Solution 4 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
3 2 2 2
r13=( - a11 *n3*r214 + 2*a11 *a22*n3*r214 - a11*a22 *n3*r214 + a11*n1 *n3*r4119
2 1 2 1 2 4
- a11*n2 *n3*r4119 - ---*a22*n1 *n3*r4119 + ---*a22*n2 *n3*r4119)/(a11
2 2
3 2 2 3
- 3*a11 *a22 + 3*a11 *a22 - a11*a22 )
3 2 2 2
r14=( - a11 *n2*r214 + 2*a11 *a22*n2*r214 - a11*a22 *n2*r214 + a11*n1 *n2*r4119
3 1 2 1 3 4
- a11*n2 *r4119 - ---*a22*n1 *n2*r4119 + ---*a22*n2 *r4119)/(a11
2 2
3 2 2 3
- 3*a11 *a22 + 3*a11 *a22 - a11*a22 )
3 2 2 3
r15=( - a11 *n1*r214 + 2*a11 *a22*n1*r214 - a11*a22 *n1*r214 + a11*n1 *r4119
2 1 3 1 2 4
- a11*n1*n2 *r4119 - ---*a22*n1 *r4119 + ---*a22*n1*n2 *r4119)/(a11
2 2
3 2 2 3
- 3*a11 *a22 + 3*a11 *a22 - a11*a22 )
r20=0
r21=0
r22=0
r23=0
r24=0
r26=0
r27=0
r28=0
4 3 2 2 2 2
r29=(a11 *r214 - 2*a11 *a22*r214 + a11 *a22 *r214 + a11 *n2 *r4119
2 2 2 1 2
- a11 *n3 *r4119 - a11*a22*n1 *r4119 - ---*a11*a22*n2 *r4119
2
3 2 1 2 2 1 2 2 4
+ ---*a11*a22*n3 *r4119 + ---*a22 *n1 *r4119 - ---*a22 *n3 *r4119)/(a11
2 2 2
3 2 2 3
- 3*a11 *a22 + 3*a11 *a22 - a11*a22 )
r210=0
r212=0
- 2*a11*n2*n3*r4119 + a22*n2*n3*r4119
r213=----------------------------------------
3 2 2
a11 - 2*a11 *a22 + a11*a22
r215=0
r216=0
r217=0
- 2*a11*n1*n3*r4119 + a22*n1*n3*r4119
r218=----------------------------------------
3 2 2
a11 - 2*a11 *a22 + a11*a22
- 2*a11*n1*n2*r4119 + a22*n1*n2*r4119
r219=----------------------------------------
3 2 2
a11 - 2*a11 *a22 + a11*a22
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
- n3*r483
r310=------------
a11 - a22
n2*r483
r311=-----------
a11 - a22
- n3*r483
r312=------------
a11 - a22
r313=0
r314=0
- n3*r483
r315=------------
a11 - a22
r316=0
r317=0
r318=0
a11*n3*r4119
r319=-------------------------
2 2
a11 - 2*a11*a22 + a22
- n2*r483
r320=------------
a11 - a22
r323=0
- n2*r483
r325=------------
a11 - a22
r326=0
r328=0
a11*n2*r4119
r329=-------------------------
2 2
a11 - 2*a11*a22 + a22
r330=0
r332=0
n3*r4119
r333=----------
a11
n2*r4119
r334=----------
a11
- n1*r483
r335=------------
a11 - a22
r336=0
- n1*r483
r337=------------
a11 - a22
r338=0
n2*r483
r339=-----------
a11 - a22
- n1*r483
r340=------------
a11 - a22
r341=0
r342=0
r343=0
a11*n1*r4119
r344=-------------------------
2 2
a11 - 2*a11*a22 + a22
r345=0
r347=0
r348=0
n1*r4119
r349=----------
a11
r350=0
r351=0
r352=0
- n3*r4119
r353=-------------
a11 - a22
- n2*r4119
r354=-------------
a11 - a22
- n1*r4119
r355=-------------
a11 - a22
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
a11*r483
r425=-----------
a11 - a22
r426=0
1 1
---*a11*r483 + ---*a22*r483
2 2
r427=-----------------------------
a11 - a22
r428=0
r429=0
1 1
---*a11*r483 + ---*a22*r483
2 2
r430=-----------------------------
a11 - a22
r431=0
r432=0
r433=0
1
- ---*a11*a22*r4119
2
r434=-------------------------
2 2
a11 - 2*a11*a22 + a22
r435=0
r439=0
r442=0
r444=0
r445=0
r448=0
r450=0
r451=0
r453=0
r454=0
1
r455=---*r483
2
r458=0
1
r460=---*r483
2
r461=0
r463=0
r464=0
r465=0
r467=0
r468=0
1
---*a22*r4119
2
r469=---------------
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
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
1
r4105= - ---*r483
2
r4106=0
1
r4107= - ---*r483
2
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
a11*r4119
r4114=-----------
a11 - a22
r4115=0
r4117=0
r4118=0
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
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:
r214, r483, r4119, n1, n2, n3, 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.
