Solution 1 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
a11
1 2 1 2 1 2
r11=( - ---*a11 *b33*n2*r29 + ---*b33*n1 *n2*r434 - ---*b33*n1 *n2*r464
2 2 2
1 2 1 2 4
+ ---*b33*n2*n3 *r434 - ---*b33*n2*n3 *r464)/a11
2 2
1 2 1 3 1 3
r12=( - ---*a11 *b33*n1*r29 + ---*b33*n1 *r434 - ---*b33*n1 *r464
2 2 2
1 2 1 2 4
+ ---*b33*n1*n3 *r434 - ---*b33*n1*n3 *r464)/a11
2 2
2 2 2 3 3
- a11 *n3*r29 - n1 *n3*r434 + n1 *n3*r464 + n3 *r434 - n3 *r464
r13=------------------------------------------------------------------
3
a11
2 2 2 2 2
- a11 *n2*r29 - n1 *n2*r434 + n1 *n2*r464 + n2*n3 *r434 - n2*n3 *r464
r14=------------------------------------------------------------------------
3
a11
2 3 3 2 2
- a11 *n1*r29 - n1 *r434 + n1 *r464 + n1*n3 *r434 - n1*n3 *r464
r15=------------------------------------------------------------------
3
a11
1 2 2
- ---*b33 *n1 *r464
2
r20=----------------------
4
a11
1 2
---*b33 *n2*n3*r464
2
r21=---------------------
4
a11
1 2 2 1 2 2 1 2 2
r22=( - ---*b33 *n1 *r434 - ---*b33 *n1 *r464 + ---*b33 *n2 *r434
4 4 4
1 2 2 4
+ ---*b33 *n2 *r464)/a11
4
1 2
---*b33 *n1*n3*r464
2
r23=---------------------
4
a11
1 2 1 2
---*b33 *n1*n2*r434 + ---*b33 *n1*n2*r464
2 2
r24=-------------------------------------------
4
a11
2 2 2 2
- b33*n1 *r434 + b33*n1 *r464 - b33*n2 *r434 + b33*n3 *r464
r26=--------------------------------------------------------------
3
a11
b33*n2*n3*r434
r27=----------------
3
a11
b33*n1*n3*r434
r28=----------------
3
a11
b33*n2*n3*r464
r210=----------------
3
a11
b33*n1*n2*r434
r212=----------------
3
a11
2*n2*n3*r434 - 2*n2*n3*r464
r213=-----------------------------
2
a11
2 2 2 2
- n1 *r434 + n1 *r464 + n2 *r434 - n2 *r464
r214=----------------------------------------------
2
a11
b33*n1*n3*r464
r215=----------------
3
a11
b33*n1*n2*r434
r216=----------------
3
a11
2 2
b33*n1 *r434 - b33*n2 *r434
r217=-----------------------------
3
a11
2*n1*n3*r434 - 2*n1*n3*r464
r218=-----------------------------
2
a11
2*n1*n2*r434 - 2*n1*n2*r464
r219=-----------------------------
2
a11
1
---*b33*n3*r483
2
r30=-----------------
2
a11
1 1
---*a11*b33*r320 + ---*b33*n2*r483
2 2
r31=------------------------------------
2
a11
1
---*b33*n3*r483
2
r32=-----------------
2
a11
1 3 1 2 1 3
---*a11 *b33*r320 + ---*a11 *b33*n2*r483 - ---*b33 *n2*r464
2 2 8
r33=-------------------------------------------------------------
