Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
2 2
- a22*b33*n1 *n3*r14 - b33*n1 *n2*n3*r29
r10=-------------------------------------------
2 2 2 2
a22 *n1 *n2 - a22 *n2*n3
1 2 1 2 2
- ---*a22*b33*n1 *r14 - ---*a22*b33*n3 *r14 - b33*n1 *n2*r29
2 2
r11=---------------------------------------------------------------
2 2 2 2
a22 *n1 - a22 *n3
1 3 1 2 3
- ---*a22*b33*n1 *r14 - ---*a22*b33*n1*n3 *r14 - b33*n1 *n2*r29
2 2
r12=------------------------------------------------------------------
2 2 2 2
a22 *n1 *n2 - a22 *n2*n3
n3*r14
r13=--------
n2
n1*r14
r15=--------
n2
1 2 3 1 2 2
r20=( - ---*a22*b33 *n1 *r14 - ---*a22*b33 *n1*n2 *r14
4 4
1 2 2 1 2 2
- ---*a22*b33*n1 *n2 *r212 + ---*a22*b33*n2 *n3 *r212
2 2
1 2 3 1 2 3 2 3 2 2
- ---*b33 *n1 *n2*r29 - ---*b33 *n1*n2 *r29)/(a22 *n1 *n2 - a22 *n1*n2*n3
4 4
)
1 2 1 2 1 3
r21=(---*a22*b33 *n1*n3*r14 + ---*a22*b33*n1 *n3*r212 - ---*a22*b33*n3 *r212
2 2 2
1 2 2 3 2 2
+ ---*b33 *n1*n2*n3*r29)/(a22 *n1 - a22 *n1*n3 )
2
1 2 1 2 1 3
r23=(---*a22*b33 *n1*n3*r14 + ---*a22*b33*n1 *n3*r212 - ---*a22*b33*n3 *r212
2 2 2
1 2 2 2 2 2
+ ---*b33 *n1*n2*n3*r29)/(a22 *n1 *n2 - a22 *n2*n3 )
2
r24=
1 2 2 2 1 2
---*a22*b33 *n1*r14 + a22*b33*n1 *r212 - a22*b33*n3 *r212 + ---*b33 *n1*n2*r29
2 2
--------------------------------------------------------------------------------
2 2 2 2
a22 *n1 - a22 *n3
1 2 3 1 2 2 1 4
r25=(---*a22*b33 *n1 *r14 - ---*a22*b33 *n1*n2 *r14 + ---*a22*b33*n1 *r212
4 4 2
1 2 2 1 2 2
- ---*a22*b33*n1 *n2 *r212 - ---*a22*b33*n1 *n3 *r212
2 2
1 2 2 1 2 3 1 2 3
+ ---*a22*b33*n2 *n3 *r212 + ---*b33 *n1 *n2*r29 - ---*b33 *n1*n2 *r29)/(
2 4 4
2 3 2 2
a22 *n1 *n2 - a22 *n1*n2*n3 )
3 2 2 2 2 2
r26=(a22*b33*n1 *r14 + a22*b33*n1*n3 *r14 - a22*n1 *n2 *r212 + a22*n1 *n3 *r212
2 2 4 3 2
+ a22*n2 *n3 *r212 - a22*n3 *r212 + b33*n1 *n2*r29 + b33*n1*n2*n3 *r29)/(
3 2
a22*n1 *n2 - a22*n1*n2*n3 )
n3*r212
r27=---------
n1
n3*r212
r28=---------
n2
2 3
a22*b33*n1*n3*r14 + a22*n1 *n3*r212 - a22*n3 *r212 + b33*n1*n2*n3*r29
r210=-----------------------------------------------------------------------
3 2
a22*n1 - a22*n1*n3
- 2*a22*n3*r14 - 2*n2*n3*r29
r213=-------------------------------
2 2
n1 - n3
2 2 2 3
a22*n1 *r14 - a22*n2 *r14 + n1 *n2*r29 - n2 *r29
r214=--------------------------------------------------
2 2
n1 *n2 - n2*n3
2 3
a22*b33*n1*n3*r14 + a22*n1 *n3*r212 - a22*n3 *r212 + b33*n1*n2*n3*r29
r215=-----------------------------------------------------------------------
2 2
a22*n1 *n2 - a22*n2*n3
r216=r212
2 2
n1 *r212 - n2 *r212
r217=---------------------
n1*n2
- 2*a22*n1*n3*r14 - 2*n1*n2*n3*r29
r218=-------------------------------------
2 2
n1 *n2 - n2*n3
- 2*a22*n1*r14 - 2*n1*n2*r29
r219=-------------------------------
2 2
n1 - n3
r30=0
1
- ---*b33*n2*r425
2
r31=--------------------
2
a22
r32=0
1 3 1 2 2 1 2 2
r33=( - ---*a22*b33 *n1*r14 - ---*a22*b33 *n1 *r212 + ---*a22*b33 *n3 *r212
8 8 8
1 3 1 3 1 2
- ---*b33 *n1*n2*r29 - ---*b33*n1 *n2*r425 + ---*b33*n1*n2*n3 *r425)/(
8 2 2
2 3 2 2
a22 *n1 - a22 *n1*n3 )
1
- ---*b33*n1*r425
2
r34=--------------------
2
a22
r35=0
1 3 1 2 2 1 2 2
r36=( - ---*a22*b33 *n1*r14 - ---*a22*b33 *n1 *r212 + ---*a22*b33 *n3 *r212
8 8 8
1 3 1 3 1 2
- ---*b33 *n1*n2*r29 - ---*b33*n1 *n2*r425 + ---*b33*n1*n2*n3 *r425)/(
