Solution 4 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
2 2
b33*n1 *n3*r434 - b33*n1 *n3*r464
r10=--------------------------------------------
4 3 2 2 3
a22 - 3*a22 *a33 + 3*a22 *a33 - a22*a33
1 3 3 2 3 2
r11=(---*a22 *b33*r14 - ---*a22 *a33*b33*r14 + ---*a22*a33 *b33*r14
2 2 2
1 3 2 2 4 3
- ---*a33 *b33*r14 + b33*n1 *n2*r434 - b33*n1 *n2*r464)/(a22 - 3*a22 *a33
2
2 2 3
+ 3*a22 *a33 - a22*a33 )
1 3 3 2 3 2
r12=(---*a22 *b33*n1*r14 - ---*a22 *a33*b33*n1*r14 + ---*a22*a33 *b33*n1*r14
2 2 2
1 3 3 3 4
- ---*a33 *b33*n1*r14 + b33*n1 *n2*r434 - b33*n1 *n2*r464)/(a22 *n2
2
3 2 2 3
- 3*a22 *a33*n2 + 3*a22 *a33 *n2 - a22*a33 *n2)
n3*r14
r13=--------
n2
n1*r14
r15=--------
n2
r20
1 2 2 1 2 2 1 2 2 1 2 2
---*b33 *n1 *r434 - ---*b33 *n1 *r464 - ---*b33 *n2 *r434 - ---*b33 *n2 *r464
4 4 4 4
=-------------------------------------------------------------------------------
4 3 2 2
a22 - 2*a22 *a33 + a22 *a33
1 2
---*b33 *n2*n3*r464
2
r21=-------------------------------
4 3 2 2
a22 - 2*a22 *a33 + a22 *a33
1 2
---*b33 *n1*n3*r464
2
r23=-------------------------------
4 3 2 2
a22 - 2*a22 *a33 + a22 *a33
1 2 1 2
---*b33 *n1*n2*r434 + ---*b33 *n1*n2*r464
2 2
r24=-------------------------------------------
4 3 2 2
a22 - 2*a22 *a33 + a22 *a33
r25
1 2 2 1 2 2 1 2 2 1 2 2
---*b33 *n1 *r434 + ---*b33 *n1 *r464 - ---*b33 *n2 *r434 - ---*b33 *n2 *r464
4 4 4 4
=-------------------------------------------------------------------------------
4 3 2 2
a22 - 2*a22 *a33 + a22 *a33
2 2 2 2
- b33*n1 *r434 + b33*n1 *r464 - b33*n2 *r434 + b33*n3 *r464
r26=--------------------------------------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
b33*n2*n3*r434
r27=------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
b33*n1*n3*r434
r28=------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
3 2 2 3 2
r29=( - a22 *r14 + 3*a22 *a33*r14 - 3*a22*a33 *r14 + a33 *r14 - n1 *n2*r434
2 2 2 2
+ n1 *n2*r464 + n2*n3 *r434 - n2*n3 *r464)/(a22 *n2 - 2*a22*a33*n2
2
+ a33 *n2)
b33*n2*n3*r464
r210=------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
b33*n1*n2*r434
r212=------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
2*n2*n3*r434 - 2*n2*n3*r464
r213=-----------------------------
2 2
a22 - 2*a22*a33 + a33
2 2 2 2
- n1 *r434 + n1 *r464 + n2 *r434 - n2 *r464
r214=----------------------------------------------
2 2
a22 - 2*a22*a33 + a33
b33*n1*n3*r464
r215=------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
b33*n1*n2*r434
r216=------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
2 2
b33*n1 *r434 - b33*n2 *r434
r217=------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
2*n1*n3*r434 - 2*n1*n3*r464
r218=-----------------------------
2 2
a22 - 2*a22*a33 + a33
2*n1*n2*r434 - 2*n1*n2*r464
r219=-----------------------------
2 2
a22 - 2*a22*a33 + a33
r30=0
1
---*b33*n2*r310
2
r31=-----------------
a22*n3
r32=0
1 4 3 1 2 2
r33=(---*a22 *b33*n2*r310 - a22 *a33*b33*n2*r310 + ---*a22 *a33 *b33*n2*r310
2 2
1 3 1 3 5
- ---*a22*b33 *n2*n3*r464 + ---*a33*b33 *n2*n3*r464)/(a22 *n3
8 8
4 3 2
- 2*a22 *a33*n3 + a22 *a33 *n3)
1
---*b33*n1*r310
2
r34=-----------------
a22*n3
r35=0
1 4 3 1 2 2
r36=(---*a22 *b33*n1*r310 - a22 *a33*b33*n1*r310 + ---*a22 *a33 *b33*n1*r310
2 2
1 3 1 3 5
- ---*a22*b33 *n1*n3*r464 + ---*a33*b33 *n1*n3*r464)/(a22 *n3
8 8
4 3 2
- 2*a22 *a33*n3 + a22 *a33 *n3)
r37=0
1 3 1 2 1 3
---*a22 *b33*n2*r310 - ---*a22 *a33*b33*n2*r310 - ---*b33 *n2*n3*r464
2 2 8
r38=-----------------------------------------------------------------------
4 3
a22 *n3 - a22 *a33*n3
1 3 1 2 1 3
---*a22 *b33*n1*r310 - ---*a22 *a33*b33*n1*r310 - ---*b33 *n1*n3*r464
2 2 8
r39=-----------------------------------------------------------------------
4 3
a22 *n3 - a22 *a33*n3
1 1 2
- ---*a22*b33*n2*r488 - ---*b33 *n2*r464
2 2
r311=-------------------------------------------
3 2
a22 - a22 *a33
3 2 1 2
a22 *r310 - a22 *a33*r310 - ---*b33 *n3*r464
4
r312=----------------------------------------------