4
a11
1 1
---*a11*b33*n1*r320 + ---*b33*n1*n2*r483
2 2
r34=------------------------------------------
2
a11 *n2
r35=0
1 3 1 2 1 3
---*a11 *b33*n1*r320 + ---*a11 *b33*n1*n2*r483 - ---*b33 *n1*n2*r464
2 2 8
r36=----------------------------------------------------------------------
4
a11 *n2
1
---*b33*n3*r483
2
r37=-----------------
2
a11
1 3 1 2 1 3
---*a11 *b33*r320 + ---*a11 *b33*n2*r483 - ---*b33 *n2*r464
2 2 8
r38=-------------------------------------------------------------
4
a11
1 3 1 2 1 3
---*a11 *b33*n1*r320 + ---*a11 *b33*n1*n2*r483 - ---*b33 *n1*n2*r464
2 2 8
r39=----------------------------------------------------------------------
4
a11 *n2
n3*r320
r310=---------
n2
3 3 1 2
- a11 *r320 + a11 *r322 - ---*b33 *n2*r464
4
r311=---------------------------------------------
3
a11
3 1 2
a11 *n3*r320 - ---*b33 *n2*n3*r464
4
r312=------------------------------------
3
a11 *n2
3 3 1 2
- a11 *n1*r320 + a11 *n1*r322 - ---*b33 *n1*n2*r464
4
r313=------------------------------------------------------
3
a11 *n2
r314=0
3 1 2
a11 *n3*r320 - ---*b33 *n2*n3*r464
4
r315=------------------------------------
3
a11 *n2
3 3 3 2
- 2*a11 *n3*r320 + 2*a11 *n3*r322 - ---*b33 *n2*n3*r464
2
r316=----------------------------------------------------------
2
a11 *b33*n2
1
- b33*n2*r434 - ---*b33*n2*r464
2
r317=----------------------------------
2
a11
1
- b33*n1*r434 - ---*b33*n1*r464
2
r318=----------------------------------
2
a11
- 2*n3*r434 + n3*r464
r319=------------------------
a11
r321=0
r323=0
3 3 1 2
- a11 *n1*r320 + a11 *n1*r322 + ---*b33 *n1*n2*r464
4
r324=------------------------------------------------------
3
a11 *n2
3 1 2
a11 *r320 - ---*b33 *n2*r464
4
r325=------------------------------
3
a11
3 3 1 2
- 2*a11 *r320 + 2*a11 *r322 - ---*b33 *n2*r464
2
r326=-------------------------------------------------
2
a11 *b33
3 3 1 2
- 2*a11 *n3*r320 + 2*a11 *n3*r322 + ---*b33 *n2*n3*r464
2
r327=----------------------------------------------------------
2
a11 *b33*n2
r328=0
- 2*n2*r434 + n2*r464
r329=------------------------
a11
r330=0
- 2*a11*r320 + 2*a11*r322
r331=----------------------------
b33
1
- ---*b33*n1*r464
2
r332=--------------------
2
a11
- n3*r464
r333=------------
a11
- n2*r464
r334=------------
a11
n1*r320
r335=---------
n2
r336=0
3 1 2
a11 *n1*r320 - ---*b33 *n1*n2*r464
4
