Solution 28 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
i*m3*r216
r10=-----------
b33
r11=0
r12=0
r13=0
r14=0
r15=0
2
- 2*b33*c12*r216 + 2*i*b33*c33*r216 - i*m3 *r448
r20=---------------------------------------------------
2
2*b33
- i*m2*r216 + m2*r26
r21=-----------------------
n3
r22=0
m2*r216 + i*m2*r26
r23=--------------------
n3
r24=0
r27=0
r28=0
r210=0
r212= - r216
r213=0
r214=0
r215=0
r217=0
r218=0
r219=0
r220=0
2 2
r30=( - b33 *c12*r216 - i*b33 *c12*r26 + i*b33*m3*n3*r420 + c12*m3*n3*r448
2
- i*c33*m3*n3*r448)/(b33 *n3)
c12*m2*r216 + i*c12*m2*r26 - i*c33*m2*r216 + c33*m2*r26
r31=---------------------------------------------------------
m3*n3
- 2*b33*c12*r216 - 2*i*b33*c12*r26 + i*m3*n3*r420
r32=----------------------------------------------------
b33*n3
r33=0
i*c12*m2*r216 - c12*m2*r26 + c33*m2*r216 + i*c33*m2*r26
r34=---------------------------------------------------------
m3*n3
- 2*i*c12*r216 + 2*c12*r26
r35=-----------------------------
n3
r36=0
i*m3*r420
r37=-----------
b33
r38=0
r39=0
2
b33*c12*r216 + i*b33*c12*r26 - i*b33*c33*r216 + b33*c33*r26 - i*m3 *r448
r310=--------------------------------------------------------------------------
b33*m3
- i*b33*m2*r216 + b33*m2*r26
r311=-------------------------------
m3*n3
r312=0
r313=0
r314=0
r315=0
- i*a33*m3*r216 + a33*m3*r26 - i*b33*n3*r216 + b33*n3*r26
r316=------------------------------------------------------------
m3*n3
r317=0
r318=0
r319=0
r320=0
m3*r448
r323=---------
b33
i*b33*m2*r216 - b33*m2*r26
r325=----------------------------
m3*n3
r326=0
b33*r216 + i*b33*r26
r328=----------------------
m3
r329=0
r330=0
r332=0
r333=0
r334=0
r335=0
- m3*r448
r336=------------
b33
- b33*m2*r216 - i*b33*m2*r26
r337=-------------------------------
m3*n3
r338=0
- i*b33*m2*r216 + b33*m2*r26
r339=-------------------------------
m3*n3
- b33*m2*r216 - i*b33*m2*r26
r340=-------------------------------
m3*n3
r341=0
- b33*r216 - i*b33*r26
r342=-------------------------
m3
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
2 2 2 2 2
r40=( - 2*i*b33 *c12 *r216 + 2*b33 *c12 *r26 - 2*b33 *c12*c33*r216
2
- 2*i*b33 *c12*c33*r26 - 2*b33*c12*m3*n3*r420 + 2*i*b33*c33*m3*n3*r420
2 2 2
+ i*c12 *m3*n3*r448 + 2*c12*c33*m3*n3*r448 - i*c33 *m3*n3*r448)/(2*b33 *m3
*n3)
r41=0
2 2
r42=( - 2*i*b33*c12 *r216 + 2*b33*c12 *r26 - 2*b33*c12*c33*r216