{n3,a11 - a22,a22,a11,n2}
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 3 first integrals:
3 3 2 2
FI=u1 *( - a11 *n1 + 2*a11 *a22*n1 - a11*a22 *n1)
2 2 4 3 2 2 3
+ u1 *u2 *(a11 - 3*a11 *a22 + 3*a11 *a22 - a11*a22 )
2 3 2 2
+ u1 *u2*( - a11 *n2 + 2*a11 *a22*n2 - a11*a22 *n2)
2 2 4 3 2 2
+ u1 *u3 *(a11 - 2*a11 *a22 + a11 *a22 )
2 3 2 2
+ u1 *u3*( - a11 *n3 + 2*a11 *a22*n3 - a11*a22 *n3)
2 3 2 2 3
+ u1*u2 *(a11 *n1 - 3*a11 *a22*n1 + 3*a11*a22 *n1 - a22 *n1)
2 2
+ u1*u2*( - 2*a11 *n1*n2 + 3*a11*a22*n1*n2 - a22 *n1*n2)
2 3 2
+ u1*u3 *(a11 *n1 - a11 *a22*n1)
2 2
+ u1*u3*( - 2*a11 *n1*n3 + 3*a11*a22*n1*n3 - a22 *n1*n3)
3 2 1 3 1 2
+ u1*(a11*n1 - a11*n1*n2 - ---*a22*n1 + ---*a22*n1*n2 )
2 2
4 1 3 3 2 2 3 3 1 4
+ u2 *(---*a11 *a22 - ---*a11 *a22 + ---*a11*a22 - ---*a22 )
2 2 2 2
3 3 2 2 3
+ u2 *(a11 *n2 - 3*a11 *a22*n2 + 3*a11*a22 *n2 - a22 *n2)
2 3 2 2 3
+ u2 *u3*(a11 *n3 - 3*a11 *a22*n3 + 3*a11*a22 *n3 - a22 *n3)
2 3 2
+ u2*u3 *(a11 *n2 - a11 *a22*n2)
2 2
+ u2*u3*( - 2*a11 *n2*n3 + 3*a11*a22*n2*n3 - a22 *n2*n3)
2 3 1 2 1 3
+ u2*(a11*n1 *n2 - a11*n2 - ---*a22*n1 *n2 + ---*a22*n2 )
2 2
4 1 3 1 2 2 3 3 2 2
+ u3 *( - ---*a11 *a22 + ---*a11 *a22 ) + u3 *(a11 *n3 - a11 *a22*n3) + u3 *
2 2
2 2 2 2 2 1 2 3 2
(a11 *n2 - a11 *n3 - a11*a22*n1 - ---*a11*a22*n2 + ---*a11*a22*n3
2 2
1 2 2 1 2 2
+ ---*a22 *n1 - ---*a22 *n3 )
2 2
2 2 1 2 1 2
+ u3*(a11*n1 *n3 - a11*n2 *n3 - ---*a22*n1 *n3 + ---*a22*n2 *n3)
2 2
which the program can not factorize further.