3 2
a22 - a22 *a33
1 1 2
- ---*a22*b33*n1*r488 - ---*b33 *n1*r464
2 2
r313=-------------------------------------------
3 2
a22 - a22 *a33
r314=0
3 2 1 2
a22 *r310 - a22 *a33*r310 - ---*b33 *n3*r464
4
r315=----------------------------------------------
3 2
a22 - a22 *a33
- a22*n3*r488 - 2*b33*n3*r464
r316=--------------------------------
2
a22 - a22*a33
1
- b33*n2*r434 - ---*b33*n2*r464
2
r317=----------------------------------
2
a22 - a22*a33
1
- b33*n1*r434 - ---*b33*n1*r464
2
r318=----------------------------------
2
a22 - a22*a33
- 2*n3*r434 + n3*r464
r319=------------------------
a22 - a33
n2*r310
r320=---------
n3
r321=0
r322=
3 2 1 1 2
a22 *n2*r310 - a22 *a33*n2*r310 - ---*a22*b33*n2*n3*r488 - ---*b33 *n2*n3*r464
2 4
--------------------------------------------------------------------------------
3 2
a22 *n3 - a22 *a33*n3
r323=0
1
- ---*b33*n1*r488
2
r324=--------------------
2
a22 - a22*a33
3 2 1 2
a22 *n2*r310 - a22 *a33*n2*r310 - ---*b33 *n2*n3*r464
4
r325=-------------------------------------------------------
3 2
a22 *n3 - a22 *a33*n3
- a22*n2*r488 - b33*n2*r464
r326=------------------------------
2
a22 - a22*a33
- n3*r488
r327=------------
a22 - a33
r328=0
- 2*n2*r434 + n2*r464
r329=------------------------
a22 - a33
r330=0
1
- a22*n2*r488 - ---*b33*n2*r464
2
r331=----------------------------------
2
a22 - a22*a33
1
- ---*b33*n1*r464
2
r332=--------------------
2
a22 - a22*a33
- n3*r464
r333=------------
a22 - a33
- n2*r464
r334=------------
a22 - a33
n1*r310
r335=---------
n3
r336=0
3 2 1 2
a22 *n1*r310 - a22 *a33*n1*r310 - ---*b33 *n1*n3*r464
4
r337=-------------------------------------------------------
3 2
a22 *n3 - a22 *a33*n3
r338=0
1
- ---*b33*n2*r488
2
r339=--------------------
2
a22 - a22*a33
r340=
3 2 1 1 2
a22 *n1*r310 - a22 *a33*n1*r310 - ---*a22*b33*n1*n3*r488 - ---*b33 *n1*n3*r464
2 4
--------------------------------------------------------------------------------
3 2
a22 *n3 - a22 *a33*n3
- a22*n1*r488 - b33*n1*r464
r341=------------------------------
2
a22 - a22*a33
r342=0
- n3*r488
r343=------------
a22 - a33
- 2*n1*r434 + n1*r464
r344=------------------------
a22 - a33
r345=0
- n1*r488
r346=------------
a22 - a33
- n2*r488
r347=------------
a22 - a33
r348=0
- n1*r464
r349=------------
a22 - a33
r350=0
1
- ---*b33*n2*r464
2
r351=--------------------
2
a22 - a22*a33
1
- a22*n1*r488 - ---*b33*n1*r464
2
r352=----------------------------------
2
a22 - a22*a33
- n3*r464
r353=------------
a22 - a33
- n2*r464
r354=------------
a22 - a33
- n1*r464
r355=------------
a22 - a33
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
- a22*r310 + a33*r310
r425=------------------------
n3
r426=0
3 2 1 2
- a22 *r310 + a22 *a33*r310 + ---*b33 *n3*r464
4
r427=-------------------------------------------------
2
a22 *n3
r428=0
r429=0
3 2 1 2
- a22 *r310 + a22 *a33*r310 + ---*b33 *n3*r464
4
r430=-------------------------------------------------
2
a22 *n3
a22*r488 + b33*r464
r431=---------------------
a22
r432=0
r433=0
r435=0
r436=0
r437=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r446=0
r447=0
r448=0
r449=0
r450=0
r451=0
r452=r488
r453=0
r454=0
r455=0
r456=0
r458=0
r459=0
r460=0
r461=0
r462=0
r463=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
r483=0
r484=0
r485=0
r486=0
r487=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=0
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
r4113=0
r4114=r464
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
- b33*n3
m3=-----------
a22 - a33
1 1
- ---*a22*b33*n2 - ---*a33*b33*n2
2 2
m2=------------------------------------
2
a22 - a22*a33
1 1
- ---*a22*b33*n1 - ---*a33*b33*n1
2 2
m1=------------------------------------
2
a22 - a22*a33
1 2
- ---*b33
4
c33=-------------
a22
c23=0
c22=0
c13=0
c12=0
c11=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:
r434, r14, r310, r488, r464, b33, n3, n1, n2, 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.
