Solution 3 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
3 2 2 3
r13=( - 2*a11 *n3*r29 + 2*a11 *a22*n3*r29 + 2*a11*n1 *n3*r4114 - 2*a11*n3 *r4114
2 3 4 3
- a22*n1 *n3*r4114 + a22*n3 *r4114)/(2*a11 - 2*a11 *a22)
3 2 3 2
r15=( - 2*a11 *n1*r29 + 2*a11 *a22*n1*r29 + 2*a11*n1 *r4114 - 2*a11*n1*n3 *r4114
3 2 4 3
- a22*n1 *r4114 + a22*n1*n3 *r4114)/(2*a11 - 2*a11 *a22)
r20=0
r21=0
r22=0
r23=0
r24=0
r26=0
r27=0
r28=0
r210=0
r212=0
- 2*a11*n2*n3*r4114 + a22*n2*n3*r4114
r213=----------------------------------------
3 2
a11 - a11 *a22
4 3 2 2 2 2
r214=(2*a11 *r29 - 4*a11 *a22*r29 + 2*a11 *a22 *r29 - 2*a11 *n2 *r4114
2 2 2 2
+ 2*a11 *n3 *r4114 + 2*a11*a22*n1 *r4114 + a11*a22*n2 *r4114
2 2 2 2 2 4
- 3*a11*a22*n3 *r4114 - a22 *n1 *r4114 + a22 *n3 *r4114)/(2*a11
3
- 2*a11 *a22)
r215=0
r216=0
r217=0
- 2*a11*n1*n3*r4114 + a22*n1*n3*r4114
r218=----------------------------------------
3 2
a11 - a11 *a22
- 2*a11*n1*n2*r4114 + a22*n1*n2*r4114
r219=----------------------------------------
3 2
a11 - a11 *a22
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
- n3*r452
r316=------------
a11
r317=0
r318=0
n3*r4114
r319=-----------
a11 - a22
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
- n2*r452
r326=------------
a11
- n3*r452
r327=------------
a11
r328=0
n2*r4114
r329=-----------
a11 - a22
r330=0
- n2*r452
r331=------------
a11
r332=0
a11*n3*r4114 - a22*n3*r4114
r333=-----------------------------
2
a11
a11*n2*r4114 - a22*n2*r4114
r334=-----------------------------
2
a11
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- n1*r452
r341=------------
a11
r342=0
- n3*r452
r343=------------
a11
n1*r4114
r344=-----------
a11 - a22
r345=0
- n1*r452
r346=------------
a11
- n2*r452
r347=------------
a11
r348=0
a11*n1*r4114 - a22*n1*r4114
r349=-----------------------------
2
a11
r350=0
r351=0
- n1*r452
r352=------------
a11
- n3*r4114
r353=-------------
a11
- n2*r4114
r354=-------------
a11
- n1*r4114
r355=-------------
a11
r40=0
r41=0
r42=0
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
r49=0
r410=0
r411=0
r412=0
r413=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
r425=0
r426=0
- r446
r427=---------
2
r428=0
r429=0
- r446
r430=---------
2
r431=r452
r432=0
r433=0
- a22*r4114
r434=---------------
2*a11 - 2*a22
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
r447=0
r448=0
r449=0
r450=0
r451=0
r453=0
r454=0
- r446
r455=---------
2
r456=0
r458=0
r459=0
- r446
r460=---------
2
a11*r452 - a22*r452
r461=---------------------
a11
r462=0
r463=0
r464=0
r465=0
a11*r452 - a22*r452
r466=---------------------
a11
r467=0
r468=0
2
a11*a22*r4114 - a22 *r4114
r469=----------------------------
2
2*a11
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=r446
r484=0
r485=0
r486=0
r487=0
r488=r452
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
a11*r452 - a22*r452
r4102=---------------------
a11
r4103=0
r4104=0
- r446
r4105=---------
2
r4106=0
- r446
r4107=---------
2
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
c33=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
3 2 2
r14=( - 2*a11 *n2*r29 + 2*a11 *a22*n2*r29 + 2*a11*n1 *n2*r4114
2 2 2 4
- 2*a11*n2*n3 *r4114 - a22*n1 *n2*r4114 + a22*n2*n3 *r4114)/(2*a11
