Solution 20 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
r13=0
r14=0
r15=0
1
2*a33*c12*r212 - ---*b11*b21*r212
4
r20=-----------------------------------
a33*b11
r21=0
r22=0
r23=0
1
---*b11*r212
4
r24=--------------
a33
r26=0
r27=0
r28=0
r210=0
r213=0
r214=0
r215=0
r216= - r212
r217=0
r218=0
r219=0
r220=0
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=0
r311=0
r312=0
r313=0
r314=0
r315=0
r316=0
r317=0
r318=0
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
r326=0
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
3 3 2 1 2 2
r40=(2*a33 *b11*c12*r441 - 2*a33 *c12 *r494 - ---*a33 *b11 *b21*r441
4
1 2 2 1 2
- ---*a33 *b11 *c12*r448 + ---*a33 *b11*b21*c12*r494
2 2
1 3 3 3 1 2 2
+ ----*a33*b11 *b21*r448 + ---*a33*b11 *c12*r453 - ----*a33*b11 *b21 *r494
16 8 32
3 4 3 2
- ----*b11 *b21*r453)/(a33 *b11 )
64
r41=0
3 1 2 1 2
r42=(2*a33 *c12*r441 - ---*a33 *b11*b21*r441 - ---*a33 *b11*c12*r448
4 2
1 2 1 2 1 3
+ ----*a33*b11 *b21*r448 + ---*a33*b11 *c12*r453 - ----*b11 *b21*r453)/(
16 4 32
3
a33 *b11)
r43=0
r44=0
r45=0
1 2 2 1 2 2 2
r46=(---*a33 *b11 *r441 - ---*a33 *b11*c12*r494 + 4*a33 *c12 *r453
4 2
1 3 1 2
- ----*a33*b11 *r448 + ----*a33*b11 *b21*r494 - a33*b11*b21*c12*r453
16 16
3 4 1 2 2 3
+ ----*b11 *r453 + ----*b11 *b21 *r453)/(a33 *b11)
64 16
r47=0
1 2 1 2 1 3
---*a33 *b11*r441 - ----*a33*b11 *r448 + ----*b11 *r453
4 16 32
r48=---------------------------------------------------------
3
a33
3 1 2 1 2
r49=(2*a33 *c12*r441 - ---*a33 *b11*b21*r441 - ---*a33 *b11*c12*r448
4 2
1 2 1 2 1 3
+ ----*a33*b11 *b21*r448 + ---*a33*b11 *c12*r453 - ----*b11 *b21*r453)/(
16 4 32
3
a33 *b11)
r410=0
1 2 1 1 2
- ----*a33*b11 *r494 + ---*a33*b11*c12*r453 - ----*b11 *b21*r453
32 2 16
r411=-------------------------------------------------------------------
3
a33
r412=0
1 2 1 2 1 3
---*a33 *b11*r441 - ----*a33*b11 *r448 + ----*b11 *r453
4 16 32
r413=---------------------------------------------------------
3
a33
2 1 2 1
r415=(2*a33 *c12*r448 + ---*a33*b11 *r494 - ---*a33*b11*b21*r448
4 4
1 2 2
- 2*a33*b11*c12*r453 + ---*b11 *b21*r453)/(a33 *b11)
4
r416=0
1 1
---*a33*b11*r494 - 2*a33*c12*r453 + ---*b11*b21*r453
4 4
r417=------------------------------------------------------
2
a33
r418=0
r419=0
1
---*b11*r448
4
r420=--------------
a33
r421=0
1 1
---*a33*b11*r494 - 2*a33*c12*r453 + ---*b11*b21*r453
4 4
r422=------------------------------------------------------
2
a33
r423=0
r424=0
1
2*a33*c12*r453 - ---*b11*b21*r453
4
r425=-----------------------------------
a33*b11
r426=0
r427=0
r428=0
1