r337=------------------------------------
3
a11 *n2
r338=0
3 3 1 2
- a11 *r320 + a11 *r322 + ---*b33 *n2*r464
4
r339=---------------------------------------------
3
a11
n1*r322
r340=---------
n2
3 3 1 2
- 2*a11 *n1*r320 + 2*a11 *n1*r322 - ---*b33 *n1*n2*r464
2
r341=----------------------------------------------------------
2
a11 *b33*n2
r342=0
3 3 1 2
- 2*a11 *n3*r320 + 2*a11 *n3*r322 + ---*b33 *n2*n3*r464
2
r343=----------------------------------------------------------
2
a11 *b33*n2
- 2*n1*r434 + n1*r464
r344=------------------------
a11
r345=0
3 3 1 2
- 2*a11 *n1*r320 + 2*a11 *n1*r322 + ---*b33 *n1*n2*r464
2
r346=----------------------------------------------------------
2
a11 *b33*n2
3 3 1 2
- 2*a11 *r320 + 2*a11 *r322 + ---*b33 *n2*r464
2
r347=-------------------------------------------------
2
a11 *b33
r348=0
- n1*r464
r349=------------
a11
r350=0
1
- ---*b33*n2*r464
2
r351=--------------------
2
a11
- 2*a11*n1*r320 + 2*a11*n1*r322
r352=----------------------------------
b33*n2
- n3*r464
r353=------------
a11
- n2*r464
r354=------------
a11
- n1*r464
r355=------------
a11
1 2
---*b33 *r483
8
r40=---------------
2
a11
r41=0
1 2
---*b33 *r483
8
r42=---------------
2
a11
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
1 2
---*b33 *r483
8
r49=---------------
2
a11
r410=0
r411=0
r412=0
r413=0
1
- ---*b33*r483
2
r415=-----------------
a11
r416=0
1
- ---*b33*r483
2
r417=-----------------
a11
r418=0
r419=0
r420=0
r421=0
1
- ---*b33*r483
2
r422=-----------------
a11
r423=0
r424=0
- a11*r320
r425=-------------
n2
r426=0
3 1 2 1 2
- a11 *r320 - ---*a11 *n2*r483 + ---*b33 *n2*r464
2 4
r427=----------------------------------------------------
2
a11 *n2
r428=0
r429=0
3 1 2 1 2
- a11 *r320 - ---*a11 *n2*r483 + ---*b33 *n2*r464
2 4
r430=----------------------------------------------------
2
a11 *n2
3 3 1 2
2*a11 *r320 - 2*a11 *r322 + ---*b33 *n2*r464
2
r431=----------------------------------------------
a11*b33*n2
r432=0
r433=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
r446=r483
r447=0
r448=0
r449=0
r450=0
r451=0
3 3 1 2
2*a11 *r320 - 2*a11 *r322 - ---*b33 *n2*r464
2
r452=----------------------------------------------
a11*b33*n2
r453=0
r454=0
1
r455= - ---*r483
2
r456=0
r458=0
r459=0
1
r460= - ---*r483
2
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
r484=0
r485=0
r486=0