{HAM,FI} = 0
2 2 1 3 3 2 3 2 1 3
FI=u1 *v2 *( - ---*a11 + ---*a11 *a22 - ---*a11*a22 + ---*a22 )
2 2 2 2
2 2 1 3 3 2 3 2 1 3
+ u1 *v3 *( - ---*a11 + ---*a11 *a22 - ---*a11*a22 + ---*a22 )
2 2 2 2
3 2 2 3
+ u1*u3*v1*v3*(a11 - 3*a11 *a22 + 3*a11*a22 - a22 )
2 2 2
+ u1*v1 *( - a11 *n1 + 2*a11*a22*n1 - a22 *n1)
2 2
+ u1*v1*v2*(a11 *n2 - 2*a11*a22*n2 + a22 *n2)
2 2 2
+ u1*v2 *( - a11 *n1 + 2*a11*a22*n1 - a22 *n1)
2 2 2
+ u1*v3 *( - a11 *n1 + 2*a11*a22*n1 - a22 *n1)
2 2 1 3 3 2 3 2 1 3
+ u2 *v1 *(---*a11 - ---*a11 *a22 + ---*a11*a22 - ---*a22 )
2 2 2 2
2 2 1 3 3 2 3 2 1 3
+ u2 *v3 *(---*a11 - ---*a11 *a22 + ---*a11*a22 - ---*a22 )
2 2 2 2
2 2 2
+ u2*v1 *( - a11 *n2 + 2*a11*a22*n2 - a22 *n2)
2 2 2
+ u2*v3 *( - a11 *n2 + 2*a11*a22*n2 - a22 *n2)
2 2 1 3 1 2 1 2 1 3
+ u3 *v1 *(---*a11 - ---*a11 *a22 - ---*a11*a22 + ---*a22 )
2 2 2 2
2 2 1 3 1 2 1 2 1 3
+ u3 *v2 *(---*a11 - ---*a11 *a22 - ---*a11*a22 + ---*a22 )
2 2 2 2
2 2 3 2 2
+ u3 *v3 *(a11 - 2*a11 *a22 + a11*a22 )
2 2 2
+ u3*v1 *( - a11 *n3 + 2*a11*a22*n3 - a22 *n3)
2 2 2
+ u3*v2 *( - a11 *n3 + 2*a11*a22*n3 - a22 *n3)
2 2
+ u3*v2*v3*(a11 *n2 - 2*a11*a22*n2 + a22 *n2)
2 2 2
+ u3*v3 *( - a11 *n3 + 2*a11*a22*n3 - a22 *n3)
1
= a product of the elements of: { - ---,
2
a11 - a22,
a11 - a22,
2 2 2 2
u1 *v2 *(a11 - a22) + u1 *v3 *(a11 - a22) + u1*u3*v1*v3*( - 2*a11 + 2*a22)
2 2 2
+ 2*u1*v1 *n1 - 2*u1*v1*v2*n2 + 2*u1*v2 *n1 + 2*u1*v3 *n1
2 2 2 2 2
+ u2 *v1 *( - a11 + a22) + u2 *v3 *( - a11 + a22) + 2*u2*v1 *n2
2 2 2 2 2
+ 2*u2*v3 *n2 + u3 *v1 *( - a11 - a22) + u3 *v2 *( - a11 - a22)
2 2 2 2 2
- 2*u3 *v3 *a11 + 2*u3*v1 *n3 + 2*u3*v2 *n3 - 2*u3*v2*v3*n2 + 2*u3*v3 *n3}
{HAM,FI} = {2,
- a11 + a22,
a11 - a22,
u1*v1 + u2*v2 + u3*v3,
2 2
u1*u2*v3*(a11 - a11*a22) + u1*u3*v2*(a11 - a11*a22)
1 1
+ u1*v2*( - ---*a11*n3 + ---*a22*n3) - u1*v3*a11*n2
2 2
1 1
+ u2*v1*( - ---*a11*n3 + ---*a22*n3)
2 2
1 1 1 1
+ u2*v3*(---*a11*n1 - ---*a22*n1) + u3*v2*(---*a11*n1 - ---*a22*n1)
2 2 2 2
1 1
+ ---*v1*n2*n3 - ---*v3*n1*n2}
2 2
2 2
FI=u1*( - a11 *n1 + 2*a11*a22*n1 - a22 *n1)
2 3 2 2 3
+ u2 *(a11 - 3*a11 *a22 + 3*a11*a22 - a22 )
2 2
+ u2*( - a11 *n2 + 2*a11*a22*n2 - a22 *n2)
2 3 2 2
+ u3 *(a11 - 2*a11 *a22 + a11*a22 )
