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*r216 + 4*a22*n1 *n3*r311 - 4*n1 *n2*n3*r425
r10=-----------------------------------------------------------
2
a22 *b33*n2
1 2 1 2 3
r11=(---*a22*b33*n1 *r216 + ---*a22*b33*n3 *r216 + 2*a22*n1 *r311
2 2
2 1 2 3
+ 2*a22*n1*n3 *r311 - ---*b33 *n1*n2*r29 - 2*n1 *n2*r425
2
2 2
- 2*n1*n2*n3 *r425)/(a22 *b33*n1)
1 2 1 2 3
r12=(---*a22*b33*n1 *r216 + ---*a22*b33*n3 *r216 + 2*a22*n1 *r311
2 2
2 1 2 3
+ 2*a22*n1*n3 *r311 - ---*b33 *n1*n2*r29 - 2*n1 *n2*r425
2
2 2
- 2*n1*n2*n3 *r425)/(a22 *b33*n2)
2 3 3
r13=( - a22*b33*n1 *n3*r216 + a22*b33*n3 *r216 - 4*a22*n1 *n3*r311
3 2 3
+ 4*a22*n1*n3 *r311 - b33 *n1*n2*n3*r29 + 4*n1 *n2*n3*r425
3 2
- 4*n1*n2*n3 *r425)/(a22*b33 *n1*n2)
2 2 3 2
r14=( - a22*b33*n1 *r216 + a22*b33*n3 *r216 - 4*a22*n1 *r311 + 4*a22*n1*n3 *r311
2 3 2 2
- b33 *n1*n2*r29 + 4*n1 *n2*r425 - 4*n1*n2*n3 *r425)/(a22*b33 *n1)
2 2 3 2
r15=( - a22*b33*n1 *r216 + a22*b33*n3 *r216 - 4*a22*n1 *r311 + 4*a22*n1*n3 *r311
2 3 2 2
- b33 *n1*n2*r29 + 4*n1 *n2*r425 - 4*n1*n2*n3 *r425)/(a22*b33 *n2)
1 2 1 2 3
r20=(---*a22*b33*n1 *r216 - ---*a22*b33*n2 *r216 + a22*n1 *r311
4 4
2 3 3 2
+ a22*n1*n2 *r311 - n1 *n2*r425 - n1*n2 *r425)/(a22 *n1*n2)
- 2*a22*n3*r311 + 2*n2*n3*r425
r21=---------------------------------
2
a22
- 2*a22*n1*n3*r311 + 2*n1*n2*n3*r425
r23=---------------------------------------
2
a22 *n2
1
---*a22*b33*r216 - 2*a22*n1*r311 + 2*n1*n2*r425
2
r24=-------------------------------------------------
2
a22
1 2 1 2 3
r25=(---*a22*b33*n1 *r216 - ---*a22*b33*n2 *r216 - a22*n1 *r311
4 4
2 3 3 2
+ a22*n1*n2 *r311 + n1 *n2*r425 - n1*n2 *r425)/(a22 *n1*n2)
2 2 3 2
r26=( - a22*b33*n1 *r216 - a22*b33*n2 *r216 - 4*a22*n1 *r311 - 4*a22*n1*n3 *r311
3 2
+ 4*n1 *n2*r425 + 4*n1*n2*n3 *r425)/(a22*b33*n1*n2)
n3*r216
r27=---------
n1
n3*r216
r28=---------
n2
- 4*a22*n3*r311 + 4*n2*n3*r425
r210=---------------------------------
a22*b33
r212=r216
2*a22*b33*n3*r216 + 8*a22*n1*n3*r311 - 8*n1*n2*n3*r425
r213=--------------------------------------------------------
2
b33 *n1
2 2 3
r214=( - a22*b33*n1 *r216 + a22*b33*n2 *r216 - 4*a22*n1 *r311
2 3 3 2
+ 4*a22*n1*n2 *r311 + 4*n1 *n2*r425 - 4*n1*n2 *r425)/(b33 *n1*n2)
- 4*a22*n1*n3*r311 + 4*n1*n2*n3*r425
r215=---------------------------------------
a22*b33*n2
2 2
n1 *r216 - n2 *r216
r217=---------------------
n1*n2
2*a22*b33*n3*r216 + 8*a22*n1*n3*r311 - 8*n1*n2*n3*r425
r218=--------------------------------------------------------
2
b33 *n2
