + 3*a33*b21*m2*n2 + 2*a33*c12*n2
2 2 2
- 2*i*b21 *n2 ) + v2*v3*( - 2*i*a33*m2*n2*n3 - 2*b21*n2 *n3)
4 2 2
+ v2*( - 2*i*n2 + 2*i*n2 *n3 )
4 2 2
+ v3 *(i*a33 *c12*m1 - a33 *c12*m2 + 2*i*a33*b21*c12*n2)
3 2
+ 2*v3 *a33*c12*n2*n3 + v3
2 2 2 2 2
*( - a33*m1*n2 - a33*m1*n3 + i*a33*m2*n2 - i*a33*m2*n3 - 2*b21*n2*n3 )
3 3
+ v3*( - 2*i*n2 *n3 + 2*i*n2*n3 )
which the program can not factorize further.
{HAM,FI} = { - 8*i,
u1*v1 + u2*v2 + u3*v3,
n2,
a33,
2
- i*u1*v3*a33*b21*n2 - i*u2*u3*a33 *n2 u2*v3*a33*b21*n2
----------------------- + -------------------- + ------------------
4 2 4
- i*u2*a33*n2*n3 2 2 2
+ ------------------- + u3*v1 *a33 *c12 + 2*i*u3*v1*v2*a33 *c12
4
i*u3*v1*a33*b21*n2 2 2
+ -------------------- - u3*v2 *a33 *c12
4
2 2
2*i*a33 *m2 + 3*a33*b21*n2 i*u3*a33*n2
+ u3*v2*---------------------------- + --------------
4 4
2 2
- i*v1 *v3*a33*b21*c12 v1 *a33*c12*n3
+ ------------------------- + ---------------- + i*v1*v2*a33*c12*n3
2 2
2
- i*a33*b21*m2 - 4*i*a33*c12*n2 - b21 *n2
+ v1*v3*--------------------------------------------
4
2 2
- i*v2 *v3*a33*b21*c12 - v2 *a33*c12*n3
+ ------------------------- + -------------------
2 2
2
i*a33*b21*m1 - 2*a33*b21*m2 + 3*i*b21 *n2
+ v2*v3*-------------------------------------------
4
2
i*a33*m2*n3 + 2*b21*n2*n3 - v3*b21*n2
+ v2*--------------------------- + ---------------}
4 4
And again in machine readable form:
HAM= - u1*v2*b21 + i*u1*n2 + u2*v1*b21 + u2*n2 + u3**2*a33 + i*u3*v3*b21 + u3*n3
+ 2*v1*v2*c12 + v1*m1 + 2*i*v2**2*c12 + v2*m2 + v3**2*( - a33*b21*m1 - i*a33*
b21*m2 + 2*i*a33*c12*n2 - 2*b21**2*n2)/(2*a33*n2) + (i*v3*b21*n3)/a33$
FI= - 8*i*u1**2*v1**2*a33**2*n2**2 - 4*i*u1**2*v2**2*a33**2*n2**2 - 8*i*u1*u3*v1
*v3*a33**2*n2**2 - 8*u1*u3*v2*v3*a33**2*n2**2 - 8*u1*v1**2*a33*n2**3 - 8*i*u1*v1
*v2*a33*n2**3 - 8*u1*v2**2*a33*n2**3 + u1*v2*v3**2*( - 4*i*a33**2*m1*n2 + 4*a33