{a22 + a33,a22,a33,a22 - a33,a11,n2,n3}
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,
c11,
c12,
c13,
c22,
c23,
a22*c33 + 1/4*b33**2,
a22**2*m1 - a22*a33*m1 + 1/2*a22*b33*n1 + 1/2*a33*b33*n1,
a22**2*m2 - a22*a33*m2 + 1/2*a22*b33*n2 + 1/2*a33*b33*n2,
a22*m3 - a33*m3 + b33*n3}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a22 + u1*n1 + u2 *a22 + u2*n2 + u3 *a33 + u3*v3*b33 + u3*n3
1 1
- ---*a22*b33*n1 - ---*a33*b33*n1
2 2
+ v1*------------------------------------
2
a22 - a22*a33
1 1 1 2 2
- ---*a22*b33*n2 - ---*a33*b33*n2 - ---*v3 *b33
2 2 4
+ v2*------------------------------------ + -----------------
2 a22
a22 - a22*a33
- v3*b33*n3
+ --------------
a22 - a33
has apart from the Hamiltonian and Casimirs the following 5 first integrals:
3 5 4 3 2
FI=u1 *( - a22 *n1 + 2*a22 *a33*n1 - a22 *a33 *n1)
2 5 4 3 2
+ u1 *u2*( - a22 *n2 + 2*a22 *a33*n2 - a22 *a33 *n2)
2 2 6 5 4 2 3 3
+ u1 *u3 *(a22 - 3*a22 *a33 + 3*a22 *a33 - a22 *a33 )
2 5 4 3 2
+ u1 *u3*( - a22 *n3 + 2*a22 *a33*n3 - a22 *a33 *n3)
2 1 4 3 1 2 2
+ u1 *v1*( - ---*a22 *b33*n1 + a22 *a33*b33*n1 - ---*a22 *a33 *b33*n1)
2 2
2 1 4 3 1 2 2
+ u1 *v2*( - ---*a22 *b33*n2 + a22 *a33*b33*n2 - ---*a22 *a33 *b33*n2)
2 2
2 5 4 3 2
+ u1*u2 *( - a22 *n1 + 2*a22 *a33*n1 - a22 *a33 *n1)
4 3
+ u1*u2*( - 2*a22 *n1*n2 + 2*a22 *a33*n1*n2)
2 5 4 3 2
+ u1*u3 *(a22 *n1 - 2*a22 *a33*n1 + a22 *a33 *n1)
4 3 2 2
+ u1*u3*v3*( - a22 *b33*n1 + 2*a22 *a33*b33*n1 - a22 *a33 *b33*n1)
4 3
+ u1*u3*( - 2*a22 *n1*n3 + 2*a22 *a33*n1*n3)
2 1 3 2 1 2 2 1 2 2
+ u1*v1 *( - ---*a22 *b33 *n1 + ---*a22 *a33*b33 *n1 - ---*a22*a33 *b33 *n1)
4 2 4
2 1 3 2 1 2 2 1 2 2
+ u1*v2 *( - ---*a22 *b33 *n1 + ---*a22 *a33*b33 *n1 - ---*a22*a33 *b33 *n1)
4 2 4
3 2
+ u1*v3*(a22 *b33*n1*n3 - a22 *a33*b33*n1*n3)
3 5 4 3 2
+ u2 *( - a22 *n2 + 2*a22 *a33*n2 - a22 *a33 *n2)
2 2 6 5 4 2 3 3
+ u2 *u3 *(a22 - 3*a22 *a33 + 3*a22 *a33 - a22 *a33 )
2 5 4 3 2
+ u2 *u3*( - a22 *n3 + 2*a22 *a33*n3 - a22 *a33 *n3)
2 1 4 3 1 2 2
+ u2 *v1*( - ---*a22 *b33*n1 + a22 *a33*b33*n1 - ---*a22 *a33 *b33*n1)
2 2
2 1 4 3 1 2 2
+ u2 *v2*( - ---*a22 *b33*n2 + a22 *a33*b33*n2 - ---*a22 *a33 *b33*n2)
2 2
2 4 2 4 2 3 2 3 2
+ u2 *(a22 *n1 - a22 *n2 - a22 *a33*n1 + a22 *a33*n2 )
2 5 4 3 2
+ u2*u3 *(a22 *n2 - 2*a22 *a33*n2 + a22 *a33 *n2)
4 3 2 2
+ u2*u3*v3*( - a22 *b33*n2 + 2*a22 *a33*b33*n2 - a22 *a33 *b33*n2)
4 3
+ u2*u3*( - 2*a22 *n2*n3 + 2*a22 *a33*n2*n3)
2 1 3 2 1 2 2 1 2 2