- 2*i*b33*c12*c33*r26 - c12*m3*n3*r420 + i*c33*m3*n3*r420)/(b33*m3*n3)
r43=0
r44=0
r45=0
2 2
2*c12 *r216 + 2*i*c12 *r26 - 2*i*c12*c33*r216 + 2*c12*c33*r26
r46=---------------------------------------------------------------
m3*n3
r47=0
r48=0
- c12*r420 + i*c33*r420
r49=--------------------------
b33
r410=0
r411=0
r412=0
r413=0
2 2
r415=( - b33 *c12*r216 - i*b33 *c12*r26 + i*b33*m3*n3*r420 + c12*m3*n3*r448
- i*c33*m3*n3*r448)/(b33*m3*n3)
r416=0
- 2*b33*c12*r216 - 2*i*b33*c12*r26 + i*m3*n3*r420
r417=----------------------------------------------------
m3*n3
r418=0
r419=0
r421=0
- 2*b33*c12*r216 - 2*i*b33*c12*r26 + i*m3*n3*r420
r422=----------------------------------------------------
m3*n3
r423=0
r424=0
r425=(2*a33*c12*r216 + 2*i*a33*c12*r26 - 2*i*a33*c33*r216 + 2*a33*c33*r26
- i*m3*n3*r448)/(2*m3*n3)
r426=0
r427=0
r428=0
r429=0
- i*a33*b33*r216 + a33*b33*r26
r431=---------------------------------
m3*n3
r432=0
r433=0
r435=0
2 2
r439=( - i*b33 *c12*r216 + b33 *c12*r26 - b33*m3*n3*r420 + i*c12*m3*n3*r448
+ c33*m3*n3*r448)/(b33*m3*n3)
r442=0
r444= - r420
r445=0
r450=0
r451=0
r454=0
r455=0
r458=0
i*r448
r460=--------
2
r461=0
r463=0
r464=0
r465=0
r467=0
r468=0
r469=0
r470=0
2 2
r471=(i*b33 *c12*r216 - b33 *c12*r26 + b33*m3*n3*r420 - i*c12*m3*n3*r448
- c33*m3*n3*r448)/(b33*m3*n3)
r472=0
2*i*b33*c12*r216 - 2*b33*c12*r26 + m3*n3*r420
r473=-----------------------------------------------
m3*n3
r474=0
r475=0
- 2*b33*c12*r216 - 2*i*b33*c12*r26
r476=-------------------------------------
m3*n3
r477=0
2*i*b33*c12*r216 - 2*b33*c12*r26 + 2*m3*n3*r420
r478=-------------------------------------------------
m3*n3
- 2*b33*c12*r216 - 2*i*b33*c12*r26
r479=-------------------------------------
m3*n3
r480=0
r481= - r448
r482=0
r483=i*r448
r484=0
r485=0
r486=0
- a33*b33*r216 - i*a33*b33*r26
r487=---------------------------------
m3*n3
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
i*r448
r4107=--------
2
r4108=0
r4109=0
r4110=i*r448
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m1=i*m2
n2=0
n1=0
c23=0
c22= - 2*i*c12
c13=0
c11=0
b32=0
b31=0
b23=0
b22=0
b21=i*b33
b13=0
b12= - i*b33
b11=0
a23=0
a22=0
a13=0
a12=0
a11=0
a33*b33*r216 + i*a33*b33*r26
r453=------------------------------
m3*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:
r26, r216, r420, r448, m3, c33, n3, m2, b33, 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.
{m2,a33,b33}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11,
a12,
a13,
a22,
a23,
b11,
b12 + i*b33,
b13,
b21 - i*b33,
b22,
b23,
b31,
b32,
c11,
c13,
2*i*c12 + c22,
c23,
n1,
n2,
m1 - i*m2}$
The system of equations related to the Hamiltonian HAM:
2
HAM= - i*u1*v2*b33 + i*u2*v1*b33 + u3 *a33 + u3*v3*b33 + u3*n3 + 2*v1*v2*c12
2 2
+ i*v1*m2 - 2*i*v2 *c12 + v2*m2 + v3 *c33 + v3*m3
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 2 2 2 2 2 2 2
FI=2*i*u1 *v1 *b33 + i*u1 *v2 *b33 + 2*i*u1*u3*v1*v3*b33 - 2*u1*u3*v2*v3*b33
2
+ u1*v2*v3 *( - 2*i*b33*c12 - 2*b33*c33) - 2*u1*v2*v3*b33*m3
2 2 2 2 2
+ i*u2 *v1 *b33 + 2*u2*u3*v1*v3*b33 + u2*v1*v3 *(2*i*b33*c12 + 2*b33*c33)
2 2 2 3
+ 2*u2*v1*v3*b33*m3 - i*u3 *v3 *b33 + u3*v3 *(2*b33*c12 - 2*i*b33*c33)
2 4 2 2
- 2*i*u3*v3 *b33*m3 + v3 *(i*c12 + 2*c12*c33 - i*c33 )
3 2 2
+ v3 *(2*c12*m3 - 2*i*c33*m3) - i*v3 *m3
= a product of the elements of: {2*i,
2 2 2
2 2 2 u1 *v2 *b33 2 2
u1 *v1 *b33 + -------------- + u1*u3*v1*v3*b33 + i*u1*u3*v2*v3*b33
2
2 2 2
2 u2 *v1 *b33
+ u1*v2*v3 *( - b33*c12 + i*b33*c33) + i*u1*v2*v3*b33*m3 + --------------
2
2 2
- i*u2*u3*v1*v3*b33 + u2*v1*v3 *(b33*c12 - i*b33*c33) - i*u2*v1*v3*b33*m3
2 2 2
- u3 *v3 *b33 3 2
+ ----------------- + u3*v3 *( - i*b33*c12 - b33*c33) - u3*v3 *b33*m3
2
2 2
4 c12 - 2*i*c12*c33 - c33 3
+ v3 *--------------------------- + v3 *( - i*c12*m3 - c33*m3)
2
2 2
- v3 *m3
+ ------------}
2
{HAM,FI} = {4*i,
b33,
b33,
u1*v1 + u2*v2 + u3*v3,
u1*v2*v3*b33 u1*v2*n3
u1*u3*v2*a33 + -------------- + ---------- + u2*u3*v1*a33
2 2
2
u2*v1*v3*b33 u2*v1*n3 - i*u3*v1 *b33 u3*v1*v2*b33
+ -------------- + ---------- + ----------------- + --------------
2 2 2 2
2 v1*v2*m3
- v1 *v3*c12 + v1*v2*v3*(2*i*c12 + c33) + ----------
2
- v1*v3*m2
+ -------------}
2
2 3 2 3 2
FI=2*u1*v1 *v2*b33 + u1*v2 *b33 + u1*v2*v3 *b33 - u2*v1 *b33 - u2*v1*v3 *b33
2 2 3
+ i*u3*v1 *v3*b33 + u3*v1*v2*v3*b33 + i*u3*v2 *v3*b33 + i*u3*v3 *b33
2 2 2 2 2
+ v1 *v3 *( - c12 + i*c33) + i*v1 *v3*m3 + v2 *v3 *( - c12 + i*c33)
2 4 3
+ i*v2 *v3*m3 + v3 *( - c12 + i*c33) + i*v3 *m3
= a product of the elements of: {2,
3 2 3
2 u1*v2 *b33 u1*v2*v3 *b33 - u2*v1 *b33
u1*v1 *v2*b33 + ------------ + --------------- + ---------------