---*b11*r453
4
r429=--------------
a33
r431=0
r432=0
r433=0
r435=0
2 1 2 1
r436=(2*a33 *c12*r448 + ---*a33*b11 *r494 - ---*a33*b11*b21*r448
4 4
3 2 2
- 3*a33*b11*c12*r453 + ---*b11 *b21*r453)/(a33 *b11)
8
r437=0
1 1
---*a33*b11*r494 - 2*a33*c12*r453 + ---*b11*b21*r453
4 4
r438=------------------------------------------------------
2
a33
2 2 1 2 1
r439=(a33 *b11*r441 - 2*a33 *c12*r494 - ---*a33*b11 *r448 + ---*a33*b11*b21*r494
4 4
3 3 2
+ ----*b11 *r453)/(a33 *b11)
16
r440=0
r442=0
2 1 1 2
a33 *r441 - ---*a33*b11*r448 + ---*b11 *r453
4 8
r444=----------------------------------------------
2
a33
r445=0
r446=0
r447=0
r449=0
r450=0
r451=0
r454=0
r455=0
r456=0
r458=0
1
a33*r448 - ---*b11*r453
2
r459=-------------------------
a33
1
r460= - ---*r494
2
r461=0
r462=0
r463=0
r464=0
r465=0
r466=0
r467=0
r468=0
r469=0
r470=0
2 2 1 2
r471=( - a33 *b11*r441 + 2*a33 *c12*r494 + ---*a33*b11 *r448
4
1 3 3 2
- ---*a33*b11*b21*r494 - ----*b11 *r453)/(a33 *b11)
4 16
r472=0
2 1 1 2
- a33 *r441 + ---*a33*b11*r448 - ---*b11 *r453
4 8
r473=-------------------------------------------------
2
a33
2 1 2 1
r474=(2*a33 *c12*r448 + ---*a33*b11 *r494 - ---*a33*b11*b21*r448
4 4
1 2 2
- a33*b11*c12*r453 + ---*b11 *b21*r453)/(a33 *b11)
8
r475=0
1 1
---*a33*b11*r494 - 4*a33*c12*r453 + ---*b11*b21*r453
2 2
r476=------------------------------------------------------
2
a33
r477=0
1
- a33*r441 + ---*b11*r448
2
r478=----------------------------
a33
1 1
---*a33*b11*r494 - 2*a33*c12*r453 + ---*b11*b21*r453
4 4
r479=------------------------------------------------------
2
a33
r480=0
r481= - r448
r482=0
r483=0
r484=0
r485=0
r486=0
r487= - r453
r488=0
r489=0
r490=0
r491=0
1
- a33*r448 + ---*b11*r453
2
r492=----------------------------
a33
r493=0
1
a33*r448 + ---*b11*r453
2
r495=-------------------------
a33
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
1
r4107= - ---*r494
2
r4108=0
1
- a33*r448 - ---*b11*r453
2
r4109=----------------------------
a33
r4110=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m3=0
m2=0
m1=0
n3=0
n2=0
n1=0
1 3 1 2
2*a33*b21*c12 + ----*b11 - ---*b11*b21
16 4
c33=------------------------------------------
a33*b11
c23=0
c22=0
c13=0
c11=0
1
b33=---*b11
2
b32=0
b31=0
b23=0
b22=0
b13=0
b12= - b21
a23=0
a22=0
a13=0
a12=0
a11=0
Parameters
Apart from the condition that they must not vanish to give
a non-trivial solution and a non-singular solution with
non-vanishing denominators, the following parameters are free:
r212, r441, r448, r494, r453, b11, b21, 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.