+ u2*v1 *( - ---*a22 *b33 *n2 + ---*a22 *a33*b33 *n2 - ---*a22*a33 *b33 *n2)
4 2 4
2 1 3 2 1 2 2 1 2 2
+ u2*v2 *( - ---*a22 *b33 *n2 + ---*a22 *a33*b33 *n2 - ---*a22*a33 *b33 *n2)
4 2 4
3 2
+ u2*v3*(a22 *b33*n2*n3 - a22 *a33*b33*n2*n3)
3 5 4 3 2 2 3
+ u3 *v3*(a22 *b33 - 3*a22 *a33*b33 + 3*a22 *a33 *b33 - a22 *a33 *b33)
3 5 4 3 2 2 2
+ u3 *(a22 *n3 - 2*a22 *a33*n3 + a22 *a33 *n3) + u3 *v1
1 4 2 3 3 2 3 2 2 2 1 3 2
*(---*a22 *b33 - ---*a22 *a33*b33 + ---*a22 *a33 *b33 - ---*a22*a33 *b33 )
4 4 4 4
2 1 4 3 1 2 2 2
+ u3 *v1*( - ---*a22 *b33*n1 + a22 *a33*b33*n1 - ---*a22 *a33 *b33*n1) + u3
2 2
2
*v2
1 4 2 3 3 2 3 2 2 2 1 3 2
*(---*a22 *b33 - ---*a22 *a33*b33 + ---*a22 *a33 *b33 - ---*a22*a33 *b33 )
4 4 4 4
2 1 4 3 1 2 2
+ u3 *v2*( - ---*a22 *b33*n2 + a22 *a33*b33*n2 - ---*a22 *a33 *b33*n2)
2 2
2 4 3 2 2
+ u3 *v3*( - 2*a22 *b33*n3 + 4*a22 *a33*b33*n3 - 2*a22 *a33 *b33*n3)
2 4 2 4 2 3 2 3 2
+ u3 *(a22 *n1 - a22 *n3 - a22 *a33*n1 + a22 *a33*n3 )
2 1 3 2 1 2 2 1 2 2
+ u3*v1 *( - ---*a22 *b33 *n3 + ---*a22 *a33*b33 *n3 - ---*a22*a33 *b33 *n3)
4 2 4
1 3 2 2 2 1 2 2
+ u3*v1*v3*( - ---*a22 *b33 *n1 + a22 *a33*b33 *n1 - ---*a22*a33 *b33 *n1)
2 2
2 1 3 2 1 2 2 1 2 2
+ u3*v2 *( - ---*a22 *b33 *n3 + ---*a22 *a33*b33 *n3 - ---*a22*a33 *b33 *n3)
4 2 4
1 3 2 2 2 1 2 2
+ u3*v2*v3*( - ---*a22 *b33 *n2 + a22 *a33*b33 *n2 - ---*a22*a33 *b33 *n2)
2 2
3 2 3 2 2 2 2 2
+ u3*v3*(a22 *b33*n1 + a22 *b33*n3 - a22 *a33*b33*n1 - a22 *a33*b33*n3 )
3 1 2 3 1 3 1 2 3
+ v1 *( - ---*a22 *b33 *n1 + ---*a22*a33*b33 *n1 - ---*a33 *b33 *n1)
8 4 8
2 1 2 3 1 3 1 2 3
+ v1 *v2*( - ---*a22 *b33 *n2 + ---*a22*a33*b33 *n2 - ---*a33 *b33 *n2) +
8 4 8
2 1 2 2 2 1 2 2 2 1 2 2
v1 *(---*a22 *b33 *n1 - ---*a22 *b33 *n2 - ---*a22*a33*b33 *n1
4 4 4
1 2 2
+ ---*a22*a33*b33 *n2 )
4
2 1 2 3 1 3 1 2 3
+ v1*v2 *( - ---*a22 *b33 *n1 + ---*a22*a33*b33 *n1 - ---*a33 *b33 *n1)
8 4 8
1 2 2 1 2
+ v1*v2*(---*a22 *b33 *n1*n2 - ---*a22*a33*b33 *n1*n2)
2 2
1 2 2 1 2 2 3
+ v1*v3*(---*a22 *b33 *n1*n3 - ---*a22*a33*b33 *n1*n3) - v1*a22 *b33*n1
2 2
3 1 2 3 1 3 1 2 3
+ v2 *( - ---*a22 *b33 *n2 + ---*a22*a33*b33 *n2 - ---*a33 *b33 *n2)
8 4 8
1 2 2 1 2 2 2
+ v2*v3*(---*a22 *b33 *n2*n3 - ---*a22*a33*b33 *n2*n3) - v2*a22 *b33*n1 *n2
2 2
2 1 2 2 2 1 2 2 2 1 2 2
+ v3 *( - ---*a22 *b33 *n1 - ---*a22 *b33 *n2 + ---*a22*a33*b33 *n1
4 4 4
1 2 2 2 2
+ ---*a22*a33*b33 *n2 ) - v3*a22 *b33*n1 *n3
4
which the program can not factorize further.