8 2 2
2 2 2 2
a22 *n1 *n2 - a22 *n2*n3 )
r37=0
1 3 1 2 2 1 2 2
r38=( - ---*a22*b33 *n1*r14 - ---*a22*b33 *n1 *r212 + ---*a22*b33 *n3 *r212
8 8 8
1 3 1 3 1 2
- ---*b33 *n1*n2*r29 - ---*b33*n1 *n2*r425 + ---*b33*n1*n2*n3 *r425)/(
8 2 2
2 3 2 2
a22 *n1 - a22 *n1*n3 )
1 3 1 2 2 1 2 2
r39=( - ---*a22*b33 *n1*r14 - ---*a22*b33 *n1 *r212 + ---*a22*b33 *n3 *r212
8 8 8
1 3 1 3 1 2
- ---*b33 *n1*n2*r29 - ---*b33*n1 *n2*r425 + ---*b33*n1*n2*n3 *r425)/(
8 2 2
2 2 2 2
a22 *n1 *n2 - a22 *n2*n3 )
- n3*r425
r310=------------
a22
1 2 1 2 1 2
r311=( - ---*a22*b33 *n1*r14 - ---*a22*b33*n1 *r212 + ---*a22*b33*n3 *r212
4 4 4
3 2 1 2 3
+ a22*n1 *r322 - a22*n1*n3 *r322 - ---*b33 *n1*n2*r29 + n1 *n2*r425
4
2 3 2
- n1*n2*n3 *r425)/(a22*n1 - a22*n1*n3 )
1 2 1 2 1 3
r312=( - ---*a22*b33 *n1*n3*r14 - ---*a22*b33*n1 *n3*r212 + ---*a22*b33*n3 *r212
4 4 4
1 2 3 3 3
- ---*b33 *n1*n2*n3*r29 - n1 *n2*n3*r425 + n1*n2*n3 *r425)/(a22*n1 *n2
4
2
- a22*n1*n2*n3 )
1 2 1 2 1 2
r313=( - ---*a22*b33 *n1*r14 - ---*a22*b33*n1 *r212 + ---*a22*b33*n3 *r212
4 4 4
3 2 1 2 3
+ a22*n1 *r322 - a22*n1*n3 *r322 - ---*b33 *n1*n2*r29 + n1 *n2*r425
4
2 2 2
- n1*n2*n3 *r425)/(a22*n1 *n2 - a22*n2*n3 )
r314=0
1 2 1 2 1 3
r315=( - ---*a22*b33 *n1*n3*r14 - ---*a22*b33*n1 *n3*r212 + ---*a22*b33*n3 *r212
4 4 4
1 2 3 3 3
- ---*b33 *n1*n2*n3*r29 - n1 *n2*n3*r425 + n1*n2*n3 *r425)/(a22*n1 *n2
4
2
- a22*n1*n2*n3 )
3 2 3 2 3 3
r316=( - ---*a22*b33 *n1*n3*r14 - ---*a22*b33*n1 *n3*r212 + ---*a22*b33*n3 *r212
2 2 2
3 3 3 2
+ 2*a22*n1 *n3*r322 - 2*a22*n1*n3 *r322 - ---*b33 *n1*n2*n3*r29
2
3 3 3 2
+ 2*n1 *n2*n3*r425 - 2*n1*n2*n3 *r425)/(b33*n1 *n2 - b33*n1*n2*n3 )
1 3 2 3 2
r317=( - ---*a22*b33*n1*r14 - ---*a22*n1 *r212 + ---*a22*n3 *r212
2 2 2
1 3 2
- ---*b33*n1*n2*r29)/(n1 - n1*n3 )
2
1 3 2 3 2
r318=( - ---*a22*b33*n1*r14 - ---*a22*n1 *r212 + ---*a22*n3 *r212
2 2 2
1 2 2
- ---*b33*n1*n2*r29)/(n1 *n2 - n2*n3 )
2
r319
2 2 2 2 3
a22 *b33*n1*n3*r14 - a22 *n1 *n3*r212 + a22 *n3 *r212 + a22*b33*n1*n2*n3*r29
=------------------------------------------------------------------------------
3 2
b33*n1 *n2 - b33*n1*n2*n3
- n2*r425
r320=------------
a22
r321=0
r323=0
1 2 1 2 1 2
r324=(---*a22*b33 *n1*r14 + ---*a22*b33*n1 *r212 - ---*a22*b33*n3 *r212
4 4 4
3 2 1 2 3
+ a22*n1 *r322 - a22*n1*n3 *r322 + ---*b33 *n1*n2*r29 + n1 *n2*r425
4
2 2 2
- n1*n2*n3 *r425)/(a22*n1 *n2 - a22*n2*n3 )
1 2 1 2 1 2
r325=( - ---*a22*b33 *n1*r14 - ---*a22*b33*n1 *r212 + ---*a22*b33*n3 *r212
4 4 4
1 2 3 2 3
- ---*b33 *n1*n2*r29 - n1 *n2*r425 + n1*n2*n3 *r425)/(a22*n1
4
2
- a22*n1*n3 )
1 2 1 2 1 2
r326=( - ---*a22*b33 *n1*r14 - ---*a22*b33*n1 *r212 + ---*a22*b33*n3 *r212
2 2 2
3 2 1 2 3
+ 2*a22*n1 *r322 - 2*a22*n1*n3 *r322 - ---*b33 *n1*n2*r29 + 2*n1 *n2*r425
2
2 3 2
- 2*n1*n2*n3 *r425)/(b33*n1 - b33*n1*n3 )
1 2 1 2 1 3
r327=(---*a22*b33 *n1*n3*r14 + ---*a22*b33*n1 *n3*r212 - ---*a22*b33*n3 *r212
2 2 2
3 3 1 2
+ 2*a22*n1 *n3*r322 - 2*a22*n1*n3 *r322 + ---*b33 *n1*n2*n3*r29
2
3 3 3 2
+ 2*n1 *n2*n3*r425 - 2*n1*n2*n3 *r425)/(b33*n1 *n2 - b33*n1*n2*n3 )
r328=0
2 2 2 2 2
a22 *b33*n1*r14 - a22 *n1 *r212 + a22 *n3 *r212 + a22*b33*n1*n2*r29
r329=---------------------------------------------------------------------
3 2
b33*n1 - b33*n1*n3
