Solution 2 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
a11*c33*r29 - a22*c33*r29
r20=---------------------------
a11*a22
r21=0
a11*c33*r29 - a22*c33*r29
r22=---------------------------
a11*a22
r23=0
r24=0
r26=0
r27=0
r28=0
r210=0
r212=0
r213=0
a11*r29 - a22*r29
r214=-------------------
a11
r215=0
r216=0
r217=0
r218=0
r219=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
2 2 2 2
r40=( - a11 *a22*c33*r446 - a11 *c33 *r464 + a11*a22*c33 *r4114
2 2 2 2 2
+ a11*a22*c33 *r464 - a22 *c33 *r4114)/(2*a11 *a22 )
r41=0
2 2 2
- 2*a11*a22*c33*r446 - 2*a11*c33 *r464 + a22 *c33*r446 + 2*a22*c33 *r464
r42=---------------------------------------------------------------------------
2
2*a11*a22
r43=0
2 2 2 2
r44=( - a11 *a22*c33*r446 - a11 *c33 *r464 + a11*a22 *c33*r446
2 2 2 2 2 2
- a11*a22*c33 *r4114 + a11*a22*c33 *r464 + a22 *c33 *r4114)/(2*a11 *a22 )
r45=0
r46=0
r47=0
r48=0
2 2
- a11*a22*c33*r446 - 2*a11*c33 *r464 + 2*a22*c33 *r464
r49=---------------------------------------------------------
2
2*a11*a22
r410=0
2 2 2
- a11*a22*c33*r446 - 2*a11*c33 *r464 + a22 *c33*r446 + 2*a22*c33 *r464
r411=-------------------------------------------------------------------------
2
2*a11*a22
r412=0
r413=0
c33*r4102
r415=-----------
a22
r416=0
c33*r4102
r417=-----------
a22
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
r425=0
r426=0
- a11*r446 - 2*c33*r4114
r427=---------------------------
2*a11
r428=0
r429=0
- a22*r446 - 2*c33*r464
r430=--------------------------
2*a22
a11*r4102
r431=-----------
a11 - a22
r432=0
r433=0
a11*r464 - a22*r4114
r434=----------------------
2*a11 - 2*a22
r435=0
c33*r4102
r436=-----------
a22
r437=0
c33*r4102
r438=-----------
a22
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
r447=0
r448=0
r449=0
r450=0
r451=0
a11*r4102
r452=-----------
a11 - a22
r453=0
r454=0
2
- a11 *r446 + 2*a11*c33*r4114 - 2*a22*c33*r4114
r455=--------------------------------------------------
2
2*a11
r456=0
r458=0
r459=0
- a11*a22*r446 - 2*a11*c33*r464 + 2*a22*c33*r464
r460=---------------------------------------------------
2*a11*a22
r461=r4102
r462=0
r463=0
r465=0
r466=r4102
r467=0
r468=0
2 2
a11 *r464 + a11*a22*r4114 - a11*a22*r464 - a22 *r4114
r469=-------------------------------------------------------
2
2*a11
r470=0
r471=0
r472=0
r473=0
c33*r4102
r474=-----------
a22
r475=0
c33*r4102
r476=-----------
a22
r477=0
r478=0
r479=0
r480=0
r481=0
r482=0
r483=r446
r484=0
r485=0
r486=0
r487=0
a11*r4102
r488=-----------
a11 - a22
r489=0
r490=0
r491=0
r492=0
r493=0
r494=r446
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4103=0
r4104=0
- a11*a22*r446 + 2*a11*c33*r4114 - 2*a22*c33*r4114
r4105=-----------------------------------------------------
2*a11*a22
r4106=0
- a11*a22*r446 + 2*a11*c33*r4114 - 2*a22*c33*r4114
r4107=-----------------------------------------------------
2*a11*a22
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4115=0
r4117=0
r4118=0
a11*r4114 - a22*r4114
r4119=-----------------------
a11
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
m3=0
m2=0
m1=0
n3=0
n2=0
n1=0
c23=0
c22=0
c13=0
c12=0
c11=0
b33=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
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, r4114, r4102, r446, r464, c33, a11, 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.