r487=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=r483
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
1
r4105= - ---*r483
2
r4106=0
1
r4107= - ---*r483
2
r4108=0
r4109=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=-----------
a11
1
- ---*b33*n2
2
m2=---------------
a11
1
- ---*b33*n1
2
m1=---------------
a11
1 2
- ---*b33
4
c33=-------------
a11
c23=0
c22=0
c13=0
c12=0
b32=0
b31=0
b11=0
3 3 1 2
2*a11 *r320 - 2*a11 *r322 - ---*b33 *n2*r464
2
r488=----------------------------------------------
a11*b33*n2
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:
r320, r434, r29, r322, r483, r464, n3, b33, n2, n1, 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.
{b33,n2,a11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2
HAM=(u1 *a11 + u1*a11*n1 + u2 *a11 + u2*a11*n2 + u3*v3*a11*b33 + u3*a11*n3
1 1 1 2 2
- ---*v1*b33*n1 - ---*v2*b33*n2 - ---*v3 *b33 - v3*b33*n3)/a11
2 2 4
has apart from the Hamiltonian and Casimirs the following 6 first integrals:
3 3 2 3 2 2 4 2 3
FI= - u1 *a11 *n1 - u1 *u2*a11 *n2 + u1 *u3 *a11 - u1 *u3*a11 *n3
1 2 2 2 3 1 2
- ---*u1 *v2*a11 *b33*n2 - u1*u2 *a11 *n1 + ---*u1*u2*v1*a11 *b33*n2
2 2
1 2 2 1 2 3
+ ---*u1*u2*v2*a11 *b33*n1 - 2*u1*u2*a11 *n1*n2 - ---*u1*u3 *v1*a11 *b33
2 2
2 3 1 2 1 2
+ u1*u3 *a11 *n1 + ---*u1*u3*v1*a11 *b33*n3 - ---*u1*u3*v3*a11 *b33*n1
2 2
2 1 2 1 2 2
- 2*u1*u3*a11 *n1*n3 + ---*u1*v1*v2*a11*b33 *n2 - ---*u1*v2 *a11*b33 *n1
4 4
3 2 3 3
+ u1*v3*a11*b33*n1*n3 + u1*(a11*n1 - a11*n1*n3 ) - u2 *a11 *n2
2 2 4 2 3 1 2 2
+ u2 *u3 *a11 - u2 *u3*a11 *n3 - ---*u2 *v1*a11 *b33*n1
2
2 2 2 2 2 1 2 3 2 3
+ u2 *(a11 *n1 - a11 *n2 ) - ---*u2*u3 *v2*a11 *b33 + u2*u3 *a11 *n2
2
1 2 1 2 2
+ ---*u2*u3*v2*a11 *b33*n3 - ---*u2*u3*v3*a11 *b33*n2 - 2*u2*u3*a11 *n2*n3
2 2
1 2 2 1 2
- ---*u2*v1 *a11*b33 *n2 + ---*u2*v1*v2*a11*b33 *n1 + u2*v3*a11*b33*n2*n3
4 4
2 2 1 3 3 3 3
+ u2*(a11*n1 *n2 - a11*n2*n3 ) + ---*u3 *v3*a11 *b33 + u3 *a11 *n3
2
1 2 2 2 2 1 2 2 1 2 2 2 2
+ ---*u3 *v1 *a11 *b33 - ---*u3 *v1*a11 *b33*n1 + ---*u3 *v2 *a11 *b33
4 2 4
1 2 2 3 2 2 1 2 2
- ---*u3 *v2*a11 *b33*n2 - ---*u3 *v3*a11 *b33*n3 - ---*u3*v1 *a11*b33 *n3
2 2 4
1 2 1 2 2
- ---*u3*v1*v3*a11*b33 *n1 - ---*u3*v2 *a11*b33 *n3
4 4
1 2 2 2
- ---*u3*v2*v3*a11*b33 *n2 + u3*v3*(a11*b33*n1 + a11*b33*n3 )
4
2 3 1 3 3 1 2 3
+ u3*(a11*n1 *n3 - a11*n3 ) - ---*v1 *b33 *n1 - ---*v1 *v2*b33 *n2