r330=0
2*a22*r322 + 2*n2*r425
r331=------------------------
b33
1 1 2 1 2
r332=( - ---*a22*b33*n1*r14 - ---*a22*n1 *r212 + ---*a22*n3 *r212
2 2 2
1 2 2
- ---*b33*n1*n2*r29)/(n1 *n2 - n2*n3 )
2
2 2 2 2 3
r333=( - a22 *b33*n1*n3*r14 - a22 *n1 *n3*r212 + a22 *n3 *r212
3 2
- a22*b33*n1*n2*n3*r29)/(b33*n1 *n2 - b33*n1*n2*n3 )
2 2 2 2 2
- a22 *b33*n1*r14 - a22 *n1 *r212 + a22 *n3 *r212 - a22*b33*n1*n2*r29
r334=------------------------------------------------------------------------
3 2
b33*n1 - b33*n1*n3
- n1*r425
r335=------------
a22
r336=0
1 2 1 2 1 2
r337=( - ---*a22*b33 *n1*r14 - ---*a22*b33*n1 *r212 + ---*a22*b33*n3 *r212
4 4 4
1 2 3 2 2
- ---*b33 *n1*n2*r29 - n1 *n2*r425 + n1*n2*n3 *r425)/(a22*n1 *n2
4
2
- a22*n2*n3 )
r338=0
1 2 1 2 1 2
r339=(---*a22*b33 *n1*r14 + ---*a22*b33*n1 *r212 - ---*a22*b33*n3 *r212
4 4 4
3 2 1 2 3
+ a22*n1 *r322 - a22*n1*n3 *r322 + ---*b33 *n1*n2*r29 + n1 *n2*r425
4
2 3 2
- n1*n2*n3 *r425)/(a22*n1 - a22*n1*n3 )
n1*r322
r340=---------
n2
1 2 1 2 1 2
r341=( - ---*a22*b33 *n1*r14 - ---*a22*b33*n1 *r212 + ---*a22*b33*n3 *r212
2 2 2
3 2 1 2 3
+ 2*a22*n1 *r322 - 2*a22*n1*n3 *r322 - ---*b33 *n1*n2*r29 + 2*n1 *n2*r425
2
2 2 2
- 2*n1*n2*n3 *r425)/(b33*n1 *n2 - b33*n2*n3 )
r342=0
1 2 1 2 1 3
r343=(---*a22*b33 *n1*n3*r14 + ---*a22*b33*n1 *n3*r212 - ---*a22*b33*n3 *r212
2 2 2
3 3 1 2
+ 2*a22*n1 *n3*r322 - 2*a22*n1*n3 *r322 + ---*b33 *n1*n2*n3*r29
2
3 3 3 2
+ 2*n1 *n2*n3*r425 - 2*n1*n2*n3 *r425)/(b33*n1 *n2 - b33*n1*n2*n3 )
2 2 2 2 2
a22 *b33*n1*r14 - a22 *n1 *r212 + a22 *n3 *r212 + a22*b33*n1*n2*r29
r344=---------------------------------------------------------------------
2 2
b33*n1 *n2 - b33*n2*n3
r345=0
1 2 1 2 1 2
r346=(---*a22*b33 *n1*r14 + ---*a22*b33*n1 *r212 - ---*a22*b33*n3 *r212
2 2 2
3 2 1 2 3
+ 2*a22*n1 *r322 - 2*a22*n1*n3 *r322 + ---*b33 *n1*n2*r29 + 2*n1 *n2*r425
2
2 2 2
- 2*n1*n2*n3 *r425)/(b33*n1 *n2 - b33*n2*n3 )
1 2 1 2 1 2
r347=(---*a22*b33 *n1*r14 + ---*a22*b33*n1 *r212 - ---*a22*b33*n3 *r212
2 2 2
3 2 1 2 3
+ 2*a22*n1 *r322 - 2*a22*n1*n3 *r322 + ---*b33 *n1*n2*r29 + 2*n1 *n2*r425
2
2 3 2
- 2*n1*n2*n3 *r425)/(b33*n1 - b33*n1*n3 )
r348=0
2 2 2 2 2
- a22 *b33*n1*r14 - a22 *n1 *r212 + a22 *n3 *r212 - a22*b33*n1*n2*r29
r349=------------------------------------------------------------------------
2 2
b33*n1 *n2 - b33*n2*n3
r350=0
1 1 2 1 2
r351=( - ---*a22*b33*n1*r14 - ---*a22*n1 *r212 + ---*a22*n3 *r212
2 2 2
1 3 2
- ---*b33*n1*n2*r29)/(n1 - n1*n3 )
2
2*a22*n1*r322 + 2*n1*n2*r425
r352=------------------------------
b33*n2
2 2 2 2 3
r353=( - a22 *b33*n1*n3*r14 - a22 *n1 *n3*r212 + a22 *n3 *r212
3 2
- a22*b33*n1*n2*n3*r29)/(b33*n1 *n2 - b33*n1*n2*n3 )
2 2 2 2 2
- a22 *b33*n1*r14 - a22 *n1 *r212 + a22 *n3 *r212 - a22*b33*n1*n2*r29
r354=------------------------------------------------------------------------
3 2
b33*n1 - b33*n1*n3
2 2 2 2 2
- a22 *b33*n1*r14 - a22 *n1 *r212 + a22 *n3 *r212 - a22*b33*n1*n2*r29
r355=------------------------------------------------------------------------
2 2
b33*n1 *n2 - b33*n2*n3
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
1 2 1 2 1 2
r427=(---*a22*b33 *n1*r14 + ---*a22*b33*n1 *r212 - ---*a22*b33*n3 *r212
4 4 4
1 2 3 2 3 2
+ ---*b33 *n1*n2*r29 + n1 *n2*r425 - n1*n2*n3 *r425)/(n1 *n2 - n1*n2*n3 )