{a11 - a22,a22,a11,c33}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a12,
a13,
a23,
a33,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
b33,
c11,
c12,
c13,
c22,
c23,
n1,
n2,
n3,
m1,
m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a11 + u2 *a22 + v3 *c33
has apart from the Hamiltonian and Casimirs the following 5 first integrals:
4 2 2 3 4 2 2 2 2 3
FI=u2 *(a11 *a22 - 2*a11*a22 + a22 ) + u2 *u3 *(2*a11 *a22 - 2*a11*a22 )
2 2 2 2 3 4 2 2
+ u2 *v1 *( - 2*a11 *a22*c33 + 4*a11*a22 *c33 - 2*a22 *c33) + u3 *a11 *a22
2 2 2 2
+ u3 *v1 *( - 2*a11 *a22*c33 + 2*a11*a22 *c33)
2 2 2 2 2 2 2
+ v1 *v2 *( - 2*a11 *c33 + 4*a11*a22*c33 - 2*a22 *c33 )
2 2 2 2 2 2 2
+ v1 *v3 *( - 2*a11 *c33 + 4*a11*a22*c33 - 2*a22 *c33 )
4 2 2 2 2 2
+ v2 *( - a11 *c33 + 2*a11*a22*c33 - a22 *c33 )
2 2 2 2 2 2 2
+ v2 *v3 *( - 2*a11 *c33 + 4*a11*a22*c33 - 2*a22 *c33 )
4 2 2 2 2 2
+ v3 *( - a11 *c33 + 2*a11*a22*c33 - a22 *c33 )
2 2 2
= a product of the elements of: {u2 *(a11*a22 - a22 ) + u3 *a11*a22
2 2
+ v2 *(a11*c33 - a22*c33) + v3 *(a11*c33 - a22*c33),
2 2 2 2
u2 *(a11*a22 - a22 ) + u3 *a11*a22 + v1 *( - 2*a11*c33 + 2*a22*c33)
2 2
+ v2 *( - a11*c33 + a22*c33) + v3 *( - a11*c33 + a22*c33)}
{HAM,FI} = 0
2 2 2 2
FI= - u1 *v2 *a11*a22 - u1 *v3 *a11*a22 + 2*u1*u2*v1*v2*a11*a22
2 2 2 2
+ 2*u1*u3*v1*v3*a11*a22 - u2 *v1 *a11*a22 - u2 *v3 *a11*a22
2 2 2 2
+ 2*u2*u3*v2*v3*a11*a22 - u3 *v1 *a11*a22 - u3 *v2 *a11*a22
2 2 2 2
+ v1 *v2 *( - a11*c33 + a22*c33) - v1 *v3 *a11*c33
4 2 2
+ v2 *( - a11*c33 + a22*c33) + v2 *v3 *( - 2*a11*c33 + a22*c33)
4
- v3 *a11*c33
which the program can not factorize further.
{HAM,FI} = 0
2 2 2
FI=u1*u2 *v1*(a11*a22 - a22 ) + u1*u3 *v1*a11*a22
2 2
+ u1*v1*v2 *(a11*c33 - a22*c33) + u1*v1*v3 *(a11*c33 - a22*c33)
3 2 2 2 2
+ u2 *v2*(a11*a22 - a22 ) + u2 *u3*v3*(a11*a22 - a22 ) + u2*u3 *v2*a11*a22
3 2
+ u2*v2 *(a11*c33 - a22*c33) + u2*v2*v3 *(a11*c33 - a22*c33)
3 2
+ u3 *v3*a11*a22 + u3*v2 *v3*(a11*c33 - a22*c33)
3
+ u3*v3 *(a11*c33 - a22*c33)
= a product of the elements of: {u1*v1 + u2*v2 + u3*v3,
2 2 2 2
u2 *(a11*a22 - a22 ) + u3 *a11*a22 + v2 *(a11*c33 - a22*c33)
2
+ v3 *(a11*c33 - a22*c33)}
{HAM,FI} = 0
2 2 3 2 2 3
FI=u1 *u2 *(2*a11 *a22 - 4*a11 *a22 + 2*a11*a22 )
2 2 3 2 2
+ u1 *u3 *(2*a11 *a22 - 2*a11 *a22 )
2 2 3 2 2
+ u1 *v2 *(2*a11 *c33 - 4*a11 *a22*c33 + 2*a11*a22 *c33)
2 2 3 2 2
+ u1 *v3 *(2*a11 *c33 - 4*a11 *a22*c33 + 2*a11*a22 *c33)
4 2 2 3 4
+ u2 *(a11 *a22 - 2*a11*a22 + a22 )
2 2 2 2 3 4 2 2
+ u2 *v3 *(2*a11 *a22*c33 - 4*a11*a22 *c33 + 2*a22 *c33) - u3 *a11 *a22
2 2 2 2
+ u3 *v2 *( - 2*a11 *a22*c33 + 2*a11*a22 *c33)
4 2 2 2 2 2