8 8
1 2 3 1 2 1 2
- ---*v1*v2 *b33 *n1 + ---*v1*v2*b33 *n1*n2 + ---*v1*v3*b33 *n1*n3
8 2 2
1 3 1 2 1 3 3
+ v1*( - ---*b33*n1 - ---*b33*n1*n3 ) - ---*v2 *b33 *n2
2 2 8
2 1 2 2 1 2 2 1 2
+ v2 *( - ---*b33 *n1 + ---*b33 *n2 ) + ---*v2*v3*b33 *n2*n3
4 4 2
1 2 1 2 1 2 2 2 2
+ v2*( - ---*b33*n1 *n2 - ---*b33*n2*n3 ) - ---*v3 *b33 *n1 - v3*b33*n1 *n3
2 2 2
{HAM,FI} = 0
1 2 2 2 1 2 2 2 2 2
FI= - ---*u1 *v2 *a11 - ---*u1 *v3 *a11 + u1*u2*v1*v2*a11 + u1*u3*v1*v3*a11
2 2
1 2 2 2 1 2 2 2 2 1 2 2 2
- ---*u2 *v1 *a11 - ---*u2 *v3 *a11 + u2*u3*v2*v3*a11 - ---*u3 *v1 *a11
2 2 2
1 2 2 2 1 2 1 2
- ---*u3 *v2 *a11 - ---*u3*v1 *v3*a11*b33 - ---*u3*v2 *v3*a11*b33
2 2 2
1 3 1 3 1 2 1 2 2 2
- ---*u3*v3 *a11*b33 + ---*v1 *b33*n1 + ---*v1 *v2*b33*n2 + ---*v1 *v3 *b33
2 2 2 8
1 2 1 2 1 2 1 3
+ ---*v1 *v3*b33*n3 + ---*v1*v2 *b33*n1 + ---*v1*v3 *b33*n1 + ---*v2 *b33*n2
2 2 2 2
1 2 2 2 1 2 1 2 1 4 2
+ ---*v2 *v3 *b33 + ---*v2 *v3*b33*n3 + ---*v2*v3 *b33*n2 + ---*v3 *b33
8 2 2 8
1 3
+ ---*v3 *b33*n3
2
{HAM,FI} = 0
2 2 2
FI=2*u1 *v1*a11*n1 + 2*u1*u2*v1*a11*n2 + 2*u1*u2*v2*a11*n1 - 2*u1*u3 *v1*a11
2
+ 2*u1*u3*v1*a11*n3 + 2*u1*u3*v3*a11*n1 + u1*v1 *b33*n1 + u1*v1*v2*b33*n2
2 2 2
+ 2*u2 *v2*a11*n2 - 2*u2*u3 *v2*a11 + 2*u2*u3*v2*a11*n3 + 2*u2*u3*v3*a11*n2
2 3 2 2
+ u2*v1*v2*b33*n1 + u2*v2 *b33*n2 - 2*u3 *v3*a11 + 2*u3 *v3*a11*n3
+ u3*v1*v3*b33*n1 + u3*v2*v3*b33*n2
{HAM,FI} = 0
2 2 1
FI= - u1*a11*n1 - u2*a11*n2 + u3 *a11 - u3*a11*n3 - ---*v1*b33*n1
2
1
- ---*v2*b33*n2
2
{HAM,FI} = 0
2 2 3 2
FI=2*u1*u2*a11 *n1*n2 - 2*u1*u3 *a11 *n1 + 2*u1*u3*a11 *n1*n3
2 2
+ u1*v1*(a11*b33*n1 - a11*b33*n2 ) + u1*v2*a11*b33*n1*n2
3 2 2 2 2 2 2
+ u1*( - a11*n1 + a11*n1*n3 ) + u2 *( - a11 *n1 + a11 *n2 )
2 3 2
- 2*u2*u3 *a11 *n2 + 2*u2*u3*a11 *n2*n3 + u2*v1*a11*b33*n1*n2
2 2 4 4 3 3
+ u2*( - a11*n1 *n2 + a11*n2*n3 ) + u3 *a11 - 2*u3 *a11 *n3
2 2 2 2
- u3 *v1*a11 *b33*n1 - u3 *v2*a11 *b33*n2 + u3*v1*a11*b33*n1*n3
2 2
+ u3*v2*a11*b33*n2*n3 + u3*v3*( - a11*b33*n1 - a11*b33*n2 )
2 3 1 2
+ u3*( - a11*n1 *n3 + a11*n3 ) + ---*v1*v2*b33 *n1*n2
2
1 3 1 2 2 1 2 2 1 2 2
+ v1*(---*b33*n1 + ---*b33*n1*n3 ) + v2 *( - ---*b33 *n1 + ---*b33 *n2 )
2 2 4 4
1 2 1 2 2
+ v2*(---*b33*n1 *n2 + ---*b33*n2*n3 ) + v3*b33*n1 *n3
2 2
{HAM,FI} = 0
2 2 2 2
FI= - 2*u1 *v1*a11 *n1 - 2*u1*u2*v1*a11 *n2 - 2*u1*u2*v2*a11 *n1
2 3 2 2
+ 2*u1*u3 *v1*a11 - 2*u1*u3*v1*a11 *n3 - 2*u1*u3*v3*a11 *n1