**2*m2*n2 - 8*i*a33*b21*n2**2) - 8*u1*v2*v3*a33*n2**2*n3 - 4*i*u2**2*v1**2*a33**
2*n2**2 + 8*u2*u3*v1*v3*a33**2*n2**2 + 8*i*u2*v1**2*a33*n2**3 + u2*v1*v3**2*(4*i
*a33**2*m1*n2 - 4*a33**2*m2*n2 + 8*i*a33*b21*n2**2) + 8*u2*v1*v3*a33*n2**2*n3 +
4*i*u3**2*v3**2*a33**2*n2**2 - 16*u3*v1*v3*a33*n2**3 + 8*i*u3*v2*v3*a33*n2**3 +
u3*v3**3*( - 4*a33**2*m1*n2 - 4*i*a33**2*m2*n2 - 8*a33*b21*n2**2) + 8*i*u3*v3**2
*a33*n2**2*n3 - 8*v1*v2*n2**4 + v1*v3**2*( - 4*i*a33*m1*n2**2 + 4*a33*m2*n2**2 -
8*i*b21*n2**3) - 8*v1*v3*n2**3*n3 + 8*i*v2**2*n2**4 + v2*v3**2*( - 4*a33*m1*n2
**2 - 4*i*a33*m2*n2**2 - 8*b21*n2**3) + 8*i*v2*v3*n2**3*n3 + v3**4*( - i*a33**2*
m1**2 + 2*a33**2*m1*m2 + i*a33**2*m2**2 - 4*i*a33*b21*m1*n2 + 4*a33*b21*m2*n2 -
4*i*b21**2*n2**2) + v3**3*( - 4*a33*m1*n2*n3 - 4*i*a33*m2*n2*n3 - 8*b21*n2**2*n3
) + v3**2*(4*i*n2**4 + 4*i*n2**2*n3**2)$
FI= - 2*i*u1**2*v1*a33**2*n2**2 - 2*u1**2*v2*a33**2*n2**2 + 2*u1*u2*v1*a33**2*n2
**2 + 2*i*u1*u3**2*v2*a33**3*n2 + 2*i*u1*u3*v2*a33**2*n2*n3 + 4*i*u1*v1**3*a33**
2*c12*n2 + 4*u1*v1**2*v2*a33**2*c12*n2 + u1*v1**2*(2*i*a33**2*m2*n2 + 2*a33*b21*
n2**2) + 4*i*u1*v1*v2**2*a33**2*c12*n2 + u1*v1*v2*(2*i*a33**2*m1*n2 + 2*i*a33*
b21*n2**2) - 4*u1*v1*a33*n2**3 + u1*v2**2*(2*i*a33**2*m2*n2 + 2*a33*b21*n2**2) +
2*u1*v2*v3**2*a33**2*c12*n2 + u1*v2*(4*i*a33*n2**3 - 2*i*a33*n2*n3**2) + u1*v3
**2*( - a33**2*m1*n2 - i*a33**2*m2*n2 - 2*a33*b21*n2**2) + 2*i*u1*v3*a33*n2**2*
n3 - 2*i*u2**2*v1*a33**2*n2**2 - 2*i*u2*u3**2*v1*a33**3*n2 - 2*i*u2*u3*v1*a33**2
*n2*n3 + 2*u2*u3*v3*a33**2*n2**2 - 4*u2*v1**3*a33**2*c12*n2 + u2*v1**2*( - 2*i*
a33**2*m1*n2 - 2*i*a33*b21*n2**2) - 2*u2*v1*v3**2*a33**2*c12*n2 + 2*i*u2*v1*a33*
n2*n3**2 + u2*v3**2*(i*a33**2*m1*n2 - a33**2*m2*n2 + 2*i*a33*b21*n2**2) + 2*u2*
v3*a33*n2**2*n3 + 2*u3**3*v3*a33**3*n2 + 2*i*u3**2*v1*a33**2*n2**2 + 2*u3**2*v2*
a33**2*n2**2 + u3**2*v3**2*(i*a33**3*m1 - a33**3*m2 + 2*i*a33**2*b21*n2) + 4*u3