2*a22*b33*r216 + 8*a22*n1*r311 - 8*n1*n2*r425
r219=-----------------------------------------------
2
b33
r30=0
1
- ---*b33*n2*r425
2
r31=--------------------
2
a22
r32=0
1
---*a22*b33*r311 - b33*n2*r425
2
r33=--------------------------------
2
a22
1
- ---*b33*n1*r425
2
r34=--------------------
2
a22
r35=0
1
---*a22*b33*n1*r311 - b33*n1*n2*r425
2
r36=--------------------------------------
2
a22 *n2
r37=0
1
---*a22*b33*r311 - b33*n2*r425
2
r38=--------------------------------
2
a22
1
---*a22*b33*n1*r311 - b33*n1*n2*r425
2
r39=--------------------------------------
2
a22 *n2
- n3*r425
r310=------------
a22
a22*n3*r311 - 2*n2*n3*r425
r312=----------------------------
a22*n2
2*a22*n1*r311 - 2*n1*n2*r425
r313=------------------------------
a22*n2
r314=0
a22*n3*r311 - 2*n2*n3*r425
r315=----------------------------
a22*n2
8*a22*n3*r311 - 8*n2*n3*r425
r316=------------------------------
b33*n2
- a22*b33*r216 + 2*a22*n1*r311 - 2*n1*n2*r425
r317=------------------------------------------------
b33*n1
- a22*b33*r216 + 2*a22*n1*r311 - 2*n1*n2*r425
r318=------------------------------------------------
b33*n2
2 2
- 2*a22 *b33*n3*r216 - 4*a22 *n1*n3*r311 + 4*a22*n1*n2*n3*r425
r319=-----------------------------------------------------------------
2
b33 *n1*n2
- n2*r425
r320=------------
a22
r323=0
a22*r311 - 2*n2*r425
r325=----------------------
a22
2*a22*r311 - 2*n2*r425
r326=------------------------
b33
r328=0
2 2
- 2*a22 *b33*r216 - 4*a22 *n1*r311 + 4*a22*n1*n2*r425
r329=--------------------------------------------------------
2
b33 *n1
r330=0
2*a22*n1*r311 - 2*n1*n2*r425
r332=------------------------------
b33*n2
2
4*a22 *n3*r311 - 4*a22*n2*n3*r425
r333=-----------------------------------
2
b33 *n2
2
4*a22 *r311 - 4*a22*n2*r425
r334=-----------------------------
2
b33
- n1*r425
r335=------------
a22
r336=0
a22*n1*r311 - 2*n1*n2*r425
r337=----------------------------
a22*n2
r338=0
- a22*r311 + 2*n2*r425
r339=-------------------------
a22
a22*n1*r311 - 2*n1*n2*r425
r340=----------------------------
a22*n2
4*a22*n1*r311 - 4*n1*n2*r425
r341=------------------------------
b33*n2
r342=0
r343=0
2 2
- 2*a22 *b33*r216 - 4*a22 *n1*r311 + 4*a22*n1*n2*r425
r344=--------------------------------------------------------
2
b33 *n2
r345=0
- 2*a22*r311 + 2*n2*r425
r347=---------------------------
b33
r348=0
2
4*a22 *n1*r311 - 4*a22*n1*n2*r425
r349=-----------------------------------
2
b33 *n2
r350=0
2*a22*r311 - 2*n2*r425
r351=------------------------
b33