2 2 2
2 2
- u2*v1*v3 *b33 i*u3*v1 *v3*b33 u3*v1*v2*v3*b33
+ ------------------ + ----------------- + -----------------
2 2 2
2 3
i*u3*v2 *v3*b33 i*u3*v3 *b33 2 2 - c12 + i*c33
+ ----------------- + -------------- + v1 *v3 *----------------
2 2 2
2 2
i*v1 *v3*m3 2 2 - c12 + i*c33 i*v2 *v3*m3
+ ------------- + v2 *v3 *---------------- + -------------
2 2 2
3
4 - c12 + i*c33 i*v3 *m3
+ v3 *---------------- + ----------}
2 2
{HAM,FI} = {2,
- u1*v1 - u2*v2 - u3*v3,
b33,
2 2
2 2 v1 *v3*b33 v1 *n3
u3*v1 *a33 - u3*v2 *a33 + ------------ + -------- + i*v1*v2*v3*b33
2 2
2 2
- v2 *v3*b33 - v2 *n3
+ --------------- + -----------}
2 2
2 2 2 3 2
FI= - u1*u3 *v2*a33*b33 - u1*u3*v2*b33 *n3 - 2*u1*v1 *b33 *c12
2 2 2 2 2 2
+ 2*i*u1*v1 *v2*b33 *c12 - u1*v1 *b33 *m2 - 2*u1*v1*v2 *b33 *c12
2 3 2 2 2
- i*u1*v1*v2*b33 *m2 + 2*i*u1*v2 *b33 *c12 - u1*v2 *b33 *m2
2 2 2 2
+ i*u1*v2*v3 *b33 *c12 + u1*v2*b33*m3*n3 + u2*u3 *v1*a33*b33
2 2 2 2 2
+ u2*u3*v1*b33 *n3 + i*u2*v1 *b33 *m2 - i*u2*v1*v3 *b33 *c12
3 2
- u2*v1*b33*m3*n3 - i*u3 *v3*a33*b33
2 2
+ u3 *v3 *(a33*b33*c12 - i*a33*b33*c33)
2 2 2 2
+ u3 *v3*( - i*a33*b33*m3 - i*b33 *n3) - 2*u3*v1 *v3*b33 *c12
2 2 2 3 2
- 2*u3*v2 *v3*b33 *c12 - i*u3*v2*v3*b33 *m2 - u3*v3 *b33 *c12
2
+ u3*v3 *(b33*c12*n3 - i*b33*c33*n3)
2 2
+ v1*v2*v3 *(2*b33*c12 - 2*i*b33*c12*c33) - 2*i*v1*v2*v3*b33*c12*m3
2
+ v1*v3 *(i*b33*c12*m2 + b33*c33*m2) + v1*v3*b33*m2*m3
2 2 2 2
+ v2 *v3 *( - 2*i*b33*c12 - 2*b33*c12*c33) - 2*v2 *v3*b33*c12*m3
2
+ v2*v3 *(b33*c12*m2 - i*b33*c33*m2) - i*v2*v3*b33*m2*m3
4 2 3
+ v3 *( - i*b33*c12 - b33*c12*c33) - v3 *b33*c12*m3
2 2
+ v3 *( - c12*m3*n3 + i*c33*m3*n3) + i*v3*m3 *n3
which the program can not factorize further.
{HAM,FI} = { - 4*i,
b33,
b33,
u1*v1 + u2*v2 + u3*v3,
2 2
u3*v1 *a33*c12 - 2*i*u3*v1*v2*a33*c12 - u3*v2 *a33*c12
2 2
- i*u3*v2*a33*m2 - v1 *v3*b33*c12 v1 *c12*n3
+ ------------------- + ------------------- + ------------
2 2 2
2
- v1*v3*b33*m2 - v2 *v3*b33*c12
- i*v1*v2*c12*n3 + ----------------- + -------------------
4 2
2
- v2 *c12*n3 - i*v2*v3*b33*m2 - i*v2*m2*n3
+ --------------- + ------------------- + ---------------}
2 4 4
2 3
FI= - i*u1*u3 *v2*a33*b33 - i*u1*u3*v2*b33*n3 - 2*i*u1*v1 *b33*c12
2 2 2
- 2*u1*v1 *v2*b33*c12 - i*u1*v1 *b33*m2 - 2*i*u1*v1*v2 *b33*c12
3 2 2