**2*v3*a33**2*n2*n3 + 4*u3*v1*v2*v3*a33**2*c12*n2 + u3*v1*v3*(2*a33**2*m1*n2 + 2
*i*a33**2*m2*n2 + 4*a33*b21*n2**2) + 2*i*u3*v1*a33*n2**2*n3 + u3*v2*v3*(2*a33**2
*m2*n2 - 2*i*a33*b21*n2**2) + 2*u3*v2*a33*n2**2*n3 - 2*i*u3*v3**3*a33**2*c12*n2
+ u3*v3**2*(i*a33**2*m1*n3 - a33**2*m2*n3 + 2*i*a33*b21*n2*n3) + 4*v1**3*a33*c12
*n2**2 + v1**2*v3**2*(2*a33**2*c12*m1 + 2*i*a33**2*c12*m2 + 4*a33*b21*c12*n2) -
4*i*v1**2*v3*a33*c12*n2*n3 + 4*v1*v2**2*a33*c12*n2**2 + v1*v2*v3**2*(2*i*a33**2*
c12*m1 - 2*a33**2*c12*m2 + 4*i*a33*b21*c12*n2) + 4*v1*v2*v3*a33*c12*n2*n3 + v1*
v2*(2*a33*m1*n2**2 + 2*i*a33*m2*n2**2 + 4*b21*n2**3) + v1*v3**2*(i*a33**2*m1**2
- a33**2*m1*m2 + 3*i*a33*b21*m1*n2 - a33*b21*m2*n2 + 2*a33*c12*n2**2 + 2*i*b21**
2*n2**2) + v1*v3*(2*a33*m1*n2*n3 + 2*b21*n2**2*n3) + v1*(2*i*n2**4 - 2*i*n2**2*
n3**2) + v2**2*( - 2*i*a33*m1*n2**2 + 2*a33*m2*n2**2 - 4*i*b21*n2**3) + v2*v3**2
*(i*a33**2*m1*m2 - a33**2*m2**2 + a33*b21*m1*n2 + 3*i*a33*b21*m2*n2 - 2*i*a33*
c12*n2**2 + 2*b21**2*n2**2) + v2*v3*(2*a33*m2*n2*n3 - 2*i*b21*n2**2*n3) + v2*(2*
n2**4 - 2*n2**2*n3**2) + v3**4*(a33**2*c12*m1 + i*a33**2*c12*m2 + 2*a33*b21*c12*
n2) - 2*i*v3**3*a33*c12*n2*n3 + v3**2*( - i*a33*m1*n2**2 - i*a33*m1*n3**2 - a33*
m2*n2**2 + a33*m2*n3**2 - 2*i*b21*n2*n3**2) + v3*(2*n2**3*n3 - 2*n2*n3**3)$
FI=2*i*u1*v2*a33*n2 - 2*i*u2*v1*a33*n2 + 2*u3*v3*a33*n2 + 2*i*v1*n2**2 + 2*v2*n2
**2 + v3**2*(i*a33*m1 - a33*m2 + 2*i*b21*n2) + 2*v3*n2*n3$
FI= - 2*u1**2*v1*a33**2*n2**2 + 2*i*u1**2*v2*a33**2*n2**2 - 2*i*u1*u2*v1*a33**2*
n2**2 + 2*u1*u3**2*v2*a33**3*n2 + 2*u1*u3*v2*a33**2*n2*n3 + 4*u1*v1**3*a33**2*
c12*n2 + 4*i*u1*v1**2*v2*a33**2*c12*n2 + u1*v1**2*(2*a33**2*m2*n2 - 2*i*a33*b21*
n2**2) + 4*u1*v1*v2**2*a33**2*c12*n2 + u1*v1*v2*(2*a33**2*m1*n2 + 2*a33*b21*n2**
2) + 4*i*u1*v1*a33*n2**3 + 4*i*u1*v2**3*a33**2*c12*n2 + u1*v2**2*(2*a33**2*m2*n2
- 2*i*a33*b21*n2**2) + 2*i*u1*v2*v3**2*a33**2*c12*n2 + u1*v2*(4*a33*n2**3 - 2*