2*a22*n1*r311 - 2*n1*n2*r425
r352=------------------------------
b33*n2
2
4*a22 *n3*r311 - 4*a22*n2*n3*r425
r353=-----------------------------------
2
b33 *n2
2
4*a22 *r311 - 4*a22*n2*r425
r354=-----------------------------
2
b33
2
4*a22 *n1*r311 - 4*a22*n1*n2*r425
r355=-----------------------------------
2
b33 *n2
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
- a22*r311 + 2*n2*r425
r427=-------------------------
n2
r428=0
r429=0
- a22*r311 + 2*n2*r425
r430=-------------------------
n2
2
- 4*a22 *r311 + 4*a22*n2*r425
r431=--------------------------------
b33*n2
r432=0
r433=0
3
a22 *r216
r434=-----------
b33*n1*n2
r435=0
r439=0
r442=0
r444=0
r445=0
r448=0
r450=0
r451=0
r453=0
r454=0
r455=0
r458=0
r460=0
r461=0
r463=0
r465=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
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
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
r4113=0
3 2
- 4*a22 *r311 + 4*a22 *n2*r425
r4114=---------------------------------
2
b33 *n2
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
3 2
- 4*a22 *r311 + 4*a22 *n2*r425
r464=---------------------------------
2
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:
r425, r29, r216, r311, 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 4 first integrals:
3 3 2 3 2 2 4 2 3
FI=4*u1 *a22 *n1 + 4*u1 *u2*a22 *n2 - 4*u1 *u3 *a22 + 4*u1 *u3*a22 *n3
2 2 2 2 2 3
+ 2*u1 *v1*a22 *b33*n1 + 2*u1 *v2*a22 *b33*n2 + 4*u1*u2 *a22 *n1
2 2 2 3
- 2*u1*u2*v1*a22 *b33*n2 + 8*u1*u2*a22 *n1*n2 - 4*u1*u3 *a22 *n1
2 2 2 2
+ 4*u1*u3*v3*a22 *b33*n1 + 8*u1*u3*a22 *n1*n3 + u1*v1 *a22*b33 *n1
2 2 2
- u1*v1*v2*a22*b33 *n2 + u1*v2 *a22*b33 *n1 - 4*u1*v3*a22*b33*n1*n3
3 2 3 3 2 2 4
+ u1*( - 4*a22*n1 + 4*a22*n1*n3 ) + 4*u2 *a22 *n2 - 4*u2 *u3 *a22
2 3 2 2 2 2 2 2 2
+ 4*u2 *u3*a22 *n3 + 2*u2 *v1*a22 *b33*n1 + u2 *( - 4*a22 *n1 + 4*a22 *n2 )
2 3 2 2
- 4*u2*u3 *a22 *n2 + 2*u2*u3*v3*a22 *b33*n2 + 8*u2*u3*a22 *n2*n3
2 2
+ u2*v1 *a22*b33 *n2 - 4*u2*v3*a22*b33*n2*n3
2 2 3 3 3 3
+ u2*( - 4*a22*n1 *n2 + 4*a22*n2*n3 ) - 4*u3 *v3*a22 *b33 - 4*u3 *a22 *n3
2 2 2 2 2 2 2 2 2 2
- u3 *v1 *a22 *b33 + 2*u3 *v1*a22 *b33*n1 - u3 *v2 *a22 *b33
2 2 2 2 2 2
+ 2*u3 *v2*a22 *b33*n2 + 8*u3 *v3*a22 *b33*n3 + u3*v1 *a22*b33 *n3
2 2 2 2
+ 2*u3*v1*v3*a22*b33 *n1 + u3*v2 *a22*b33 *n3 + u3*v2*v3*a22*b33 *n2
2 2
+ u3*v3*( - 4*a22*b33*n1 - 4*a22*b33*n3 )
2 3 1 3 3 1 2 3
+ u3*( - 4*a22*n1 *n3 + 4*a22*n3 ) + ---*v1 *b33 *n1 + ---*v1 *v2*b33 *n2
2 2
2 2 2 2 2 1 2 3 2
+ v1 *( - b33 *n1 + b33 *n2 ) + ---*v1*v2 *b33 *n1 - 2*v1*v2*b33 *n1*n2