+ u1*v1*v2*b33*m2 - 2*u1*v2 *b33*c12 - i*u1*v2 *b33*m2 - u1*v2*v3 *b33*c12
2 2
+ i*u2*u3 *v1*a33*b33 + i*u2*u3*v1*b33*n3 - u2*v1 *b33*m2
2 3 2 2
+ u2*v1*v3 *b33*c12 + u3 *v3*a33*b33 + u3 *v3 *(i*a33*c12 + a33*c33)
2 2 2
+ u3 *v3*(a33*m3 + b33*n3) - 2*i*u3*v1 *v3*b33*c12 - 2*i*u3*v2 *v3*b33*c12
3 2
+ u3*v2*v3*b33*m2 - i*u3*v3 *b33*c12 + u3*v3 *(i*c12*n3 + c33*n3)
2 2
+ u3*v3*m3*n3 + v1*v2*v3 *(2*i*c12 + 2*c12*c33) + 2*v1*v2*v3*c12*m3
2
+ v1*v3 *( - c12*m2 + i*c33*m2) + i*v1*v3*m2*m3
2 2 2 2
+ v2 *v3 *(2*c12 - 2*i*c12*c33) - 2*i*v2 *v3*c12*m3
2 4 2
+ v2*v3 *(i*c12*m2 + c33*m2) + v2*v3*m2*m3 + v3 *(c12 - i*c12*c33)
3
- i*v3 *c12*m3
= a product of the elements of: { - i,
2 3
u1*u3 *v2*a33*b33 + u1*u3*v2*b33*n3 + 2*u1*v1 *b33*c12
2 2 2
- 2*i*u1*v1 *v2*b33*c12 + u1*v1 *b33*m2 + 2*u1*v1*v2 *b33*c12
3 2
+ i*u1*v1*v2*b33*m2 - 2*i*u1*v2 *b33*c12 + u1*v2 *b33*m2
2 2
- i*u1*v2*v3 *b33*c12 - u2*u3 *v1*a33*b33 - u2*u3*v1*b33*n3
2 2 3
- i*u2*v1 *b33*m2 + i*u2*v1*v3 *b33*c12 + i*u3 *v3*a33*b33
2 2 2
+ u3 *v3 *( - a33*c12 + i*a33*c33) + u3 *v3*(i*a33*m3 + i*b33*n3)
2 2
+ 2*u3*v1 *v3*b33*c12 + 2*u3*v2 *v3*b33*c12 + i*u3*v2*v3*b33*m2
3 2
+ u3*v3 *b33*c12 + u3*v3 *( - c12*n3 + i*c33*n3) + i*u3*v3*m3*n3
2 2
+ v1*v2*v3 *( - 2*c12 + 2*i*c12*c33) + 2*i*v1*v2*v3*c12*m3
2
+ v1*v3 *( - i*c12*m2 - c33*m2) - v1*v3*m2*m3
2 2 2 2
+ v2 *v3 *(2*i*c12 + 2*c12*c33) + 2*v2 *v3*c12*m3
2 4 2
+ v2*v3 *( - c12*m2 + i*c33*m2) + i*v2*v3*m2*m3 + v3 *(i*c12 + c12*c33)
3
+ v3 *c12*m3}
{HAM,FI} = {4,
u1*v1 + u2*v2 + u3*v3,
b33,
2 2
u3*v1 *a33*c12 - 2*i*u3*v1*v2*a33*c12 - u3*v2 *a33*c12
2 2
- i*u3*v2*a33*m2 - v1 *v3*b33*c12 v1 *c12*n3
+ ------------------- + ------------------- + ------------
2 2 2
2
- v1*v3*b33*m2 - v2 *v3*b33*c12
- i*v1*v2*c12*n3 + ----------------- + -------------------
4 2
2
- v2 *c12*n3 - i*v2*v3*b33*m2 - i*v2*m2*n3
+ --------------- + ------------------- + ---------------}
2 4 4
And again in machine readable form:
HAM= - i*u1*v2*b33 + i*u2*v1*b33 + u3**2*a33 + u3*v3*b33 + u3*n3 + 2*v1*v2*c12 +
i*v1*m2 - 2*i*v2**2*c12 + v2*m2 + v3**2*c33 + v3*m3$
FI=2*i*u1**2*v1**2*b33**2 + i*u1**2*v2**2*b33**2 + 2*i*u1*u3*v1*v3*b33**2 - 2*u1
*u3*v2*v3*b33**2 + u1*v2*v3**2*( - 2*i*b33*c12 - 2*b33*c33) - 2*u1*v2*v3*b33*m3
+ i*u2**2*v1**2*b33**2 + 2*u2*u3*v1*v3*b33**2 + u2*v1*v3**2*(2*i*b33*c12 + 2*b33