{HAM,FI} = 0
2 3 2 2
FI=u1 *v1*( - a22 *n1 + 2*a22 *a33*n1 - a22*a33 *n1)
3 2 2
+ u1*u2*v1*( - a22 *n2 + 2*a22 *a33*n2 - a22*a33 *n2)
3 2 2
+ u1*u2*v2*( - a22 *n1 + 2*a22 *a33*n1 - a22*a33 *n1)
2 4 3 2 2 3
+ u1*u3 *v1*(a22 - 3*a22 *a33 + 3*a22 *a33 - a22*a33 )
3 2 2
+ u1*u3*v1*( - a22 *n3 + 2*a22 *a33*n3 - a22*a33 *n3)
3 2 2
+ u1*u3*v3*( - a22 *n1 + 2*a22 *a33*n1 - a22*a33 *n1)
2 1 2 1 2
+ u1*v1 *( - ---*a22 *b33*n1 + a22*a33*b33*n1 - ---*a33 *b33*n1)
2 2
1 2 1 2
+ u1*v1*v2*( - ---*a22 *b33*n2 + a22*a33*b33*n2 - ---*a33 *b33*n2)
2 2
2 3 2 2
+ u2 *v2*( - a22 *n2 + 2*a22 *a33*n2 - a22*a33 *n2)
2 4 3 2 2 3
+ u2*u3 *v2*(a22 - 3*a22 *a33 + 3*a22 *a33 - a22*a33 )
3 2 2
+ u2*u3*v2*( - a22 *n3 + 2*a22 *a33*n3 - a22*a33 *n3)
3 2 2
+ u2*u3*v3*( - a22 *n2 + 2*a22 *a33*n2 - a22*a33 *n2)
1 2 1 2
+ u2*v1*v2*( - ---*a22 *b33*n1 + a22*a33*b33*n1 - ---*a33 *b33*n1)
2 2
2 1 2 1 2
+ u2*v2 *( - ---*a22 *b33*n2 + a22*a33*b33*n2 - ---*a33 *b33*n2)
2 2
3 4 3 2 2 3
+ u3 *v3*(a22 - 3*a22 *a33 + 3*a22 *a33 - a22*a33 )
2 3 2 2
+ u3 *v3*( - a22 *n3 + 2*a22 *a33*n3 - a22*a33 *n3)
1 2 1 2
+ u3*v1*v3*( - ---*a22 *b33*n1 + a22*a33*b33*n1 - ---*a33 *b33*n1)
2 2
1 2 1 2
+ u3*v2*v3*( - ---*a22 *b33*n2 + a22*a33*b33*n2 - ---*a33 *b33*n2)
2 2
= a product of the elements of: {a22 - a33,
a22 - a33,
u1*v1 + u2*v2 + u3*v3,
2 2 1
- u1*a22*n1 - u2*a22*n2 + u3 *(a22 - a22*a33) - u3*a22*n3 - ---*v1*b33*n1
2
1
- ---*v2*b33*n2}
2
{HAM,FI} = 0
2 2 2 2 2
FI=u1*v1 *a22*n1 + u1*v2 *a22*n1 + u1*v3 *a22*n1 + u2*v1 *a22*n2 + u2*v2 *a22*n2
2 2 2 2 2 2 2
+ u2*v3 *a22*n2 + u3 *v1 *( - a22 + a22*a33) + u3 *v2 *( - a22 + a22*a33)
2 2 2 2 2
+ u3 *v3 *( - a22 + a22*a33) + u3*v1 *a22*n3 + u3*v2 *a22*n3
2 1 3 1 2 1 2
+ u3*v3 *a22*n3 + ---*v1 *b33*n1 + ---*v1 *v2*b33*n2 + ---*v1*v2 *b33*n1
2 2 2
1 2 1 3 1 2
+ ---*v1*v3 *b33*n1 + ---*v2 *b33*n2 + ---*v2*v3 *b33*n2
2 2 2
2 2 2
= a product of the elements of: { - v1 - v2 - v3 ,
2 2 1
- u1*a22*n1 - u2*a22*n2 + u3 *(a22 - a22*a33) - u3*a22*n3 - ---*v1*b33*n1
2
1
- ---*v2*b33*n2}
2
{HAM,FI} = 0
2 2 1
FI=u1*a22*n1 + u2*a22*n2 + u3 *( - a22 + a22*a33) + u3*a22*n3 + ---*v1*b33*n1
2
1
+ ---*v2*b33*n2
2
which the program can not factorize further.