2 2
- u1*v1*v2*a11*b33*n2 + u1*v2 *a11*b33*n1 + u1*v3 *a11*b33*n1
2 2 2 3 2
- 2*u2 *v2*a11 *n2 + 2*u2*u3 *v2*a11 - 2*u2*u3*v2*a11 *n3
2 2
- 2*u2*u3*v3*a11 *n2 + u2*v1 *a11*b33*n2 - u2*v1*v2*a11*b33*n1
2 3 3 2 2 2 2 2 2
+ u2*v3 *a11*b33*n2 + 2*u3 *v3*a11 - u3 *v1 *a11 *b33 - u3 *v2 *a11 *b33
2 2 2 2 2 2
- u3 *v3 *a11 *b33 - 2*u3 *v3*a11 *n3 + u3*v1 *a11*b33*n3
2
- u3*v1*v3*a11*b33*n1 + u3*v2 *a11*b33*n3 - u3*v2*v3*a11*b33*n2
2 1 3 2 1 2 2
+ u3*v3 *a11*b33*n3 + ---*v1 *b33 *n1 + ---*v1 *v2*b33 *n2
2 2
1 2 2 1 2 2 1 3 2
+ ---*v1*v2 *b33 *n1 + ---*v1*v3 *b33 *n1 + ---*v2 *b33 *n2
2 2 2
1 2 2
+ ---*v2*v3 *b33 *n2
2
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a11**2 + u1*a11*n1 + u2**2*a11**2 + u2*a11*n2 + u3*v3*a11*b33 + u3*
a11*n3 - 1/2*v1*b33*n1 - 1/2*v2*b33*n2 - 1/4*v3**2*b33**2 - v3*b33*n3)/a11$
FI= - u1**3*a11**3*n1 - u1**2*u2*a11**3*n2 + u1**2*u3**2*a11**4 - u1**2*u3*a11**
3*n3 - 1/2*u1**2*v2*a11**2*b33*n2 - u1*u2**2*a11**3*n1 + 1/2*u1*u2*v1*a11**2*b33
*n2 + 1/2*u1*u2*v2*a11**2*b33*n1 - 2*u1*u2*a11**2*n1*n2 - 1/2*u1*u3**2*v1*a11**3
*b33 + u1*u3**2*a11**3*n1 + 1/2*u1*u3*v1*a11**2*b33*n3 - 1/2*u1*u3*v3*a11**2*b33
*n1 - 2*u1*u3*a11**2*n1*n3 + 1/4*u1*v1*v2*a11*b33**2*n2 - 1/4*u1*v2**2*a11*b33**
2*n1 + u1*v3*a11*b33*n1*n3 + u1*(a11*n1**3 - a11*n1*n3**2) - u2**3*a11**3*n2 +
u2**2*u3**2*a11**4 - u2**2*u3*a11**3*n3 - 1/2*u2**2*v1*a11**2*b33*n1 + u2**2*(
a11**2*n1**2 - a11**2*n2**2) - 1/2*u2*u3**2*v2*a11**3*b33 + u2*u3**2*a11**3*n2 +
1/2*u2*u3*v2*a11**2*b33*n3 - 1/2*u2*u3*v3*a11**2*b33*n2 - 2*u2*u3*a11**2*n2*n3
- 1/4*u2*v1**2*a11*b33**2*n2 + 1/4*u2*v1*v2*a11*b33**2*n1 + u2*v3*a11*b33*n2*n3
+ u2*(a11*n1**2*n2 - a11*n2*n3**2) + 1/2*u3**3*v3*a11**3*b33 + u3**3*a11**3*n3 +
1/4*u3**2*v1**2*a11**2*b33**2 - 1/2*u3**2*v1*a11**2*b33*n1 + 1/4*u3**2*v2**2*
a11**2*b33**2 - 1/2*u3**2*v2*a11**2*b33*n2 - 3/2*u3**2*v3*a11**2*b33*n3 - 1/4*u3
*v1**2*a11*b33**2*n3 - 1/4*u3*v1*v3*a11*b33**2*n1 - 1/4*u3*v2**2*a11*b33**2*n3 -
1/4*u3*v2*v3*a11*b33**2*n2 + u3*v3*(a11*b33*n1**2 + a11*b33*n3**2) + u3*(a11*n1
**2*n3 - a11*n3**3) - 1/8*v1**3*b33**3*n1 - 1/8*v1**2*v2*b33**3*n2 - 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 + v2**2*( - 1/4*b33**2*n1**2 + 1
/4*b33**2*n2**2) + 1/2*v2*v3*b33**2*n2*n3 + v2*( - 1/2*b33*n1**2*n2 - 1/2*b33*n2
*n3**2) - 1/2*v3**2*b33**2*n1**2 - v3*b33*n1**2*n3$