*c33) + 2*u2*v1*v3*b33*m3 - i*u3**2*v3**2*b33**2 + u3*v3**3*(2*b33*c12 - 2*i*b33
*c33) - 2*i*u3*v3**2*b33*m3 + v3**4*(i*c12**2 + 2*c12*c33 - i*c33**2) + v3**3*(2
*c12*m3 - 2*i*c33*m3) - i*v3**2*m3**2$
FI=2*u1*v1**2*v2*b33 + u1*v2**3*b33 + u1*v2*v3**2*b33 - u2*v1**3*b33 - u2*v1*v3
**2*b33 + i*u3*v1**2*v3*b33 + u3*v1*v2*v3*b33 + i*u3*v2**2*v3*b33 + i*u3*v3**3*
b33 + v1**2*v3**2*( - c12 + i*c33) + i*v1**2*v3*m3 + v2**2*v3**2*( - c12 + i*c33
) + i*v2**2*v3*m3 + v3**4*( - c12 + i*c33) + i*v3**3*m3$
FI= - u1*u3**2*v2*a33*b33**2 - u1*u3*v2*b33**2*n3 - 2*u1*v1**3*b33**2*c12 + 2*i*
u1*v1**2*v2*b33**2*c12 - u1*v1**2*b33**2*m2 - 2*u1*v1*v2**2*b33**2*c12 - i*u1*v1
*v2*b33**2*m2 + 2*i*u1*v2**3*b33**2*c12 - u1*v2**2*b33**2*m2 + i*u1*v2*v3**2*b33
**2*c12 + u1*v2*b33*m3*n3 + u2*u3**2*v1*a33*b33**2 + u2*u3*v1*b33**2*n3 + i*u2*
v1**2*b33**2*m2 - i*u2*v1*v3**2*b33**2*c12 - u2*v1*b33*m3*n3 - i*u3**3*v3*a33*
b33**2 + u3**2*v3**2*(a33*b33*c12 - i*a33*b33*c33) + u3**2*v3*( - i*a33*b33*m3 -
i*b33**2*n3) - 2*u3*v1**2*v3*b33**2*c12 - 2*u3*v2**2*v3*b33**2*c12 - i*u3*v2*v3
*b33**2*m2 - u3*v3**3*b33**2*c12 + u3*v3**2*(b33*c12*n3 - i*b33*c33*n3) + v1*v2*
v3**2*(2*b33*c12**2 - 2*i*b33*c12*c33) - 2*i*v1*v2*v3*b33*c12*m3 + v1*v3**2*(i*
b33*c12*m2 + b33*c33*m2) + v1*v3*b33*m2*m3 + v2**2*v3**2*( - 2*i*b33*c12**2 - 2*
b33*c12*c33) - 2*v2**2*v3*b33*c12*m3 + v2*v3**2*(b33*c12*m2 - i*b33*c33*m2) - i*
v2*v3*b33*m2*m3 + v3**4*( - i*b33*c12**2 - b33*c12*c33) - v3**3*b33*c12*m3 + v3
**2*( - c12*m3*n3 + i*c33*m3*n3) + i*v3*m3**2*n3$
FI= - i*u1*u3**2*v2*a33*b33 - i*u1*u3*v2*b33*n3 - 2*i*u1*v1**3*b33*c12 - 2*u1*v1
**2*v2*b33*c12 - i*u1*v1**2*b33*m2 - 2*i*u1*v1*v2**2*b33*c12 + u1*v1*v2*b33*m2 -
2*u1*v2**3*b33*c12 - i*u1*v2**2*b33*m2 - u1*v2*v3**2*b33*c12 + i*u2*u3**2*v1*
a33*b33 + i*u2*u3*v1*b33*n3 - u2*v1**2*b33*m2 + u2*v1*v3**2*b33*c12 + u3**3*v3*
a33*b33 + u3**2*v3**2*(i*a33*c12 + a33*c33) + u3**2*v3*(a33*m3 + b33*n3) - 2*i*
u3*v1**2*v3*b33*c12 - 2*i*u3*v2**2*v3*b33*c12 + u3*v2*v3*b33*m2 - i*u3*v3**3*b33
*c12 + u3*v3**2*(i*c12*n3 + c33*n3) + u3*v3*m3*n3 + v1*v2*v3**2*(2*i*c12**2 + 2*
c12*c33) + 2*v1*v2*v3*c12*m3 + v1*v3**2*( - c12*m2 + i*c33*m2) + i*v1*v3*m2*m3 +
v2**2*v3**2*(2*c12**2 - 2*i*c12*c33) - 2*i*v2**2*v3*c12*m3 + v2*v3**2*(i*c12*m2
+ c33*m2) + v2*v3*m2*m3 + v3**4*(c12**2 - i*c12*c33) - i*v3**3*c12*m3$