{a33,b21,b11}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11,
a12,
a13,
a22,
a23,
b12 + b21,
b13,
b22,
b23,
b31,
b32,
- b11 + 2*b33,
c11,
c13,
c22,
c23,
16*a33*b11*c33 - 32*a33*b21*c12 - b11**3 + 4*b11*b21**2,
n1,
n2,
n3,
m1,
m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2 u3*v3*b11
HAM=u1*v1*b11 - u1*v2*b21 + u2*v1*b21 + u3 *a33 + ----------- + 2*v1*v2*c12
2
3 2
2 32*a33*b21*c12 + b11 - 4*b11*b21
+ v3 *------------------------------------
16*a33*b11
has apart from the Hamiltonian and Casimirs the following 5 first integrals:
2 2 2 2 2 2 2 2 2
FI= - 32*u1 *v1*v2*a33 *b11 + 32*u1*u2*v1 *a33 *b11 + 32*u1*u2*v2 *a33 *b11
2 3 3 2 2
- 64*u1*u3 *v2*a33 *b11 + u1*v1 *( - 128*a33 *b11*c12 + 16*a33*b11 *b21)
2 2 2
+ u1*v1*v2 *( - 256*a33 *b11*c12 + 32*a33*b11 *b21)
2 2 2 3 3
+ u1*v1*v3 *( - 64*a33 *b11*c12 + 8*a33*b11 *b21) - 8*u1*v2 *a33*b11
2 3 2 2 2 2 3
- 12*u1*v2*v3 *a33*b11 - 32*u2 *v1*v2*a33 *b11 + 64*u2*u3 *v1*a33 *b11
3 3 2 3
+ 8*u2*v1 *a33*b11 + 12*u2*v1*v3 *a33*b11
3 2 2
+ u2*v2 *( - 128*a33 *b11*c12 + 16*a33*b11 *b21)
2 2 2 2 2 2
+ u2*v2*v3 *( - 192*a33 *b11*c12 + 24*a33*b11 *b21) + 16*u3 *v1*v2*a33 *b11
2 2 3 2
+ u3 *v3 *(128*a33 *c12 - 16*a33 *b11*b21)
2 2 2
+ u3*v1 *v3*( - 128*a33 *b11*c12 + 16*a33*b11 *b21)
2 2 2
+ u3*v2 *v3*( - 128*a33 *b11*c12 + 16*a33*b11 *b21)
3 2 2 3 4
+ u3*v3 *( - 128*a33 *b11*c12 + 16*a33*b11 *b21) + 2*v1 *v2*b11
2 2 2 3
+ v1 *v2 *(32*a33*b11 *c12 - 4*b11 *b21)
2 2 2 3 3 4
+ v1 *v3 *(16*a33*b11 *c12 - 2*b11 *b21) + 2*v1*v2 *b11
2 2 2 4 2 2
+ v1*v2*v3 *(256*a33 *c12 - 64*a33*b11*b21*c12 + 3*b11 + 4*b11 *b21 )
2 2 2 3
+ v2 *v3 *(16*a33*b11 *c12 - 2*b11 *b21)
4 2 3
+ v3 *(24*a33*b11 *c12 - 3*b11 *b21)
= a product of the elements of: {64,
2 2 2 2 2 2 2 2 2
- u1 *v1*v2*a33 *b11 u1*u2*v1 *a33 *b11 u1*u2*v2 *a33 *b11
------------------------ + --------------------- + ---------------------
2 2 2
2 2
2 3 3 - 8*a33 *b11*c12 + a33*b11 *b21
- u1*u3 *v2*a33 *b11 + u1*v1 *----------------------------------
4
2 2
2 - 8*a33 *b11*c12 + a33*b11 *b21
+ u1*v1*v2 *----------------------------------
2
2 2 3 3
2 - 8*a33 *b11*c12 + a33*b11 *b21 - u1*v2 *a33*b11
+ u1*v1*v3 *---------------------------------- + --------------------
8 8
2 3 2 2 2
- 3*u1*v2*v3 *a33*b11 - u2 *v1*v2*a33 *b11 2 3
+ ------------------------- + ------------------------ + u2*u3 *v1*a33 *b11
16 2
3 3 2 3
u2*v1 *a33*b11 3*u2*v1*v3 *a33*b11