4
r428=0
r429=0
1 2 1 2 1 2
r430=(---*a22*b33 *n1*r14 + ---*a22*b33*n1 *r212 - ---*a22*b33*n3 *r212
4 4 4
1 2 3 2 3 2
+ ---*b33 *n1*n2*r29 + n1 *n2*r425 - n1*n2*n3 *r425)/(n1 *n2 - n1*n2*n3 )
4
1 2 2 1 2 2 1 2 2
r431=(---*a22 *b33 *n1*r14 + ---*a22 *b33*n1 *r212 - ---*a22 *b33*n3 *r212
2 2 2
2 3 2 2 1 2
- 2*a22 *n1 *r322 + 2*a22 *n1*n3 *r322 + ---*a22*b33 *n1*n2*r29
2
3 2 3 2
- 2*a22*n1 *n2*r425 + 2*a22*n1*n2*n3 *r425)/(b33*n1 *n2 - b33*n1*n2*n3 )
r432=0
r433=0
3
a22 *r212
r434=-----------
b33*n1*n2
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
1 2 2 1 2 2 1 2 2
r452=( - ---*a22 *b33 *n1*r14 - ---*a22 *b33*n1 *r212 + ---*a22 *b33*n3 *r212
2 2 2
2 3 2 2 1 2
- 2*a22 *n1 *r322 + 2*a22 *n1*n3 *r322 - ---*a22*b33 *n1*n2*r29
2
3 2 3 2
- 2*a22*n1 *n2*r425 + 2*a22*n1*n2*n3 *r425)/(b33*n1 *n2 - b33*n1*n2*n3 )
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
3 3 2 3 2 2
a22 *b33*n1*r14 + a22 *n1 *r212 - a22 *n3 *r212 + a22 *b33*n1*n2*r29
r4114=----------------------------------------------------------------------
3 2
b33*n1 *n2 - b33*n1*n2*n3
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
- b33*n3
m3=-----------
a22
1
- ---*b33*n2
2
m2=---------------
a22
1
- ---*b33*n1
2
m1=---------------
a22
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
a33=0
a23=0
a13=0
a12=0
a11=a22
1 2 2 1 2 2 1 2 2
r488=( - ---*a22 *b33 *n1*r14 - ---*a22 *b33*n1 *r212 + ---*a22 *b33*n3 *r212
2 2 2
2 3 2 2 1 2
- 2*a22 *n1 *r322 + 2*a22 *n1*n3 *r322 - ---*a22*b33 *n1*n2*r29
2
3 2 3 2
- 2*a22*n1 *n2*r425 + 2*a22*n1*n2*n3 *r425)/(b33*n1 *n2 - b33*n1*n2*n3 )
3 3 2 3 2 2
a22 *b33*n1*r14 + a22 *n1 *r212 - a22 *n3 *r212 + a22 *b33*n1*n2*r29
r464=----------------------------------------------------------------------
3 2
b33*n1 *n2 - b33*n1*n2*n3
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, r14, r212, r322, n3, b33, n2, n1, 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,a22,b33}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11 - a22,
a12,
a13,
a23,
a33,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
c11,
c12,
c13,
c22,
c23,
a22*c33 + 1/4*b33**2,
a22*m1 + 1/2*b33*n1,
a22*m2 + 1/2*b33*n2,
a22*m3 + b33*n3}$
The system of equations related to the Hamiltonian HAM:
1
- ---*v1*b33*n1
2 2 2
HAM=u1 *a22 + u1*n1 + u2 *a22 + u2*n2 + u3*v3*b33 + u3*n3 + ------------------
a22
1 1 2 2
- ---*v2*b33*n2 - ---*v3 *b33
2 4 - v3*b33*n3
+ ------------------ + ----------------- + --------------
a22 a22 a22
has apart from the Hamiltonian and Casimirs the following 5 first integrals:
2 2 2
FI=2*u1 *v1*a22*n1 + 2*u1*u2*v1*a22*n2 + 2*u1*u2*v2*a22*n1 - 2*u1*u3 *v1*a22
2
+ 2*u1*u3*v1*a22*n3 + 2*u1*u3*v3*a22*n1 + u1*v1 *b33*n1 + u1*v1*v2*b33*n2
2 2 2
+ 2*u2 *v2*a22*n2 - 2*u2*u3 *v2*a22 + 2*u2*u3*v2*a22*n3 + 2*u2*u3*v3*a22*n2
2 3 2 2
+ u2*v1*v2*b33*n1 + u2*v2 *b33*n2 - 2*u3 *v3*a22 + 2*u3 *v3*a22*n3
+ u3*v1*v3*b33*n1 + u3*v2*v3*b33*n2
= a product of the elements of: {2,
- u1*v1 - u2*v2 - u3*v3,
2 2 1
- u1*a22*n1 - u2*a22*n2 + u3 *a22 - u3*a22*n3 - ---*v1*b33*n1
2
1
- ---*v2*b33*n2}
2
{HAM,FI} = 0
3 3 2 3 2 2 4 2 3
FI= - u1 *a22 *n1 - u1 *u2*a22 *n2 + u1 *u3 *a22 - u1 *u3*a22 *n3
1 2 2 2 3 1 2
- ---*u1 *v2*a22 *b33*n2 - u1*u2 *a22 *n1 + ---*u1*u2*v1*a22 *b33*n2