2
2 3 2 1 3 3
- 2*v1*v3*b33 *n1*n3 + v1*(2*b33*n1 + 2*b33*n1*n3 ) + ---*v2 *b33 *n2
2
2 2 2
- 2*v2*v3*b33 *n2*n3 + v2*(2*b33*n1 *n2 + 2*b33*n2*n3 )
2 2 2 2 2 2
+ v3 *(b33 *n1 + b33 *n2 ) + 4*v3*b33*n1 *n3
= a product of the elements of: {4,
3 3 2 3 2 2 4 2 3
u1 *a22 *n1 + u1 *u2*a22 *n2 - u1 *u3 *a22 + u1 *u3*a22 *n3
1 2 2 1 2 2 2 3
+ ---*u1 *v1*a22 *b33*n1 + ---*u1 *v2*a22 *b33*n2 + u1*u2 *a22 *n1
2 2
1 2 2 2 3
- ---*u1*u2*v1*a22 *b33*n2 + 2*u1*u2*a22 *n1*n2 - u1*u3 *a22 *n1
2
2 2 1 2 2
+ u1*u3*v3*a22 *b33*n1 + 2*u1*u3*a22 *n1*n3 + ---*u1*v1 *a22*b33 *n1
4
1 2 1 2 2
- ---*u1*v1*v2*a22*b33 *n2 + ---*u1*v2 *a22*b33 *n1 - u1*v3*a22*b33*n1*n3
4 4
3 2 3 3 2 2 4 2 3
+ u1*( - a22*n1 + a22*n1*n3 ) + u2 *a22 *n2 - u2 *u3 *a22 + u2 *u3*a22 *n3
1 2 2 2 2 2 2 2 2 3
+ ---*u2 *v1*a22 *b33*n1 + u2 *( - a22 *n1 + a22 *n2 ) - u2*u3 *a22 *n2
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
2 2 3 3
- u2*v3*a22*b33*n2*n3 + u2*( - a22*n1 *n2 + a22*n2*n3 ) - u3 *v3*a22 *b33
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 2 2
- ---*u3 *v2 *a22 *b33 + ---*u3 *v2*a22 *b33*n2 + 2*u3 *v3*a22 *b33*n3
4 2
1 2 2 1 2 1 2 2
+ ---*u3*v1 *a22*b33 *n3 + ---*u3*v1*v3*a22*b33 *n1 + ---*u3*v2 *a22*b33 *n3
4 2 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 1 2
+ v1*(---*b33*n1 + ---*b33*n1*n3 ) + ---*v2 *b33 *n2 - ---*v2*v3*b33 *n2*n3
2 2 8 2
1 2 1 2 2 1 2 2 1 2 2
+ v2*(---*b33*n1 *n2 + ---*b33*n2*n3 ) + v3 *(---*b33 *n1 + ---*b33 *n2 )
2 2 4 4
2
+ v3*b33*n1 *n3}
{HAM,FI} = {4,
- u1*v1 - u2*v2 - u3*v3,
n2,
b33,
a22,
2 1 1 1
u1*u3*a22 - ---*u1*a22*n3 - ---*u3*v1*a22*b33 + ---*u3*a22*n1
2 2 2
1
+ ---*v1*b33*n3}
4
2 2 3 2
FI=2*u1*u2*a22 *n1*n2 - 2*u1*u3 *a22 *n1 + 2*u1*u3*a22 *n1*n3
2 2
+ u1*v1*(a22*b33*n1 - a22*b33*n2 ) + u1*v2*a22*b33*n1*n2
3 2 2 2 2 2 2
+ u1*( - a22*n1 + a22*n1*n3 ) + u2 *( - a22 *n1 + a22 *n2 )
2 3 2
- 2*u2*u3 *a22 *n2 + 2*u2*u3*a22 *n2*n3 + u2*v1*a22*b33*n1*n2
2 2 4 4 3 3
+ u2*( - a22*n1 *n2 + a22*n2*n3 ) + u3 *a22 - 2*u3 *a22 *n3
2 2 2 2
- u3 *v1*a22 *b33*n1 - u3 *v2*a22 *b33*n2 + u3*v1*a22*b33*n1*n3
2 2
+ u3*v2*a22*b33*n2*n3 + u3*v3*( - a22*b33*n1 - a22*b33*n2 )
2 3 2 1 2 2 1 2 2
+ u3*( - a22*n1 *n3 + a22*n3 ) + v1 *(---*b33 *n1 - ---*b33 *n2 )
4 4
1 2 1 3 1 2
+ ---*v1*v2*b33 *n1*n2 + v1*(---*b33*n1 + ---*b33*n1*n3 )
2 2 2
1 2 1 2 2 1 2 2 1 2 2
+ v2*(---*b33*n1 *n2 + ---*b33*n2*n3 ) + v3 *(---*b33 *n1 - ---*b33 *n2 )
2 2 4 4
2
+ v3*b33*n1 *n3
which the program can not factorize further.