{HAM,FI} = 0
3 2
FI=u1*u2*(2*a22 *n1*n2 - 2*a22 *a33*n1*n2)
2 4 3 2 2
+ u1*u3 *( - 2*a22 *n1 + 4*a22 *a33*n1 - 2*a22 *a33 *n1)
3 2
+ u1*u3*(2*a22 *n1*n3 - 2*a22 *a33*n1*n3)
2 2 2 2 2 2
+ u1*v1*(a22 *b33*n1 - a22 *b33*n2 - a22*a33*b33*n1 + a22*a33*b33*n2 )
2
+ u1*v2*(a22 *b33*n1*n2 - a22*a33*b33*n1*n2)
2 3 2 3 2 2 2 2 2
+ u2 *( - a22 *n1 + a22 *n2 + a22 *a33*n1 - a22 *a33*n2 )
2 4 3 2 2
+ u2*u3 *( - 2*a22 *n2 + 4*a22 *a33*n2 - 2*a22 *a33 *n2)
3 2
+ u2*u3*(2*a22 *n2*n3 - 2*a22 *a33*n2*n3)
2
+ u2*v1*(a22 *b33*n1*n2 - a22*a33*b33*n1*n2)
4 5 4 3 2 2 3
+ u3 *(a22 - 3*a22 *a33 + 3*a22 *a33 - a22 *a33 )
3 4 3 2 2
+ u3 *( - 2*a22 *n3 + 4*a22 *a33*n3 - 2*a22 *a33 *n3)
2 3 2 2
+ u3 *v1*( - a22 *b33*n1 + 2*a22 *a33*b33*n1 - a22*a33 *b33*n1)
2 3 2 2
+ u3 *v2*( - a22 *b33*n2 + 2*a22 *a33*b33*n2 - a22*a33 *b33*n2)
2 3 2 3 2 2 2 2 2
+ u3 *( - a22 *n1 + a22 *n3 + a22 *a33*n1 - a22 *a33*n3 )
2
+ u3*v1*(a22 *b33*n1*n3 - a22*a33*b33*n1*n3)
2
+ u3*v2*(a22 *b33*n2*n3 - a22*a33*b33*n2*n3)
2 2 2 2 2 2
+ u3*v3*( - a22 *b33*n1 - a22 *b33*n2 + a22*a33*b33*n1 + a22*a33*b33*n2 )
2
+ v1
1 2 2 1 2 2 1 2 2 1 2 2
*(---*a22*b33 *n1 - ---*a22*b33 *n2 - ---*a33*b33 *n1 + ---*a33*b33 *n2 )
4 4 4 4
1 2 1 2 3
+ v1*v2*(---*a22*b33 *n1*n2 - ---*a33*b33 *n1*n2) + v1*a22*b33*n1
2 2
2 2
+ v2*a22*b33*n1 *n2 + v3
1 2 2 1 2 2 1 2 2 1 2 2
*(---*a22*b33 *n1 - ---*a22*b33 *n2 - ---*a33*b33 *n1 + ---*a33*b33 *n2 )
4 4 4 4
2
+ v3*a22*b33*n1 *n3
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a22 + u1*n1 + u2**2*a22 + u2*n2 + u3**2*a33 + u3*v3*b33 + u3*n3 + v1*(
- 1/2*a22*b33*n1 - 1/2*a33*b33*n1)/(a22**2 - a22*a33) + v2*( - 1/2*a22*b33*n2 -
1/2*a33*b33*n2)/(a22**2 - a22*a33) + ( - 1/4*v3**2*b33**2)/a22 + ( - v3*b33*n3)
/(a22 - a33)$
FI=u1**3*( - a22**5*n1 + 2*a22**4*a33*n1 - a22**3*a33**2*n1) + u1**2*u2*( - a22
**5*n2 + 2*a22**4*a33*n2 - a22**3*a33**2*n2) + u1**2*u3**2*(a22**6 - 3*a22**5*
a33 + 3*a22**4*a33**2 - a22**3*a33**3) + u1**2*u3*( - a22**5*n3 + 2*a22**4*a33*
n3 - a22**3*a33**2*n3) + u1**2*v1*( - 1/2*a22**4*b33*n1 + a22**3*a33*b33*n1 - 1/
2*a22**2*a33**2*b33*n1) + u1**2*v2*( - 1/2*a22**4*b33*n2 + a22**3*a33*b33*n2 - 1
/2*a22**2*a33**2*b33*n2) + u1*u2**2*( - a22**5*n1 + 2*a22**4*a33*n1 - a22**3*a33
**2*n1) + u1*u2*( - 2*a22**4*n1*n2 + 2*a22**3*a33*n1*n2) + u1*u3**2*(a22**5*n1 -
2*a22**4*a33*n1 + a22**3*a33**2*n1) + u1*u3*v3*( - a22**4*b33*n1 + 2*a22**3*a33
*b33*n1 - a22**2*a33**2*b33*n1) + u1*u3*( - 2*a22**4*n1*n3 + 2*a22**3*a33*n1*n3)
+ u1*v1**2*( - 1/4*a22**3*b33**2*n1 + 1/2*a22**2*a33*b33**2*n1 - 1/4*a22*a33**2
*b33**2*n1) + u1*v2**2*( - 1/4*a22**3*b33**2*n1 + 1/2*a22**2*a33*b33**2*n1 - 1/4
*a22*a33**2*b33**2*n1) + u1*v3*(a22**3*b33*n1*n3 - a22**2*a33*b33*n1*n3) + u2**3
*( - a22**5*n2 + 2*a22**4*a33*n2 - a22**3*a33**2*n2) + u2**2*u3**2*(a22**6 - 3*
a22**5*a33 + 3*a22**4*a33**2 - a22**3*a33**3) + u2**2*u3*( - a22**5*n3 + 2*a22**