+ ----------------- + ----------------------
8 16
2 2
3 - 8*a33 *b11*c12 + a33*b11 *b21
+ u2*v2 *----------------------------------
4
2 2 2 2 2
2 - 24*a33 *b11*c12 + 3*a33*b11 *b21 u3 *v1*v2*a33 *b11
+ u2*v2*v3 *------------------------------------- + ---------------------
8 4
3 2
2 2 8*a33 *c12 - a33 *b11*b21
+ u3 *v3 *---------------------------
4
2 2
2 - 8*a33 *b11*c12 + a33*b11 *b21
+ u3*v1 *v3*----------------------------------
4
2 2
2 - 8*a33 *b11*c12 + a33*b11 *b21
+ u3*v2 *v3*----------------------------------
4
2 2 3 4
3 - 8*a33 *b11*c12 + a33*b11 *b21 v1 *v2*b11
+ u3*v3 *---------------------------------- + -------------
4 32
2 3 2 3
2 2 8*a33*b11 *c12 - b11 *b21 2 2 8*a33*b11 *c12 - b11 *b21
+ v1 *v2 *--------------------------- + v1 *v3 *---------------------------
16 32
3 4
v1*v2 *b11
+ -------------
32
2 2 4 2 2
2 256*a33 *c12 - 64*a33*b11*b21*c12 + 3*b11 + 4*b11 *b21
+ v1*v2*v3 *-----------------------------------------------------------
64
2 3 2 3
2 2 8*a33*b11 *c12 - b11 *b21 4 24*a33*b11 *c12 - 3*b11 *b21
+ v2 *v3 *--------------------------- + v3 *------------------------------}
32 64
{HAM,FI} = 0
2 2 2 2 2 2 3 3
FI= - 16*u1 *v2 *a33 *b11 + 32*u1*u2*v1*v2*a33 *b11 + 8*u1*v1 *a33*b11
2 3 2 3
+ 16*u1*v1*v2 *a33*b11 + 8*u1*v1*v3 *a33*b11
2 2 2 2 2 2 2
+ u1*v2*v3 *(64*a33 *b11*c12 - 8*a33*b11 *b21) - 16*u2 *v1 *a33 *b11
2 2 2 3 3
+ u2*v1*v3 *( - 64*a33 *b11*c12 + 8*a33*b11 *b21) + 8*u2*v2 *a33*b11
2 3 2 3 2 3
+ 8*u2*v2*v3 *a33*b11 + 8*u3*v1 *v3*a33*b11 + 8*u3*v2 *v3*a33*b11
3 3 2 2 4
+ 8*u3*v3 *a33*b11 - v1 *v2 *b11
2 2 3
+ v1*v2*v3 *( - 16*a33*b11 *c12 + 2*b11 *b21)
4 2 2 2 2
+ v3 *( - 64*a33 *c12 + 16*a33*b11*b21*c12 - b11 *b21 )
= a product of the elements of: {16,
3 3
2 2 2 2 2 2 u1*v1 *a33*b11
- u1 *v2 *a33 *b11 + 2*u1*u2*v1*v2*a33 *b11 + -----------------
2
2 3
2 3 u1*v1*v3 *a33*b11
+ u1*v1*v2 *a33*b11 + --------------------
2
2 2
2 8*a33 *b11*c12 - a33*b11 *b21 2 2 2 2
+ u1*v2*v3 *------------------------------- - u2 *v1 *a33 *b11
2
2 2 3 3
2 - 8*a33 *b11*c12 + a33*b11 *b21 u2*v2 *a33*b11
+ u2*v1*v3 *---------------------------------- + -----------------
2 2
2 3 2 3 2 3
u2*v2*v3 *a33*b11 u3*v1 *v3*a33*b11 u3*v2 *v3*a33*b11
+ -------------------- + -------------------- + --------------------
2 2 2
3 3 2 2 4
u3*v3 *a33*b11 - v1 *v2 *b11
+ ----------------- + -----------------
2 16
2 3
2 - 8*a33*b11 *c12 + b11 *b21