3
- 2*a11 *a22)
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, r446, r4114, r452, n3, n1, n2, 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}
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,
c33,
m1,
m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2 2
HAM=u1 *a11 + u1*n1 + u2 *a22 + u2*n2 + u3*n3
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 2
FI= - u1 *v1*n1 + u1*u2 *v1*(a11 - a22) - u1*u2*v1*n2 - u1*u2*v2*n1
2 3
+ u1*u3 *v1*a11 - u1*u3*v1*n3 - u1*u3*v3*n1 + u2 *v2*(a11 - a22)
2 2 2
+ u2 *u3*v3*(a11 - a22) - u2 *v2*n2 + u2*u3 *v2*a11 - u2*u3*v2*n3
3 2
- u2*u3*v3*n2 + u3 *v3*a11 - u3 *v3*n3
= a product of the elements of: {u1*v1 + u2*v2 + u3*v3,
2 2
- u1*n1 + u2 *(a11 - a22) - u2*n2 + u3 *a11 - u3*n3}
{HAM,FI} = 0
3 3 2
FI=u1 *( - 2*a11 *n1 + 2*a11 *a22*n1)
2 2 4 3 2 2
+ u1 *u2 *(2*a11 - 4*a11 *a22 + 2*a11 *a22 )
2 3 2 2 2 4 3
+ u1 *u2*( - 2*a11 *n2 + 2*a11 *a22*n2) + u1 *u3 *(2*a11 - 2*a11 *a22)
2 3 2
+ u1 *u3*( - 2*a11 *n3 + 2*a11 *a22*n3)
2 3 2 2
+ u1*u2 *(2*a11 *n1 - 4*a11 *a22*n1 + 2*a11*a22 *n1)
2 2 3
+ u1*u2*( - 4*a11 *n1*n2 + 2*a11*a22*n1*n2) + 2*u1*u3 *a11 *n1
2
+ u1*u3*( - 4*a11 *n1*n3 + 2*a11*a22*n1*n3)
3 2 3 2
+ u1*(2*a11*n1 - 2*a11*n1*n3 - a22*n1 + a22*n1*n3 )
4 3 2 2 3
+ u2 *(a11 *a22 - 2*a11 *a22 + a11*a22 )
3 3 2 2
+ u2 *(2*a11 *n2 - 4*a11 *a22*n2 + 2*a11*a22 *n2)
2 3 2 2 2 2 2
+ u2 *u3*(2*a11 *n3 - 4*a11 *a22*n3 + 2*a11*a22 *n3) + u2 *( - 2*a11 *n2
2 2 2 2 2 2 2
+ 2*a11 *n3 + 2*a11*a22*n1 + a11*a22*n2 - 3*a11*a22*n3 - a22 *n1
2 2 2 3
+ a22 *n3 ) + 2*u2*u3 *a11 *n2
2
+ u2*u3*( - 4*a11 *n2*n3 + 2*a11*a22*n2*n3)
2 2 2 2 4 3
+ u2*(2*a11*n1 *n2 - 2*a11*n2*n3 - a22*n1 *n2 + a22*n2*n3 ) - u3 *a11 *a22
3 3 2 3 2 3
+ 2*u3 *a11 *n3 + u3*(2*a11*n1 *n3 - 2*a11*n3 - a22*n1 *n3 + a22*n3 )
= a product of the elements of: {2,
3 3 2 2 2 4 3 2 2
u1 *( - a11 *n1 + a11 *a22*n1) + u1 *u2 *(a11 - 2*a11 *a22 + a11 *a22 )
2 3 2 2 2 4 3
+ u1 *u2*( - a11 *n2 + a11 *a22*n2) + u1 *u3 *(a11 - a11 *a22)
2 3 2
+ u1 *u3*( - a11 *n3 + a11 *a22*n3)
2 3 2 2
+ u1*u2 *(a11 *n1 - 2*a11 *a22*n1 + a11*a22 *n1)
2 2 3
+ u1*u2*( - 2*a11 *n1*n2 + a11*a22*n1*n2) + u1*u3 *a11 *n1
2
+ u1*u3*( - 2*a11 *n1*n3 + a11*a22*n1*n3)
3 2 3 2
2*a11*n1 - 2*a11*n1*n3 - a22*n1 + a22*n1*n3
+ u1*-------------------------------------------------
2
3 2 2 3
4 a11 *a22 - 2*a11 *a22 + a11*a22
+ u2 *-----------------------------------
2
3 3 2 2
+ u2 *(a11 *n2 - 2*a11 *a22*n2 + a11*a22 *n2)
2 3 2 2 2 2 2
+ u2 *u3*(a11 *n3 - 2*a11 *a22*n3 + a11*a22 *n3) + u2 *( - 2*a11 *n2
2 2 2 2 2 2 2
+ 2*a11 *n3 + 2*a11*a22*n1 + a11*a22*n2 - 3*a11*a22*n3 - a22 *n1
2 2 2 3 2
+ a22 *n3 )/2 + u2*u3 *a11 *n2 + u2*u3*( - 2*a11 *n2*n3 + a11*a22*n2*n3)
2 2 2 2
2*a11*n1 *n2 - 2*a11*n2*n3 - a22*n1 *n2 + a22*n2*n3
+ u2*-------------------------------------------------------
2
4 3
- u3 *a11 *a22 3 3
+ ----------------- + u3 *a11 *n3
2
2 3 2 3
2*a11*n1 *n3 - 2*a11*n3 - a22*n1 *n3 + a22*n3
+ u3*-------------------------------------------------}
2
{HAM,FI} = 0
2 2 2 2 2 2 2 2
FI= - u1 *v2 - u1 *v3 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2 *v1 - u2 *v3
2 2 2 2
+ 2*u2*u3*v2*v3 - u3 *v1 - u3 *v2
which the program can not factorize further.