4*a33*n3 - a22**3*a33**2*n3) + u2**2*v1*( - 1/2*a22**4*b33*n1 + a22**3*a33*b33*
n1 - 1/2*a22**2*a33**2*b33*n1) + u2**2*v2*( - 1/2*a22**4*b33*n2 + a22**3*a33*b33
*n2 - 1/2*a22**2*a33**2*b33*n2) + u2**2*(a22**4*n1**2 - a22**4*n2**2 - a22**3*
a33*n1**2 + a22**3*a33*n2**2) + u2*u3**2*(a22**5*n2 - 2*a22**4*a33*n2 + a22**3*
a33**2*n2) + u2*u3*v3*( - a22**4*b33*n2 + 2*a22**3*a33*b33*n2 - a22**2*a33**2*
b33*n2) + u2*u3*( - 2*a22**4*n2*n3 + 2*a22**3*a33*n2*n3) + u2*v1**2*( - 1/4*a22
**3*b33**2*n2 + 1/2*a22**2*a33*b33**2*n2 - 1/4*a22*a33**2*b33**2*n2) + u2*v2**2*
( - 1/4*a22**3*b33**2*n2 + 1/2*a22**2*a33*b33**2*n2 - 1/4*a22*a33**2*b33**2*n2)
+ u2*v3*(a22**3*b33*n2*n3 - a22**2*a33*b33*n2*n3) + u3**3*v3*(a22**5*b33 - 3*a22
**4*a33*b33 + 3*a22**3*a33**2*b33 - a22**2*a33**3*b33) + u3**3*(a22**5*n3 - 2*
a22**4*a33*n3 + a22**3*a33**2*n3) + u3**2*v1**2*(1/4*a22**4*b33**2 - 3/4*a22**3*
a33*b33**2 + 3/4*a22**2*a33**2*b33**2 - 1/4*a22*a33**3*b33**2) + u3**2*v1*( - 1/
2*a22**4*b33*n1 + a22**3*a33*b33*n1 - 1/2*a22**2*a33**2*b33*n1) + u3**2*v2**2*(1
/4*a22**4*b33**2 - 3/4*a22**3*a33*b33**2 + 3/4*a22**2*a33**2*b33**2 - 1/4*a22*
a33**3*b33**2) + u3**2*v2*( - 1/2*a22**4*b33*n2 + a22**3*a33*b33*n2 - 1/2*a22**2
*a33**2*b33*n2) + u3**2*v3*( - 2*a22**4*b33*n3 + 4*a22**3*a33*b33*n3 - 2*a22**2*
a33**2*b33*n3) + u3**2*(a22**4*n1**2 - a22**4*n3**2 - a22**3*a33*n1**2 + a22**3*
a33*n3**2) + u3*v1**2*( - 1/4*a22**3*b33**2*n3 + 1/2*a22**2*a33*b33**2*n3 - 1/4*
a22*a33**2*b33**2*n3) + u3*v1*v3*( - 1/2*a22**3*b33**2*n1 + a22**2*a33*b33**2*n1
- 1/2*a22*a33**2*b33**2*n1) + u3*v2**2*( - 1/4*a22**3*b33**2*n3 + 1/2*a22**2*
a33*b33**2*n3 - 1/4*a22*a33**2*b33**2*n3) + u3*v2*v3*( - 1/2*a22**3*b33**2*n2 +
a22**2*a33*b33**2*n2 - 1/2*a22*a33**2*b33**2*n2) + u3*v3*(a22**3*b33*n1**2 + a22
**3*b33*n3**2 - a22**2*a33*b33*n1**2 - a22**2*a33*b33*n3**2) + v1**3*( - 1/8*a22
**2*b33**3*n1 + 1/4*a22*a33*b33**3*n1 - 1/8*a33**2*b33**3*n1) + v1**2*v2*( - 1/8
*a22**2*b33**3*n2 + 1/4*a22*a33*b33**3*n2 - 1/8*a33**2*b33**3*n2) + v1**2*(1/4*
a22**2*b33**2*n1**2 - 1/4*a22**2*b33**2*n2**2 - 1/4*a22*a33*b33**2*n1**2 + 1/4*
a22*a33*b33**2*n2**2) + v1*v2**2*( - 1/8*a22**2*b33**3*n1 + 1/4*a22*a33*b33**3*
n1 - 1/8*a33**2*b33**3*n1) + v1*v2*(1/2*a22**2*b33**2*n1*n2 - 1/2*a22*a33*b33**2
*n1*n2) + v1*v3*(1/2*a22**2*b33**2*n1*n3 - 1/2*a22*a33*b33**2*n1*n3) - v1*a22**2
*b33*n1**3 + v2**3*( - 1/8*a22**2*b33**3*n2 + 1/4*a22*a33*b33**3*n2 - 1/8*a33**2
*b33**3*n2) + v2*v3*(1/2*a22**2*b33**2*n2*n3 - 1/2*a22*a33*b33**2*n2*n3) - v2*
a22**2*b33*n1**2*n2 + v3**2*( - 1/4*a22**2*b33**2*n1**2 - 1/4*a22**2*b33**2*n2**
2 + 1/4*a22*a33*b33**2*n1**2 + 1/4*a22*a33*b33**2*n2**2) - v3*a22**2*b33*n1**2*
n3$
FI=u1**2*v1*( - a22**3*n1 + 2*a22**2*a33*n1 - a22*a33**2*n1) + u1*u2*v1*( - a22
**3*n2 + 2*a22**2*a33*n2 - a22*a33**2*n2) + u1*u2*v2*( - a22**3*n1 + 2*a22**2*
a33*n1 - a22*a33**2*n1) + u1*u3**2*v1*(a22**4 - 3*a22**3*a33 + 3*a22**2*a33**2 -