2 2
1 2 1 2 3 2 3
+ ---*u1*u2*v2*a22 *b33*n1 - ---*u1*u3 *v1*a22 *b33 - u1*u3 *a22 *n1
2 2
1 2 1 2
+ ---*u1*u3*v1*a22 *b33*n3 - ---*u1*u3*v3*a22 *b33*n1
2 2
1 2 2 2
+ ---*u1*v1*v2*a22*b33 *n2 + u1*v1*(a22*b33*n1 - a22*b33*n2 )
4
1 2 2
- ---*u1*v2 *a22*b33 *n1 + u1*v2*a22*b33*n1*n2 + u1*v3*a22*b33*n1*n3
4
3 3 2 2 4 2 3 1 2 2
- u2 *a22 *n2 + u2 *u3 *a22 - u2 *u3*a22 *n3 - ---*u2 *v1*a22 *b33*n1
2
1 2 3 2 3 1 2
- ---*u2*u3 *v2*a22 *b33 - u2*u3 *a22 *n2 + ---*u2*u3*v2*a22 *b33*n3
2 2
1 2 1 2 2
- ---*u2*u3*v3*a22 *b33*n2 - ---*u2*v1 *a22*b33 *n2
2 4
1 2
+ ---*u2*v1*v2*a22*b33 *n1 + u2*v1*a22*b33*n1*n2 + u2*v3*a22*b33*n2*n3
4
4 4 1 3 3 3 3 1 2 2 2 2
+ u3 *a22 + ---*u3 *v3*a22 *b33 - u3 *a22 *n3 + ---*u3 *v1 *a22 *b33
2 4
3 2 2 1 2 2 2 2 3 2 2
- ---*u3 *v1*a22 *b33*n1 + ---*u3 *v2 *a22 *b33 - ---*u3 *v2*a22 *b33*n2
2 4 2
3 2 2 1 2 2 1 2
- ---*u3 *v3*a22 *b33*n3 - ---*u3*v1 *a22*b33 *n3 - ---*u3*v1*v3*a22*b33 *n1
2 4 4
1 2 2 1 2
+ u3*v1*a22*b33*n1*n3 - ---*u3*v2 *a22*b33 *n3 - ---*u3*v2*v3*a22*b33 *n2
4 4
2 2
+ u3*v2*a22*b33*n2*n3 + u3*v3*( - a22*b33*n2 + a22*b33*n3 )
1 3 3 1 2 3 2 1 2 2 1 2 2
- ---*v1 *b33 *n1 - ---*v1 *v2*b33 *n2 + v1 *(---*b33 *n1 - ---*b33 *n2 )
8 8 2 2
1 2 3 2 1 2
- ---*v1*v2 *b33 *n1 + v1*v2*b33 *n1*n2 + ---*v1*v3*b33 *n1*n3
8 2
1 3 3 1 2 1 2 2 2
- ---*v2 *b33 *n2 + ---*v2*v3*b33 *n2*n3 - ---*v3 *b33 *n2
8 2 2
which the program can not factorize further.
{HAM,FI} = 0
3 3 2 3 2 2 4 2 3
FI= - u1 *a22 *n1 - u1 *u2*a22 *n2 + u1 *u3 *a22 - u1 *u3*a22 *n3
1 2 2 2 3 1 2
- ---*u1 *v2*a22 *b33*n2 - u1*u2 *a22 *n1 + ---*u1*u2*v1*a22 *b33*n2
2 2
1 2 2 1 2 3
+ ---*u1*u2*v2*a22 *b33*n1 - 2*u1*u2*a22 *n1*n2 - ---*u1*u3 *v1*a22 *b33
2 2
2 3 1 2 1 2
+ u1*u3 *a22 *n1 + ---*u1*u3*v1*a22 *b33*n3 - ---*u1*u3*v3*a22 *b33*n1
2 2
2 1 2 1 2 2
- 2*u1*u3*a22 *n1*n3 + ---*u1*v1*v2*a22*b33 *n2 - ---*u1*v2 *a22*b33 *n1
4 4
3 2 3 3
+ u1*v3*a22*b33*n1*n3 + u1*(a22*n1 - a22*n1*n3 ) - u2 *a22 *n2
2 2 4 2 3 1 2 2
+ u2 *u3 *a22 - u2 *u3*a22 *n3 - ---*u2 *v1*a22 *b33*n1
2
2 2 2 2 2 1 2 3 2 3
+ u2 *(a22 *n1 - a22 *n2 ) - ---*u2*u3 *v2*a22 *b33 + u2*u3 *a22 *n2
2
1 2 1 2 2
+ ---*u2*u3*v2*a22 *b33*n3 - ---*u2*u3*v3*a22 *b33*n2 - 2*u2*u3*a22 *n2*n3
2 2
1 2 2 1 2
- ---*u2*v1 *a22*b33 *n2 + ---*u2*v1*v2*a22*b33 *n1 + u2*v3*a22*b33*n2*n3
4 4
2 2 1 3 3 3 3
+ u2*(a22*n1 *n2 - a22*n2*n3 ) + ---*u3 *v3*a22 *b33 + u3 *a22 *n3
2
1 2 2 2 2 1 2 2 1 2 2 2 2
+ ---*u3 *v1 *a22 *b33 - ---*u3 *v1*a22 *b33*n1 + ---*u3 *v2 *a22 *b33
4 2 4
1 2 2 3 2 2 1 2 2
- ---*u3 *v2*a22 *b33*n2 - ---*u3 *v3*a22 *b33*n3 - ---*u3*v1 *a22*b33 *n3
2 2 4
1 2 1 2 2
- ---*u3*v1*v3*a22*b33 *n1 - ---*u3*v2 *a22*b33 *n3
4 4
1 2 2 2
- ---*u3*v2*v3*a22*b33 *n2 + u3*v3*(a22*b33*n1 + a22*b33*n3 )
4
2 3 1 3 3 1 2 3
+ u3*(a22*n1 *n3 - a22*n3 ) - ---*v1 *b33 *n1 - ---*v1 *v2*b33 *n2
8 8
2 1 2 2 1 2 2 1 2 3
+ v1 *(---*b33 *n1 - ---*b33 *n2 ) - ---*v1*v2 *b33 *n1
4 4 8
1 2 1 2
+ ---*v1*v2*b33 *n1*n2 + ---*v1*v3*b33 *n1*n3
2 2
1 3 1 2 1 3 3
+ v1*( - ---*b33*n1 - ---*b33*n1*n3 ) - ---*v2 *b33 *n2
2 2 8
1 2 1 2 1 2
+ ---*v2*v3*b33 *n2*n3 + v2*( - ---*b33*n1 *n2 - ---*b33*n2*n3 )
2 2 2
2 1 2 2 1 2 2 2
+ v3 *( - ---*b33 *n1 - ---*b33 *n2 ) - v3*b33*n1 *n3
4 4
which the program can not factorize further.