{HAM,FI} = 0
2 2 1
FI= - u1*a22*n1 - u2*a22*n2 + u3 *a22 - u3*a22*n3 - ---*v1*b33*n1
2
1
- ---*v2*b33*n2
2
which the program can not factorize further.
{HAM,FI} = 0
3 3 2 3 2 2 4 2 3
FI= - 4*u1 *a22 *n1 - 4*u1 *u2*a22 *n2 + 4*u1 *u3 *a22 - 4*u1 *u3*a22 *n3
2 2 2 2 2 3
- 2*u1 *v1*a22 *b33*n1 - 2*u1 *v2*a22 *b33*n2 - 4*u1*u2 *a22 *n1
2 2 2 3
+ 2*u1*u2*v1*a22 *b33*n2 - 8*u1*u2*a22 *n1*n2 + 4*u1*u3 *a22 *n1
2 2 2 2
- 4*u1*u3*v3*a22 *b33*n1 - 8*u1*u3*a22 *n1*n3 - 2*u1*v1 *a22*b33 *n1
2 2 2 2 2
+ 2*u1*v1*v2*a22*b33 *n2 - 2*u1*v2 *a22*b33 *n1 - u1*v3 *a22*b33 *n1
3 2 3 3
+ 4*u1*v3*a22*b33*n1*n3 + u1*(4*a22*n1 - 4*a22*n1*n3 ) - 4*u2 *a22 *n2
2 2 4 2 3 2 2
+ 4*u2 *u3 *a22 - 4*u2 *u3*a22 *n3 - 2*u2 *v1*a22 *b33*n1
2 2 2 2 2 2 3 2
+ u2 *(4*a22 *n1 - 4*a22 *n2 ) + 4*u2*u3 *a22 *n2 - 2*u2*u3*v3*a22 *b33*n2
2 2 2 2 2
- 8*u2*u3*a22 *n2*n3 - 2*u2*v1 *a22*b33 *n2 - u2*v3 *a22*b33 *n2
2 2
+ 4*u2*v3*a22*b33*n2*n3 + u2*(4*a22*n1 *n2 - 4*a22*n2*n3 )
3 3 3 3 2 2 2 2
+ 4*u3 *v3*a22 *b33 + 4*u3 *a22 *n3 + 2*u3 *v1 *a22 *b33
2 2 2 2 2 2 2 2
- 2*u3 *v1*a22 *b33*n1 + 2*u3 *v2 *a22 *b33 - 2*u3 *v2*a22 *b33*n2
2 2 2 2 2 2 2 2
+ u3 *v3 *a22 *b33 - 8*u3 *v3*a22 *b33*n3 - 2*u3*v1 *a22*b33 *n3
2 2 2 2 2
- 2*u3*v1*v3*a22*b33 *n1 - 2*u3*v2 *a22*b33 *n3 - u3*v3 *a22*b33 *n3
2 2 2 3
+ u3*v3*(4*a22*b33*n1 + 4*a22*b33*n3 ) + u3*(4*a22*n1 *n3 - 4*a22*n3 )
3 3 2 3 2 2 2 2 2 2 3
- v1 *b33 *n1 - v1 *v2*b33 *n2 + v1 *(b33 *n1 - b33 *n2 ) - v1*v2 *b33 *n1
2 1 2 3 2
+ 2*v1*v2*b33 *n1*n2 - ---*v1*v3 *b33 *n1 + 2*v1*v3*b33 *n1*n3
2
3 2 3 3 1 2 3
+ v1*( - 2*b33*n1 - 2*b33*n1*n3 ) - v2 *b33 *n2 - ---*v2*v3 *b33 *n2
2
2 2 2
+ 2*v2*v3*b33 *n2*n3 + v2*( - 2*b33*n1 *n2 - 2*b33*n2*n3 )
2 2 2 2 2 2
+ v3 *( - b33 *n1 - b33 *n2 ) - 4*v3*b33*n1 *n3
= a product of the elements of: {4,
3 3 2 3 2 2 4 2 3
- u1 *a22 *n1 - u1 *u2*a22 *n2 + u1 *u3 *a22 - u1 *u3*a22 *n3
1 2 2 1 2 2 2 3