+ v2 *( - a11 *c33 + 2*a11*a22*c33 - a22 *c33 )
4 2 2 2 2 2
+ v3 *(a11 *c33 - 2*a11*a22*c33 + a22 *c33 )
= a product of the elements of: {2,
2 2 2 2
u2 *(a11*a22 - a22 ) + u3 *a11*a22 + v2 *(a11*c33 - a22*c33)
2
+ v3 *(a11*c33 - a22*c33),
2 2
2 2 2 a11*a22 - a22 - u3 *a11*a22
u1 *(a11 - a11*a22) + u2 *---------------- + ----------------
2 2
2 - a11*c33 + a22*c33 2 a11*c33 - a22*c33
+ v2 *---------------------- + v3 *-------------------}
2 2
{HAM,FI} = 0
2 2 2 2
FI=u2 *(a11*a22 - a22 ) + u3 *a11*a22 + v2 *(a11*c33 - a22*c33)
2
+ v3 *(a11*c33 - a22*c33)
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u2**2*a22 + v3**2*c33$
FI=u2**4*(a11**2*a22**2 - 2*a11*a22**3 + a22**4) + u2**2*u3**2*(2*a11**2*a22**2
- 2*a11*a22**3) + u2**2*v1**2*( - 2*a11**2*a22*c33 + 4*a11*a22**2*c33 - 2*a22**3
*c33) + u3**4*a11**2*a22**2 + u3**2*v1**2*( - 2*a11**2*a22*c33 + 2*a11*a22**2*
c33) + v1**2*v2**2*( - 2*a11**2*c33**2 + 4*a11*a22*c33**2 - 2*a22**2*c33**2) +
v1**2*v3**2*( - 2*a11**2*c33**2 + 4*a11*a22*c33**2 - 2*a22**2*c33**2) + v2**4*(
- a11**2*c33**2 + 2*a11*a22*c33**2 - a22**2*c33**2) + v2**2*v3**2*( - 2*a11**2*
c33**2 + 4*a11*a22*c33**2 - 2*a22**2*c33**2) + v3**4*( - a11**2*c33**2 + 2*a11*
a22*c33**2 - a22**2*c33**2)$
FI= - u1**2*v2**2*a11*a22 - u1**2*v3**2*a11*a22 + 2*u1*u2*v1*v2*a11*a22 + 2*u1*
u3*v1*v3*a11*a22 - u2**2*v1**2*a11*a22 - u2**2*v3**2*a11*a22 + 2*u2*u3*v2*v3*a11
*a22 - u3**2*v1**2*a11*a22 - u3**2*v2**2*a11*a22 + v1**2*v2**2*( - a11*c33 + a22
*c33) - v1**2*v3**2*a11*c33 + v2**4*( - a11*c33 + a22*c33) + v2**2*v3**2*( - 2*
a11*c33 + a22*c33) - v3**4*a11*c33$
FI=u1*u2**2*v1*(a11*a22 - a22**2) + u1*u3**2*v1*a11*a22 + u1*v1*v2**2*(a11*c33 -
a22*c33) + u1*v1*v3**2*(a11*c33 - a22*c33) + u2**3*v2*(a11*a22 - a22**2) + u2**
2*u3*v3*(a11*a22 - a22**2) + u2*u3**2*v2*a11*a22 + u2*v2**3*(a11*c33 - a22*c33)
+ u2*v2*v3**2*(a11*c33 - a22*c33) + u3**3*v3*a11*a22 + u3*v2**2*v3*(a11*c33 -
a22*c33) + u3*v3**3*(a11*c33 - a22*c33)$
FI=u1**2*u2**2*(2*a11**3*a22 - 4*a11**2*a22**2 + 2*a11*a22**3) + u1**2*u3**2*(2*
a11**3*a22 - 2*a11**2*a22**2) + u1**2*v2**2*(2*a11**3*c33 - 4*a11**2*a22*c33 + 2
*a11*a22**2*c33) + u1**2*v3**2*(2*a11**3*c33 - 4*a11**2*a22*c33 + 2*a11*a22**2*
c33) + u2**4*(a11**2*a22**2 - 2*a11*a22**3 + a22**4) + u2**2*v3**2*(2*a11**2*a22
*c33 - 4*a11*a22**2*c33 + 2*a22**3*c33) - u3**4*a11**2*a22**2 + u3**2*v2**2*( -
2*a11**2*a22*c33 + 2*a11*a22**2*c33) + v2**4*( - a11**2*c33**2 + 2*a11*a22*c33**
2 - a22**2*c33**2) + v3**4*(a11**2*c33**2 - 2*a11*a22*c33**2 + a22**2*c33**2)$
FI=u2**2*(a11*a22 - a22**2) + u3**2*a11*a22 + v2**2*(a11*c33 - a22*c33) + v3**2*
(a11*c33 - a22*c33)$