{HAM,FI} = 0
2 2 2 2 2 3
FI=2*u1 *v1*a22 *n1 + 2*u1*u2*v1*a22 *n2 + 2*u1*u2*v2*a22 *n1 - 2*u1*u3 *v1*a22
2 2
+ 2*u1*u3*v1*a22 *n3 + 2*u1*u3*v3*a22 *n1 + u1*v1*v2*a22*b33*n2
2 2 2 2
- u1*v2 *a22*b33*n1 - u1*v3 *a22*b33*n1 + 2*u2 *v2*a22 *n2
2 3 2 2
- 2*u2*u3 *v2*a22 + 2*u2*u3*v2*a22 *n3 + 2*u2*u3*v3*a22 *n2
2 2
- u2*v1 *a22*b33*n2 + u2*v1*v2*a22*b33*n1 - u2*v3 *a22*b33*n2
3 3 2 2 2 2 2 2 2 2 2
- 2*u3 *v3*a22 + u3 *v1 *a22 *b33 + u3 *v2 *a22 *b33 + u3 *v3 *a22 *b33
2 2 2
+ 2*u3 *v3*a22 *n3 - u3*v1 *a22*b33*n3 + u3*v1*v3*a22*b33*n1
2 2
- u3*v2 *a22*b33*n3 + u3*v2*v3*a22*b33*n2 - u3*v3 *a22*b33*n3
1 3 2 1 2 2 1 2 2
- ---*v1 *b33 *n1 - ---*v1 *v2*b33 *n2 - ---*v1*v2 *b33 *n1
2 2 2
1 2 2 1 3 2 1 2 2
- ---*v1*v3 *b33 *n1 - ---*v2 *b33 *n2 - ---*v2*v3 *b33 *n2
2 2 2
= a product of the elements of: {2,
1 2 1 2
- u1*v1*a22 - u2*v2*a22 - u3*v3*a22 + ---*v1 *b33 + ---*v2 *b33
2 2
1 2
+ ---*v3 *b33,
2
2 2 1
- u1*a22*n1 - u2*a22*n2 + u3 *a22 - u3*a22*n3 - ---*v1*b33*n1
2
1
- ---*v2*b33*n2}
2
{HAM,FI} = 0
3 3 2 3 2 2 4 2 3
FI= - u1 *a22 *n1 - u1 *u2*a22 *n2 + u1 *u3 *a22 - u1 *u3*a22 *n3
1 2 2 2 3 1 2
- ---*u1 *v2*a22 *b33*n2 - u1*u2 *a22 *n1 + ---*u1*u2*v1*a22 *b33*n2
2 2
1 2 2 1 2 3
+ ---*u1*u2*v2*a22 *b33*n1 - 2*u1*u2*a22 *n1*n2 - ---*u1*u3 *v1*a22 *b33
2 2
2 3 1 2 1 2
+ u1*u3 *a22 *n1 + ---*u1*u3*v1*a22 *b33*n3 - ---*u1*u3*v3*a22 *b33*n1
2 2
2 1 2 1 2 2
- 2*u1*u3*a22 *n1*n3 + ---*u1*v1*v2*a22*b33 *n2 - ---*u1*v2 *a22*b33 *n1
4 4
3 3 2 2 4 2 3
+ u1*v3*a22*b33*n1*n3 - u2 *a22 *n2 + u2 *u3 *a22 - u2 *u3*a22 *n3
1 2 2 2 2 2 2 2
- ---*u2 *v1*a22 *b33*n1 + u2 *(a22 *n1 - a22 *n2 )
2
1 2 3 2 3 1 2
- ---*u2*u3 *v2*a22 *b33 + u2*u3 *a22 *n2 + ---*u2*u3*v2*a22 *b33*n3
2 2
1 2 2 1 2 2
- ---*u2*u3*v3*a22 *b33*n2 - 2*u2*u3*a22 *n2*n3 - ---*u2*v1 *a22*b33 *n2
2 4
1 2 1 3 3
+ ---*u2*v1*v2*a22*b33 *n1 + u2*v3*a22*b33*n2*n3 + ---*u3 *v3*a22 *b33
4 2
3 3 1 2 2 2 2 1 2 2
+ u3 *a22 *n3 + ---*u3 *v1 *a22 *b33 - ---*u3 *v1*a22 *b33*n1
4 2
1 2 2 2 2 1 2 2 3 2 2
+ ---*u3 *v2 *a22 *b33 - ---*u3 *v2*a22 *b33*n2 - ---*u3 *v3*a22 *b33*n3
4 2 2
2 2 2 2 2 1 2 2
+ u3 *(a22 *n1 - a22 *n3 ) - ---*u3*v1 *a22*b33 *n3
4
1 2 1 2 2
- ---*u3*v1*v3*a22*b33 *n1 - ---*u3*v2 *a22*b33 *n3
4 4
1 2 2 2
- ---*u3*v2*v3*a22*b33 *n2 + u3*v3*(a22*b33*n1 + a22*b33*n3 )
4
1 3 3 1 2 3 2 1 2 2 1 2 2
- ---*v1 *b33 *n1 - ---*v1 *v2*b33 *n2 + v1 *(---*b33 *n1 - ---*b33 *n2 )
8 8 4 4
1 2 3 1 2 1 2
- ---*v1*v2 *b33 *n1 + ---*v1*v2*b33 *n1*n2 + ---*v1*v3*b33 *n1*n3
8 2 2
3 1 3 3 1 2 2
- v1*b33*n1 - ---*v2 *b33 *n2 + ---*v2*v3*b33 *n2*n3 - v2*b33*n1 *n2
8 2
2 1 2 2 1 2 2 2
+ v3 *( - ---*b33 *n1 - ---*b33 *n2 ) - v3*b33*n1 *n3
4 4
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*v3*b33 + u3*n3 + ( - 1/2*v1*b33*
n1)/a22 + ( - 1/2*v2*b33*n2)/a22 + ( - 1/4*v3**2*b33**2)/a22 + ( - v3*b33*n3)/
a22$