{HAM,FI} = 0
2 2
FI= - u1*n1 + u2 *(a11 - a22) - u2*n2 + u3 *a11 - u3*n3
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*n1 + u2**2*a22 + u2*n2 + u3*n3$
FI= - u1**2*v1*n1 + u1*u2**2*v1*(a11 - a22) - u1*u2*v1*n2 - u1*u2*v2*n1 + u1*u3
**2*v1*a11 - u1*u3*v1*n3 - u1*u3*v3*n1 + u2**3*v2*(a11 - a22) + u2**2*u3*v3*(a11
- a22) - u2**2*v2*n2 + u2*u3**2*v2*a11 - u2*u3*v2*n3 - u2*u3*v3*n2 + u3**3*v3*
a11 - u3**2*v3*n3$
FI=u1**3*( - 2*a11**3*n1 + 2*a11**2*a22*n1) + u1**2*u2**2*(2*a11**4 - 4*a11**3*
a22 + 2*a11**2*a22**2) + u1**2*u2*( - 2*a11**3*n2 + 2*a11**2*a22*n2) + u1**2*u3
**2*(2*a11**4 - 2*a11**3*a22) + u1**2*u3*( - 2*a11**3*n3 + 2*a11**2*a22*n3) + u1
*u2**2*(2*a11**3*n1 - 4*a11**2*a22*n1 + 2*a11*a22**2*n1) + u1*u2*( - 4*a11**2*n1
*n2 + 2*a11*a22*n1*n2) + 2*u1*u3**2*a11**3*n1 + u1*u3*( - 4*a11**2*n1*n3 + 2*a11
*a22*n1*n3) + u1*(2*a11*n1**3 - 2*a11*n1*n3**2 - a22*n1**3 + a22*n1*n3**2) + u2
**4*(a11**3*a22 - 2*a11**2*a22**2 + a11*a22**3) + u2**3*(2*a11**3*n2 - 4*a11**2*
a22*n2 + 2*a11*a22**2*n2) + u2**2*u3*(2*a11**3*n3 - 4*a11**2*a22*n3 + 2*a11*a22
**2*n3) + u2**2*( - 2*a11**2*n2**2 + 2*a11**2*n3**2 + 2*a11*a22*n1**2 + a11*a22*
n2**2 - 3*a11*a22*n3**2 - a22**2*n1**2 + a22**2*n3**2) + 2*u2*u3**2*a11**3*n2 +
u2*u3*( - 4*a11**2*n2*n3 + 2*a11*a22*n2*n3) + u2*(2*a11*n1**2*n2 - 2*a11*n2*n3**
2 - a22*n1**2*n2 + a22*n2*n3**2) - u3**4*a11**3*a22 + 2*u3**3*a11**3*n3 + u3*(2*
a11*n1**2*n3 - 2*a11*n3**3 - a22*n1**2*n3 + a22*n3**3)$
FI= - u1**2*v2**2 - u1**2*v3**2 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2**2*v1**2 -
u2**2*v3**2 + 2*u2*u3*v2*v3 - u3**2*v1**2 - u3**2*v2**2$
FI= - u1*n1 + u2**2*(a11 - a22) - u2*n2 + u3**2*a11 - u3*n3$