+ v1*v2*v3 *------------------------------
8
2 2 2 2
4 - 64*a33 *c12 + 16*a33*b11*b21*c12 - b11 *b21
+ v3 *--------------------------------------------------}
16
{HAM,FI} = 0
2 2 2 2 2 2
FI= - 16*u1 *v1*v2*a33 *b11 + 16*u1*u2*v1 *a33 *b11 - 16*u1*u2*v2 *a33 *b11
2 2 2
- 16*u1*u3*v2*v3*a33 *b11 + 8*u1*v1 *v2*a33*b11
2 2 3 2
+ u1*v1*v3 *(32*a33 *c12 - 4*a33*b11*b21) + 4*u1*v2 *a33*b11
2 2 2 2 2
+ 4*u1*v2*v3 *a33*b11 + 16*u2 *v1*v2*a33 *b11 + 16*u2*u3*v1*v3*a33 *b11
3 2 2 2
- 4*u2*v1 *a33*b11 - 4*u2*v1*v3 *a33*b11
2 2 2
+ u2*v2*v3 *(32*a33 *c12 - 4*a33*b11*b21) + 4*u3*v1*v2*v3*a33*b11
3 2 3 3
+ u3*v3 *(32*a33 *c12 - 4*a33*b11*b21) - v1 *v2*b11
2 2 2 3 3 2 3
+ v1 *v3 *( - 8*a33*b11*c12 + b11 *b21) - v1*v2 *b11 - v1*v2*v3 *b11
2 2 2 4 2
+ v2 *v3 *( - 8*a33*b11*c12 + b11 *b21) + v3 *( - 8*a33*b11*c12 + b11 *b21)
= a product of the elements of: {16,
2 2 2
v1 *b11 v2 *b11 v3 *b11
- u1*v1*a33 - u2*v2*a33 - u3*v3*a33 + --------- + --------- + ---------,
4 4 4
2
- v1*v2*b11
u1*v2*a33*b11 - u2*v1*a33*b11 + ---------------
4
2 - 8*a33*c12 + b11*b21
+ v3 *------------------------}
4
{HAM,FI} = 0
2 3 2
FI= - 4*u1*v1 *v2*a33*b11 - 4*u1*v2 *a33*b11 - 4*u1*v2*v3 *a33*b11
3 2 2 3 2
+ 4*u2*v1 *a33*b11 + 4*u2*v1*v2 *a33*b11 + 4*u2*v1*v3 *a33*b11 + v1 *v2*b11
2 2 3 2 2 2
+ v1 *v3 *(8*a33*c12 - b11*b21) + v1*v2 *b11 + v1*v2*v3 *b11
2 2 4
+ v2 *v3 *(8*a33*c12 - b11*b21) + v3 *(8*a33*c12 - b11*b21)
= a product of the elements of: {4,
2 2 2
- v1 - v2 - v3 ,
2
- v1*v2*b11
u1*v2*a33*b11 - u2*v1*a33*b11 + ---------------
4
2 - 8*a33*c12 + b11*b21
+ v3 *------------------------}
4
{HAM,FI} = 0
2 2
FI= - 4*u1*v2*a33*b11 + 4*u2*v1*a33*b11 + v1*v2*b11 + v3 *(8*a33*c12 - b11*b21)
= a product of the elements of: {4,
2
v1*v2*b11 2 8*a33*c12 - b11*b21
- u1*v2*a33*b11 + u2*v1*a33*b11 + ------------ + v3 *---------------------}
4 4
{HAM,FI} = 0
And again in machine readable form:
HAM=u1*v1*b11 - u1*v2*b21 + u2*v1*b21 + u3**2*a33 + (u3*v3*b11)/2 + 2*v1*v2*c12
+ v3**2*(32*a33*b21*c12 + b11**3 - 4*b11*b21**2)/(16*a33*b11)$
FI= - 32*u1**2*v1*v2*a33**2*b11**2 + 32*u1*u2*v1**2*a33**2*b11**2 + 32*u1*u2*v2
**2*a33**2*b11**2 - 64*u1*u3**2*v2*a33**3*b11 + u1*v1**3*( - 128*a33**2*b11*c12
+ 16*a33*b11**2*b21) + u1*v1*v2**2*( - 256*a33**2*b11*c12 + 32*a33*b11**2*b21) +
u1*v1*v3**2*( - 64*a33**2*b11*c12 + 8*a33*b11**2*b21) - 8*u1*v2**3*a33*b11**3 -
12*u1*v2*v3**2*a33*b11**3 - 32*u2**2*v1*v2*a33**2*b11**2 + 64*u2*u3**2*v1*a33**