a33*n2*n3**2) + u1*v3**2*(i*a33**2*m1*n2 - a33**2*m2*n2 + 2*i*a33*b21*n2**2) + 2
*u1*v3*a33*n2**2*n3 - 2*u2**2*v1*a33**2*n2**2 - 2*u2*u3**2*v1*a33**3*n2 - 2*u2*
u3*v1*a33**2*n2*n3 - 2*i*u2*u3*v3*a33**2*n2**2 + u2*v1**2*( - 2*a33**2*m1*n2 - 2
*a33*b21*n2**2) - 2*i*u2*v1*v3**2*a33**2*c12*n2 + 2*u2*v1*a33*n2*n3**2 + u2*v3**
2*(a33**2*m1*n2 + i*a33**2*m2*n2 + 2*a33*b21*n2**2) - 2*i*u2*v3*a33*n2**2*n3 - 2
*i*u3**3*v3*a33**3*n2 + 2*u3**2*v1*a33**2*n2**2 - 2*i*u3**2*v2*a33**2*n2**2 + u3
**2*v3**2*(a33**3*m1 + i*a33**3*m2 + 2*a33**2*b21*n2) - 4*i*u3**2*v3*a33**2*n2*
n3 + 4*u3*v1**2*v3*a33**2*c12*n2 + u3*v1*v3*( - 2*i*a33**2*m1*n2 + 2*a33**2*m2*
n2 - 4*i*a33*b21*n2**2) + 2*u3*v1*a33*n2**2*n3 + 4*u3*v2**2*v3*a33**2*c12*n2 +
u3*v2*v3*( - 2*i*a33**2*m2*n2 - 2*a33*b21*n2**2) - 2*i*u3*v2*a33*n2**2*n3 + 2*u3
*v3**3*a33**2*c12*n2 + u3*v3**2*(a33**2*m1*n3 + i*a33**2*m2*n3 + 2*a33*b21*n2*n3
) + 4*v1**2*v2*a33*c12*n2**2 + v1*v2*v3**2*(2*a33**2*c12*m1 + 2*i*a33**2*c12*m2
+ 4*a33*b21*c12*n2) - 4*i*v1*v2*v3*a33*c12*n2*n3 + v1*v2*( - 2*i*a33*m1*n2**2 +
2*a33*m2*n2**2 - 4*i*b21*n2**3) + v1*v3**2*(a33**2*m1**2 + i*a33**2*m1*m2 + 3*
a33*b21*m1*n2 + i*a33*b21*m2*n2 + 2*i*a33*c12*n2**2 + 2*b21**2*n2**2) + v1*v3*(
- 2*i*a33*m1*n2*n3 - 2*i*b21*n2**2*n3) + v1*(2*n2**4 - 2*n2**2*n3**2) + 4*v2**3*
a33*c12*n2**2 + v2**2*v3**2*(2*i*a33**2*c12*m1 - 2*a33**2*c12*m2 + 4*i*a33*b21*
c12*n2) + 4*v2**2*v3*a33*c12*n2*n3 + v2**2*( - 2*a33*m1*n2**2 - 2*i*a33*m2*n2**2
- 4*b21*n2**3) + v2*v3**2*(a33**2*m1*m2 + i*a33**2*m2**2 - i*a33*b21*m1*n2 + 3*
a33*b21*m2*n2 + 2*a33*c12*n2**2 - 2*i*b21**2*n2**2) + v2*v3*( - 2*i*a33*m2*n2*n3
- 2*b21*n2**2*n3) + v2*( - 2*i*n2**4 + 2*i*n2**2*n3**2) + v3**4*(i*a33**2*c12*
m1 - a33**2*c12*m2 + 2*i*a33*b21*c12*n2) + 2*v3**3*a33*c12*n2*n3 + v3**2*( - a33
*m1*n2**2 - a33*m1*n3**2 + i*a33*m2*n2**2 - i*a33*m2*n3**2 - 2*b21*n2*n3**2) +
v3*( - 2*i*n2**3*n3 + 2*i*n2*n3**3)$