FI= - 1/2*u1**2*v2**2*a11**2 - 1/2*u1**2*v3**2*a11**2 + u1*u2*v1*v2*a11**2 + u1*
u3*v1*v3*a11**2 - 1/2*u2**2*v1**2*a11**2 - 1/2*u2**2*v3**2*a11**2 + u2*u3*v2*v3*
a11**2 - 1/2*u3**2*v1**2*a11**2 - 1/2*u3**2*v2**2*a11**2 - 1/2*u3*v1**2*v3*a11*
b33 - 1/2*u3*v2**2*v3*a11*b33 - 1/2*u3*v3**3*a11*b33 + 1/2*v1**3*b33*n1 + 1/2*v1
**2*v2*b33*n2 + 1/8*v1**2*v3**2*b33**2 + 1/2*v1**2*v3*b33*n3 + 1/2*v1*v2**2*b33*
n1 + 1/2*v1*v3**2*b33*n1 + 1/2*v2**3*b33*n2 + 1/8*v2**2*v3**2*b33**2 + 1/2*v2**2
*v3*b33*n3 + 1/2*v2*v3**2*b33*n2 + 1/8*v3**4*b33**2 + 1/2*v3**3*b33*n3$
FI=2*u1**2*v1*a11*n1 + 2*u1*u2*v1*a11*n2 + 2*u1*u2*v2*a11*n1 - 2*u1*u3**2*v1*a11
**2 + 2*u1*u3*v1*a11*n3 + 2*u1*u3*v3*a11*n1 + u1*v1**2*b33*n1 + u1*v1*v2*b33*n2
+ 2*u2**2*v2*a11*n2 - 2*u2*u3**2*v2*a11**2 + 2*u2*u3*v2*a11*n3 + 2*u2*u3*v3*a11*
n2 + u2*v1*v2*b33*n1 + u2*v2**2*b33*n2 - 2*u3**3*v3*a11**2 + 2*u3**2*v3*a11*n3 +
u3*v1*v3*b33*n1 + u3*v2*v3*b33*n2$
FI= - u1*a11*n1 - u2*a11*n2 + u3**2*a11**2 - u3*a11*n3 - 1/2*v1*b33*n1 - 1/2*v2*
b33*n2$
FI=2*u1*u2*a11**2*n1*n2 - 2*u1*u3**2*a11**3*n1 + 2*u1*u3*a11**2*n1*n3 + u1*v1*(
a11*b33*n1**2 - a11*b33*n2**2) + u1*v2*a11*b33*n1*n2 + u1*( - a11*n1**3 + a11*n1
*n3**2) + u2**2*( - a11**2*n1**2 + a11**2*n2**2) - 2*u2*u3**2*a11**3*n2 + 2*u2*
u3*a11**2*n2*n3 + u2*v1*a11*b33*n1*n2 + u2*( - a11*n1**2*n2 + a11*n2*n3**2) + u3
**4*a11**4 - 2*u3**3*a11**3*n3 - u3**2*v1*a11**2*b33*n1 - u3**2*v2*a11**2*b33*n2
+ u3*v1*a11*b33*n1*n3 + u3*v2*a11*b33*n2*n3 + u3*v3*( - a11*b33*n1**2 - a11*b33
*n2**2) + u3*( - a11*n1**2*n3 + a11*n3**3) + 1/2*v1*v2*b33**2*n1*n2 + v1*(1/2*
b33*n1**3 + 1/2*b33*n1*n3**2) + v2**2*( - 1/4*b33**2*n1**2 + 1/4*b33**2*n2**2) +
v2*(1/2*b33*n1**2*n2 + 1/2*b33*n2*n3**2) + v3*b33*n1**2*n3$
FI= - 2*u1**2*v1*a11**2*n1 - 2*u1*u2*v1*a11**2*n2 - 2*u1*u2*v2*a11**2*n1 + 2*u1*
u3**2*v1*a11**3 - 2*u1*u3*v1*a11**2*n3 - 2*u1*u3*v3*a11**2*n1 - u1*v1*v2*a11*b33
*n2 + u1*v2**2*a11*b33*n1 + u1*v3**2*a11*b33*n1 - 2*u2**2*v2*a11**2*n2 + 2*u2*u3
**2*v2*a11**3 - 2*u2*u3*v2*a11**2*n3 - 2*u2*u3*v3*a11**2*n2 + u2*v1**2*a11*b33*
n2 - u2*v1*v2*a11*b33*n1 + u2*v3**2*a11*b33*n2 + 2*u3**3*v3*a11**3 - u3**2*v1**2
*a11**2*b33 - u3**2*v2**2*a11**2*b33 - u3**2*v3**2*a11**2*b33 - 2*u3**2*v3*a11**
2*n3 + u3*v1**2*a11*b33*n3 - u3*v1*v3*a11*b33*n1 + u3*v2**2*a11*b33*n3 - u3*v2*
v3*a11*b33*n2 + u3*v3**2*a11*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$