FI=2*u1**2*v1*a22*n1 + 2*u1*u2*v1*a22*n2 + 2*u1*u2*v2*a22*n1 - 2*u1*u3**2*v1*a22
**2 + 2*u1*u3*v1*a22*n3 + 2*u1*u3*v3*a22*n1 + u1*v1**2*b33*n1 + u1*v1*v2*b33*n2
+ 2*u2**2*v2*a22*n2 - 2*u2*u3**2*v2*a22**2 + 2*u2*u3*v2*a22*n3 + 2*u2*u3*v3*a22*
n2 + u2*v1*v2*b33*n1 + u2*v2**2*b33*n2 - 2*u3**3*v3*a22**2 + 2*u3**2*v3*a22*n3 +
u3*v1*v3*b33*n1 + u3*v2*v3*b33*n2$
FI= - u1**3*a22**3*n1 - u1**2*u2*a22**3*n2 + u1**2*u3**2*a22**4 - u1**2*u3*a22**
3*n3 - 1/2*u1**2*v2*a22**2*b33*n2 - u1*u2**2*a22**3*n1 + 1/2*u1*u2*v1*a22**2*b33
*n2 + 1/2*u1*u2*v2*a22**2*b33*n1 - 1/2*u1*u3**2*v1*a22**3*b33 - u1*u3**2*a22**3*
n1 + 1/2*u1*u3*v1*a22**2*b33*n3 - 1/2*u1*u3*v3*a22**2*b33*n1 + 1/4*u1*v1*v2*a22*
b33**2*n2 + u1*v1*(a22*b33*n1**2 - a22*b33*n2**2) - 1/4*u1*v2**2*a22*b33**2*n1 +
u1*v2*a22*b33*n1*n2 + u1*v3*a22*b33*n1*n3 - u2**3*a22**3*n2 + u2**2*u3**2*a22**
4 - u2**2*u3*a22**3*n3 - 1/2*u2**2*v1*a22**2*b33*n1 - 1/2*u2*u3**2*v2*a22**3*b33
- u2*u3**2*a22**3*n2 + 1/2*u2*u3*v2*a22**2*b33*n3 - 1/2*u2*u3*v3*a22**2*b33*n2
- 1/4*u2*v1**2*a22*b33**2*n2 + 1/4*u2*v1*v2*a22*b33**2*n1 + u2*v1*a22*b33*n1*n2
+ u2*v3*a22*b33*n2*n3 + u3**4*a22**4 + 1/2*u3**3*v3*a22**3*b33 - u3**3*a22**3*n3
+ 1/4*u3**2*v1**2*a22**2*b33**2 - 3/2*u3**2*v1*a22**2*b33*n1 + 1/4*u3**2*v2**2*
a22**2*b33**2 - 3/2*u3**2*v2*a22**2*b33*n2 - 3/2*u3**2*v3*a22**2*b33*n3 - 1/4*u3
*v1**2*a22*b33**2*n3 - 1/4*u3*v1*v3*a22*b33**2*n1 + u3*v1*a22*b33*n1*n3 - 1/4*u3
*v2**2*a22*b33**2*n3 - 1/4*u3*v2*v3*a22*b33**2*n2 + u3*v2*a22*b33*n2*n3 + u3*v3*
( - a22*b33*n2**2 + a22*b33*n3**2) - 1/8*v1**3*b33**3*n1 - 1/8*v1**2*v2*b33**3*
n2 + v1**2*(1/2*b33**2*n1**2 - 1/2*b33**2*n2**2) - 1/8*v1*v2**2*b33**3*n1 + v1*
v2*b33**2*n1*n2 + 1/2*v1*v3*b33**2*n1*n3 - 1/8*v2**3*b33**3*n2 + 1/2*v2*v3*b33**
2*n2*n3 - 1/2*v3**2*b33**2*n2**2$
FI= - u1**3*a22**3*n1 - u1**2*u2*a22**3*n2 + u1**2*u3**2*a22**4 - u1**2*u3*a22**
3*n3 - 1/2*u1**2*v2*a22**2*b33*n2 - u1*u2**2*a22**3*n1 + 1/2*u1*u2*v1*a22**2*b33
*n2 + 1/2*u1*u2*v2*a22**2*b33*n1 - 2*u1*u2*a22**2*n1*n2 - 1/2*u1*u3**2*v1*a22**3
*b33 + u1*u3**2*a22**3*n1 + 1/2*u1*u3*v1*a22**2*b33*n3 - 1/2*u1*u3*v3*a22**2*b33
*n1 - 2*u1*u3*a22**2*n1*n3 + 1/4*u1*v1*v2*a22*b33**2*n2 - 1/4*u1*v2**2*a22*b33**
2*n1 + u1*v3*a22*b33*n1*n3 + u1*(a22*n1**3 - a22*n1*n3**2) - u2**3*a22**3*n2 +
u2**2*u3**2*a22**4 - u2**2*u3*a22**3*n3 - 1/2*u2**2*v1*a22**2*b33*n1 + u2**2*(
a22**2*n1**2 - a22**2*n2**2) - 1/2*u2*u3**2*v2*a22**3*b33 + u2*u3**2*a22**3*n2 +
1/2*u2*u3*v2*a22**2*b33*n3 - 1/2*u2*u3*v3*a22**2*b33*n2 - 2*u2*u3*a22**2*n2*n3
- 1/4*u2*v1**2*a22*b33**2*n2 + 1/4*u2*v1*v2*a22*b33**2*n1 + u2*v3*a22*b33*n2*n3
+ u2*(a22*n1**2*n2 - a22*n2*n3**2) + 1/2*u3**3*v3*a22**3*b33 + u3**3*a22**3*n3 +
1/4*u3**2*v1**2*a22**2*b33**2 - 1/2*u3**2*v1*a22**2*b33*n1 + 1/4*u3**2*v2**2*
a22**2*b33**2 - 1/2*u3**2*v2*a22**2*b33*n2 - 3/2*u3**2*v3*a22**2*b33*n3 - 1/4*u3