3*b11 + 8*u2*v1**3*a33*b11**3 + 12*u2*v1*v3**2*a33*b11**3 + u2*v2**3*( - 128*a33
**2*b11*c12 + 16*a33*b11**2*b21) + u2*v2*v3**2*( - 192*a33**2*b11*c12 + 24*a33*
b11**2*b21) + 16*u3**2*v1*v2*a33**2*b11**2 + u3**2*v3**2*(128*a33**3*c12 - 16*
a33**2*b11*b21) + u3*v1**2*v3*( - 128*a33**2*b11*c12 + 16*a33*b11**2*b21) + u3*
v2**2*v3*( - 128*a33**2*b11*c12 + 16*a33*b11**2*b21) + u3*v3**3*( - 128*a33**2*
b11*c12 + 16*a33*b11**2*b21) + 2*v1**3*v2*b11**4 + v1**2*v2**2*(32*a33*b11**2*
c12 - 4*b11**3*b21) + v1**2*v3**2*(16*a33*b11**2*c12 - 2*b11**3*b21) + 2*v1*v2**
3*b11**4 + v1*v2*v3**2*(256*a33**2*c12**2 - 64*a33*b11*b21*c12 + 3*b11**4 + 4*
b11**2*b21**2) + v2**2*v3**2*(16*a33*b11**2*c12 - 2*b11**3*b21) + v3**4*(24*a33*
b11**2*c12 - 3*b11**3*b21)$
FI= - 16*u1**2*v2**2*a33**2*b11**2 + 32*u1*u2*v1*v2*a33**2*b11**2 + 8*u1*v1**3*
a33*b11**3 + 16*u1*v1*v2**2*a33*b11**3 + 8*u1*v1*v3**2*a33*b11**3 + u1*v2*v3**2*
(64*a33**2*b11*c12 - 8*a33*b11**2*b21) - 16*u2**2*v1**2*a33**2*b11**2 + u2*v1*v3
**2*( - 64*a33**2*b11*c12 + 8*a33*b11**2*b21) + 8*u2*v2**3*a33*b11**3 + 8*u2*v2*
v3**2*a33*b11**3 + 8*u3*v1**2*v3*a33*b11**3 + 8*u3*v2**2*v3*a33*b11**3 + 8*u3*v3
**3*a33*b11**3 - v1**2*v2**2*b11**4 + v1*v2*v3**2*( - 16*a33*b11**2*c12 + 2*b11
**3*b21) + v3**4*( - 64*a33**2*c12**2 + 16*a33*b11*b21*c12 - b11**2*b21**2)$
FI= - 16*u1**2*v1*v2*a33**2*b11 + 16*u1*u2*v1**2*a33**2*b11 - 16*u1*u2*v2**2*a33
**2*b11 - 16*u1*u3*v2*v3*a33**2*b11 + 8*u1*v1**2*v2*a33*b11**2 + u1*v1*v3**2*(32
*a33**2*c12 - 4*a33*b11*b21) + 4*u1*v2**3*a33*b11**2 + 4*u1*v2*v3**2*a33*b11**2
+ 16*u2**2*v1*v2*a33**2*b11 + 16*u2*u3*v1*v3*a33**2*b11 - 4*u2*v1**3*a33*b11**2
- 4*u2*v1*v3**2*a33*b11**2 + u2*v2*v3**2*(32*a33**2*c12 - 4*a33*b11*b21) + 4*u3*
v1*v2*v3*a33*b11**2 + u3*v3**3*(32*a33**2*c12 - 4*a33*b11*b21) - v1**3*v2*b11**3
+ v1**2*v3**2*( - 8*a33*b11*c12 + b11**2*b21) - v1*v2**3*b11**3 - v1*v2*v3**2*
b11**3 + v2**2*v3**2*( - 8*a33*b11*c12 + b11**2*b21) + v3**4*( - 8*a33*b11*c12 +
b11**2*b21)$
FI= - 4*u1*v1**2*v2*a33*b11 - 4*u1*v2**3*a33*b11 - 4*u1*v2*v3**2*a33*b11 + 4*u2*
v1**3*a33*b11 + 4*u2*v1*v2**2*a33*b11 + 4*u2*v1*v3**2*a33*b11 + v1**3*v2*b11**2
+ v1**2*v3**2*(8*a33*c12 - b11*b21) + v1*v2**3*b11**2 + v1*v2*v3**2*b11**2 + v2
**2*v3**2*(8*a33*c12 - b11*b21) + v3**4*(8*a33*c12 - b11*b21)$
FI= - 4*u1*v2*a33*b11 + 4*u2*v1*a33*b11 + v1*v2*b11**2 + v3**2*(8*a33*c12 - b11*
b21)$