- ---*u1 *v1*a22 *b33*n1 - ---*u1 *v2*a22 *b33*n2 - u1*u2 *a22 *n1
2 2
1 2 2 2 3
+ ---*u1*u2*v1*a22 *b33*n2 - 2*u1*u2*a22 *n1*n2 + u1*u3 *a22 *n1
2
2 2 1 2 2
- u1*u3*v3*a22 *b33*n1 - 2*u1*u3*a22 *n1*n3 - ---*u1*v1 *a22*b33 *n1
2
1 2 1 2 2 1 2 2
+ ---*u1*v1*v2*a22*b33 *n2 - ---*u1*v2 *a22*b33 *n1 - ---*u1*v3 *a22*b33 *n1
2 2 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 2 3 1 2
+ u2 *(a22 *n1 - a22 *n2 ) + u2*u3 *a22 *n2 - ---*u2*u3*v3*a22 *b33*n2
2
2 1 2 2 1 2 2
- 2*u2*u3*a22 *n2*n3 - ---*u2*v1 *a22*b33 *n2 - ---*u2*v3 *a22*b33 *n2
2 4
2 2 3 3
+ u2*v3*a22*b33*n2*n3 + u2*(a22*n1 *n2 - a22*n2*n3 ) + u3 *v3*a22 *b33
3 3 1 2 2 2 2 1 2 2
+ u3 *a22 *n3 + ---*u3 *v1 *a22 *b33 - ---*u3 *v1*a22 *b33*n1
2 2
1 2 2 2 2 1 2 2 1 2 2 2 2
+ ---*u3 *v2 *a22 *b33 - ---*u3 *v2*a22 *b33*n2 + ---*u3 *v3 *a22 *b33
2 2 4
2 2 1 2 2 1 2
- 2*u3 *v3*a22 *b33*n3 - ---*u3*v1 *a22*b33 *n3 - ---*u3*v1*v3*a22*b33 *n1
2 2
1 2 2 1 2 2
- ---*u3*v2 *a22*b33 *n3 - ---*u3*v3 *a22*b33 *n3
2 4
2 2 2 3
+ u3*v3*(a22*b33*n1 + a22*b33*n3 ) + u3*(a22*n1 *n3 - a22*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 )
4 4 4 4
1 2 3 1 2 1 2 3
- ---*v1*v2 *b33 *n1 + ---*v1*v2*b33 *n1*n2 - ---*v1*v3 *b33 *n1
4 2 8
1 2 1 3 1 2
+ ---*v1*v3*b33 *n1*n3 + v1*( - ---*b33*n1 - ---*b33*n1*n3 )
2 2 2
1 3 3 1 2 3 1 2
- ---*v2 *b33 *n2 - ---*v2*v3 *b33 *n2 + ---*v2*v3*b33 *n2*n3
4 8 2
1 2 1 2
+ v2*( - ---*b33*n1 *n2 - ---*b33*n2*n3 )
2 2
2 1 2 2 1 2 2 2
+ v3 *( - ---*b33 *n1 - ---*b33 *n2 ) - v3*b33*n1 *n3}
4 4
{HAM,FI} = {4,
u1*v1 + u2*v2 + u3*v3,
n2,
b33,
a22,
2 1 1 1
u1*u3*a22 + ---*u1*v3*a22*b33 - ---*u1*a22*n3 - ---*u3*v1*a22*b33
2 2 2
1 1 2 1
+ ---*u3*a22*n1 - ---*v1*v3*b33 + ---*v3*b33*n1}
2 4 4
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=4*u1**3*a22**3*n1 + 4*u1**2*u2*a22**3*n2 - 4*u1**2*u3**2*a22**4 + 4*u1**2*u3*