a22*a33**3) + u1*u3*v1*( - a22**3*n3 + 2*a22**2*a33*n3 - a22*a33**2*n3) + u1*u3
*v3*( - a22**3*n1 + 2*a22**2*a33*n1 - a22*a33**2*n1) + u1*v1**2*( - 1/2*a22**2*
b33*n1 + a22*a33*b33*n1 - 1/2*a33**2*b33*n1) + u1*v1*v2*( - 1/2*a22**2*b33*n2 +
a22*a33*b33*n2 - 1/2*a33**2*b33*n2) + u2**2*v2*( - a22**3*n2 + 2*a22**2*a33*n2 -
a22*a33**2*n2) + u2*u3**2*v2*(a22**4 - 3*a22**3*a33 + 3*a22**2*a33**2 - a22*a33
**3) + u2*u3*v2*( - a22**3*n3 + 2*a22**2*a33*n3 - a22*a33**2*n3) + u2*u3*v3*( -
a22**3*n2 + 2*a22**2*a33*n2 - a22*a33**2*n2) + u2*v1*v2*( - 1/2*a22**2*b33*n1 +
a22*a33*b33*n1 - 1/2*a33**2*b33*n1) + u2*v2**2*( - 1/2*a22**2*b33*n2 + a22*a33*
b33*n2 - 1/2*a33**2*b33*n2) + u3**3*v3*(a22**4 - 3*a22**3*a33 + 3*a22**2*a33**2
- a22*a33**3) + u3**2*v3*( - a22**3*n3 + 2*a22**2*a33*n3 - a22*a33**2*n3) + u3*
v1*v3*( - 1/2*a22**2*b33*n1 + a22*a33*b33*n1 - 1/2*a33**2*b33*n1) + u3*v2*v3*( -
1/2*a22**2*b33*n2 + a22*a33*b33*n2 - 1/2*a33**2*b33*n2)$
FI=u1*v1**2*a22*n1 + u1*v2**2*a22*n1 + u1*v3**2*a22*n1 + u2*v1**2*a22*n2 + u2*v2
**2*a22*n2 + u2*v3**2*a22*n2 + u3**2*v1**2*( - a22**2 + a22*a33) + u3**2*v2**2*(
- a22**2 + a22*a33) + u3**2*v3**2*( - a22**2 + a22*a33) + u3*v1**2*a22*n3 + u3*
v2**2*a22*n3 + u3*v3**2*a22*n3 + 1/2*v1**3*b33*n1 + 1/2*v1**2*v2*b33*n2 + 1/2*v1
*v2**2*b33*n1 + 1/2*v1*v3**2*b33*n1 + 1/2*v2**3*b33*n2 + 1/2*v2*v3**2*b33*n2$
FI=u1*a22*n1 + u2*a22*n2 + u3**2*( - a22**2 + a22*a33) + u3*a22*n3 + 1/2*v1*b33*
n1 + 1/2*v2*b33*n2$
FI=u1*u2*(2*a22**3*n1*n2 - 2*a22**2*a33*n1*n2) + u1*u3**2*( - 2*a22**4*n1 + 4*
a22**3*a33*n1 - 2*a22**2*a33**2*n1) + u1*u3*(2*a22**3*n1*n3 - 2*a22**2*a33*n1*n3
) + u1*v1*(a22**2*b33*n1**2 - a22**2*b33*n2**2 - a22*a33*b33*n1**2 + a22*a33*b33
*n2**2) + u1*v2*(a22**2*b33*n1*n2 - a22*a33*b33*n1*n2) + u2**2*( - a22**3*n1**2
+ a22**3*n2**2 + a22**2*a33*n1**2 - a22**2*a33*n2**2) + u2*u3**2*( - 2*a22**4*n2
+ 4*a22**3*a33*n2 - 2*a22**2*a33**2*n2) + u2*u3*(2*a22**3*n2*n3 - 2*a22**2*a33*
n2*n3) + u2*v1*(a22**2*b33*n1*n2 - a22*a33*b33*n1*n2) + u3**4*(a22**5 - 3*a22**4
*a33 + 3*a22**3*a33**2 - a22**2*a33**3) + u3**3*( - 2*a22**4*n3 + 4*a22**3*a33*
n3 - 2*a22**2*a33**2*n3) + u3**2*v1*( - a22**3*b33*n1 + 2*a22**2*a33*b33*n1 -
a22*a33**2*b33*n1) + u3**2*v2*( - a22**3*b33*n2 + 2*a22**2*a33*b33*n2 - a22*a33
**2*b33*n2) + u3**2*( - a22**3*n1**2 + a22**3*n3**2 + a22**2*a33*n1**2 - a22**2*
a33*n3**2) + u3*v1*(a22**2*b33*n1*n3 - a22*a33*b33*n1*n3) + u3*v2*(a22**2*b33*n2
*n3 - a22*a33*b33*n2*n3) + u3*v3*( - a22**2*b33*n1**2 - a22**2*b33*n2**2 + a22*
a33*b33*n1**2 + a22*a33*b33*n2**2) + v1**2*(1/4*a22*b33**2*n1**2 - 1/4*a22*b33**
2*n2**2 - 1/4*a33*b33**2*n1**2 + 1/4*a33*b33**2*n2**2) + v1*v2*(1/2*a22*b33**2*
n1*n2 - 1/2*a33*b33**2*n1*n2) + v1*a22*b33*n1**3 + v2*a22*b33*n1**2*n2 + v3**2*(
1/4*a22*b33**2*n1**2 - 1/4*a22*b33**2*n2**2 - 1/4*a33*b33**2*n1**2 + 1/4*a33*b33
**2*n2**2) + v3*a22*b33*n1**2*n3$