*v1**2*a22*b33**2*n3 - 1/4*u3*v1*v3*a22*b33**2*n1 - 1/4*u3*v2**2*a22*b33**2*n3 -
1/4*u3*v2*v3*a22*b33**2*n2 + u3*v3*(a22*b33*n1**2 + a22*b33*n3**2) + u3*(a22*n1
**2*n3 - a22*n3**3) - 1/8*v1**3*b33**3*n1 - 1/8*v1**2*v2*b33**3*n2 + v1**2*(1/4*
b33**2*n1**2 - 1/4*b33**2*n2**2) - 1/8*v1*v2**2*b33**3*n1 + 1/2*v1*v2*b33**2*n1*
n2 + 1/2*v1*v3*b33**2*n1*n3 + v1*( - 1/2*b33*n1**3 - 1/2*b33*n1*n3**2) - 1/8*v2
**3*b33**3*n2 + 1/2*v2*v3*b33**2*n2*n3 + v2*( - 1/2*b33*n1**2*n2 - 1/2*b33*n2*n3
**2) + v3**2*( - 1/4*b33**2*n1**2 - 1/4*b33**2*n2**2) - v3*b33*n1**2*n3$
FI=2*u1**2*v1*a22**2*n1 + 2*u1*u2*v1*a22**2*n2 + 2*u1*u2*v2*a22**2*n1 - 2*u1*u3
**2*v1*a22**3 + 2*u1*u3*v1*a22**2*n3 + 2*u1*u3*v3*a22**2*n1 + u1*v1*v2*a22*b33*
n2 - u1*v2**2*a22*b33*n1 - u1*v3**2*a22*b33*n1 + 2*u2**2*v2*a22**2*n2 - 2*u2*u3
**2*v2*a22**3 + 2*u2*u3*v2*a22**2*n3 + 2*u2*u3*v3*a22**2*n2 - u2*v1**2*a22*b33*
n2 + u2*v1*v2*a22*b33*n1 - u2*v3**2*a22*b33*n2 - 2*u3**3*v3*a22**3 + u3**2*v1**2
*a22**2*b33 + u3**2*v2**2*a22**2*b33 + u3**2*v3**2*a22**2*b33 + 2*u3**2*v3*a22**
2*n3 - u3*v1**2*a22*b33*n3 + u3*v1*v3*a22*b33*n1 - u3*v2**2*a22*b33*n3 + u3*v2*
v3*a22*b33*n2 - u3*v3**2*a22*b33*n3 - 1/2*v1**3*b33**2*n1 - 1/2*v1**2*v2*b33**2*
n2 - 1/2*v1*v2**2*b33**2*n1 - 1/2*v1*v3**2*b33**2*n1 - 1/2*v2**3*b33**2*n2 - 1/2
*v2*v3**2*b33**2*n2$
FI= - u1**3*a22**3*n1 - u1**2*u2*a22**3*n2 + u1**2*u3**2*a22**4 - u1**2*u3*a22**
3*n3 - 1/2*u1**2*v2*a22**2*b33*n2 - u1*u2**2*a22**3*n1 + 1/2*u1*u2*v1*a22**2*b33
*n2 + 1/2*u1*u2*v2*a22**2*b33*n1 - 2*u1*u2*a22**2*n1*n2 - 1/2*u1*u3**2*v1*a22**3
*b33 + u1*u3**2*a22**3*n1 + 1/2*u1*u3*v1*a22**2*b33*n3 - 1/2*u1*u3*v3*a22**2*b33
*n1 - 2*u1*u3*a22**2*n1*n3 + 1/4*u1*v1*v2*a22*b33**2*n2 - 1/4*u1*v2**2*a22*b33**
2*n1 + u1*v3*a22*b33*n1*n3 - u2**3*a22**3*n2 + u2**2*u3**2*a22**4 - u2**2*u3*a22
**3*n3 - 1/2*u2**2*v1*a22**2*b33*n1 + u2**2*(a22**2*n1**2 - a22**2*n2**2) - 1/2*
u2*u3**2*v2*a22**3*b33 + u2*u3**2*a22**3*n2 + 1/2*u2*u3*v2*a22**2*b33*n3 - 1/2*
u2*u3*v3*a22**2*b33*n2 - 2*u2*u3*a22**2*n2*n3 - 1/4*u2*v1**2*a22*b33**2*n2 + 1/4
*u2*v1*v2*a22*b33**2*n1 + u2*v3*a22*b33*n2*n3 + 1/2*u3**3*v3*a22**3*b33 + u3**3*
a22**3*n3 + 1/4*u3**2*v1**2*a22**2*b33**2 - 1/2*u3**2*v1*a22**2*b33*n1 + 1/4*u3
**2*v2**2*a22**2*b33**2 - 1/2*u3**2*v2*a22**2*b33*n2 - 3/2*u3**2*v3*a22**2*b33*
n3 + u3**2*(a22**2*n1**2 - a22**2*n3**2) - 1/4*u3*v1**2*a22*b33**2*n3 - 1/4*u3*
v1*v3*a22*b33**2*n1 - 1/4*u3*v2**2*a22*b33**2*n3 - 1/4*u3*v2*v3*a22*b33**2*n2 +
u3*v3*(a22*b33*n1**2 + a22*b33*n3**2) - 1/8*v1**3*b33**3*n1 - 1/8*v1**2*v2*b33**
3*n2 + v1**2*(1/4*b33**2*n1**2 - 1/4*b33**2*n2**2) - 1/8*v1*v2**2*b33**3*n1 + 1/
2*v1*v2*b33**2*n1*n2 + 1/2*v1*v3*b33**2*n1*n3 - v1*b33*n1**3 - 1/8*v2**3*b33**3*
n2 + 1/2*v2*v3*b33**2*n2*n3 - v2*b33*n1**2*n2 + v3**2*( - 1/4*b33**2*n1**2 - 1/4
*b33**2*n2**2) - v3*b33*n1**2*n3$