a22**3*n3 + 2*u1**2*v1*a22**2*b33*n1 + 2*u1**2*v2*a22**2*b33*n2 + 4*u1*u2**2*a22
**3*n1 - 2*u1*u2*v1*a22**2*b33*n2 + 8*u1*u2*a22**2*n1*n2 - 4*u1*u3**2*a22**3*n1
+ 4*u1*u3*v3*a22**2*b33*n1 + 8*u1*u3*a22**2*n1*n3 + u1*v1**2*a22*b33**2*n1 - u1*
v1*v2*a22*b33**2*n2 + u1*v2**2*a22*b33**2*n1 - 4*u1*v3*a22*b33*n1*n3 + u1*( - 4*
a22*n1**3 + 4*a22*n1*n3**2) + 4*u2**3*a22**3*n2 - 4*u2**2*u3**2*a22**4 + 4*u2**2
*u3*a22**3*n3 + 2*u2**2*v1*a22**2*b33*n1 + u2**2*( - 4*a22**2*n1**2 + 4*a22**2*
n2**2) - 4*u2*u3**2*a22**3*n2 + 2*u2*u3*v3*a22**2*b33*n2 + 8*u2*u3*a22**2*n2*n3
+ u2*v1**2*a22*b33**2*n2 - 4*u2*v3*a22*b33*n2*n3 + u2*( - 4*a22*n1**2*n2 + 4*a22
*n2*n3**2) - 4*u3**3*v3*a22**3*b33 - 4*u3**3*a22**3*n3 - u3**2*v1**2*a22**2*b33
**2 + 2*u3**2*v1*a22**2*b33*n1 - u3**2*v2**2*a22**2*b33**2 + 2*u3**2*v2*a22**2*
b33*n2 + 8*u3**2*v3*a22**2*b33*n3 + u3*v1**2*a22*b33**2*n3 + 2*u3*v1*v3*a22*b33
**2*n1 + u3*v2**2*a22*b33**2*n3 + u3*v2*v3*a22*b33**2*n2 + u3*v3*( - 4*a22*b33*
n1**2 - 4*a22*b33*n3**2) + u3*( - 4*a22*n1**2*n3 + 4*a22*n3**3) + 1/2*v1**3*b33
**3*n1 + 1/2*v1**2*v2*b33**3*n2 + v1**2*( - b33**2*n1**2 + b33**2*n2**2) + 1/2*
v1*v2**2*b33**3*n1 - 2*v1*v2*b33**2*n1*n2 - 2*v1*v3*b33**2*n1*n3 + v1*(2*b33*n1
**3 + 2*b33*n1*n3**2) + 1/2*v2**3*b33**3*n2 - 2*v2*v3*b33**2*n2*n3 + v2*(2*b33*
n1**2*n2 + 2*b33*n2*n3**2) + v3**2*(b33**2*n1**2 + b33**2*n2**2) + 4*v3*b33*n1**
2*n3$
FI=2*u1*u2*a22**2*n1*n2 - 2*u1*u3**2*a22**3*n1 + 2*u1*u3*a22**2*n1*n3 + u1*v1*(
a22*b33*n1**2 - a22*b33*n2**2) + u1*v2*a22*b33*n1*n2 + u1*( - a22*n1**3 + a22*n1
*n3**2) + u2**2*( - a22**2*n1**2 + a22**2*n2**2) - 2*u2*u3**2*a22**3*n2 + 2*u2*
u3*a22**2*n2*n3 + u2*v1*a22*b33*n1*n2 + u2*( - a22*n1**2*n2 + a22*n2*n3**2) + u3
**4*a22**4 - 2*u3**3*a22**3*n3 - u3**2*v1*a22**2*b33*n1 - u3**2*v2*a22**2*b33*n2
+ u3*v1*a22*b33*n1*n3 + u3*v2*a22*b33*n2*n3 + u3*v3*( - a22*b33*n1**2 - a22*b33
*n2**2) + u3*( - a22*n1**2*n3 + a22*n3**3) + v1**2*(1/4*b33**2*n1**2 - 1/4*b33**
2*n2**2) + 1/2*v1*v2*b33**2*n1*n2 + v1*(1/2*b33*n1**3 + 1/2*b33*n1*n3**2) + 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= - u1*a22*n1 - u2*a22*n2 + u3**2*a22**2 - u3*a22*n3 - 1/2*v1*b33*n1 - 1/2*v2*
b33*n2$
FI= - 4*u1**3*a22**3*n1 - 4*u1**2*u2*a22**3*n2 + 4*u1**2*u3**2*a22**4 - 4*u1**2*
u3*a22**3*n3 - 2*u1**2*v1*a22**2*b33*n1 - 2*u1**2*v2*a22**2*b33*n2 - 4*u1*u2**2*
a22**3*n1 + 2*u1*u2*v1*a22**2*b33*n2 - 8*u1*u2*a22**2*n1*n2 + 4*u1*u3**2*a22**3*
n1 - 4*u1*u3*v3*a22**2*b33*n1 - 8*u1*u3*a22**2*n1*n3 - 2*u1*v1**2*a22*b33**2*n1
+ 2*u1*v1*v2*a22*b33**2*n2 - 2*u1*v2**2*a22*b33**2*n1 - u1*v3**2*a22*b33**2*n1 +
4*u1*v3*a22*b33*n1*n3 + u1*(4*a22*n1**3 - 4*a22*n1*n3**2) - 4*u2**3*a22**3*n2 +
4*u2**2*u3**2*a22**4 - 4*u2**2*u3*a22**3*n3 - 2*u2**2*v1*a22**2*b33*n1 + u2**2*
(4*a22**2*n1**2 - 4*a22**2*n2**2) + 4*u2*u3**2*a22**3*n2 - 2*u2*u3*v3*a22**2*b33
*n2 - 8*u2*u3*a22**2*n2*n3 - 2*u2*v1**2*a22*b33**2*n2 - u2*v3**2*a22*b33**2*n2 +
4*u2*v3*a22*b33*n2*n3 + u2*(4*a22*n1**2*n2 - 4*a22*n2*n3**2) + 4*u3**3*v3*a22**
3*b33 + 4*u3**3*a22**3*n3 + 2*u3**2*v1**2*a22**2*b33**2 - 2*u3**2*v1*a22**2*b33*
n1 + 2*u3**2*v2**2*a22**2*b33**2 - 2*u3**2*v2*a22**2*b33*n2 + u3**2*v3**2*a22**2
*b33**2 - 8*u3**2*v3*a22**2*b33*n3 - 2*u3*v1**2*a22*b33**2*n3 - 2*u3*v1*v3*a22*
b33**2*n1 - 2*u3*v2**2*a22*b33**2*n3 - u3*v3**2*a22*b33**2*n3 + u3*v3*(4*a22*b33
*n1**2 + 4*a22*b33*n3**2) + u3*(4*a22*n1**2*n3 - 4*a22*n3**3) - v1**3*b33**3*n1
- v1**2*v2*b33**3*n2 + v1**2*(b33**2*n1**2 - b33**2*n2**2) - v1*v2**2*b33**3*n1
+ 2*v1*v2*b33**2*n1*n2 - 1/2*v1*v3**2*b33**3*n1 + 2*v1*v3*b33**2*n1*n3 + v1*( -
2*b33*n1**3 - 2*b33*n1*n3**2) - v2**3*b33**3*n2 - 1/2*v2*v3**2*b33**3*n2 + 2*v2*
v3*b33**2*n2*n3 + v2*( - 2*b33*n1**2*n2 - 2*b33*n2*n3**2) + v3**2*( - b33**2*n1
**2 - b33**2*n2**2) - 4*v3*b33*n1**2*n3$