2 2
+ u3*( - a11 *n3 + 2*a11*a22*n3 - a22 *n3)
= a product of the elements of: {a11 - a22,
a11 - a22,
2 2
- u1*n1 + u2 *(a11 - a22) - u2*n2 + u3 *a11 - u3*n3}
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*n1 + u2**2*a22 + u2*n2 + u3*n3$
FI=u1**3*( - a11**3*n1 + 2*a11**2*a22*n1 - a11*a22**2*n1) + u1**2*u2**2*(a11**4
- 3*a11**3*a22 + 3*a11**2*a22**2 - a11*a22**3) + u1**2*u2*( - a11**3*n2 + 2*a11
**2*a22*n2 - a11*a22**2*n2) + u1**2*u3**2*(a11**4 - 2*a11**3*a22 + a11**2*a22**2
) + u1**2*u3*( - a11**3*n3 + 2*a11**2*a22*n3 - a11*a22**2*n3) + u1*u2**2*(a11**3
*n1 - 3*a11**2*a22*n1 + 3*a11*a22**2*n1 - a22**3*n1) + u1*u2*( - 2*a11**2*n1*n2
+ 3*a11*a22*n1*n2 - a22**2*n1*n2) + u1*u3**2*(a11**3*n1 - a11**2*a22*n1) + u1*u3
*( - 2*a11**2*n1*n3 + 3*a11*a22*n1*n3 - a22**2*n1*n3) + u1*(a11*n1**3 - a11*n1*
n2**2 - 1/2*a22*n1**3 + 1/2*a22*n1*n2**2) + u2**4*(1/2*a11**3*a22 - 3/2*a11**2*
a22**2 + 3/2*a11*a22**3 - 1/2*a22**4) + u2**3*(a11**3*n2 - 3*a11**2*a22*n2 + 3*
a11*a22**2*n2 - a22**3*n2) + u2**2*u3*(a11**3*n3 - 3*a11**2*a22*n3 + 3*a11*a22**
2*n3 - a22**3*n3) + u2*u3**2*(a11**3*n2 - a11**2*a22*n2) + u2*u3*( - 2*a11**2*n2
*n3 + 3*a11*a22*n2*n3 - a22**2*n2*n3) + u2*(a11*n1**2*n2 - a11*n2**3 - 1/2*a22*
n1**2*n2 + 1/2*a22*n2**3) + u3**4*( - 1/2*a11**3*a22 + 1/2*a11**2*a22**2) + u3**
3*(a11**3*n3 - a11**2*a22*n3) + u3**2*(a11**2*n2**2 - a11**2*n3**2 - a11*a22*n1
**2 - 1/2*a11*a22*n2**2 + 3/2*a11*a22*n3**2 + 1/2*a22**2*n1**2 - 1/2*a22**2*n3**
2) + u3*(a11*n1**2*n3 - a11*n2**2*n3 - 1/2*a22*n1**2*n3 + 1/2*a22*n2**2*n3)$
FI=u1**2*v2**2*( - 1/2*a11**3 + 3/2*a11**2*a22 - 3/2*a11*a22**2 + 1/2*a22**3) +
u1**2*v3**2*( - 1/2*a11**3 + 3/2*a11**2*a22 - 3/2*a11*a22**2 + 1/2*a22**3) + u1*
u3*v1*v3*(a11**3 - 3*a11**2*a22 + 3*a11*a22**2 - a22**3) + u1*v1**2*( - a11**2*
n1 + 2*a11*a22*n1 - a22**2*n1) + u1*v1*v2*(a11**2*n2 - 2*a11*a22*n2 + a22**2*n2)
+ u1*v2**2*( - a11**2*n1 + 2*a11*a22*n1 - a22**2*n1) + u1*v3**2*( - a11**2*n1 +
2*a11*a22*n1 - a22**2*n1) + u2**2*v1**2*(1/2*a11**3 - 3/2*a11**2*a22 + 3/2*a11*
a22**2 - 1/2*a22**3) + u2**2*v3**2*(1/2*a11**3 - 3/2*a11**2*a22 + 3/2*a11*a22**2
- 1/2*a22**3) + u2*v1**2*( - a11**2*n2 + 2*a11*a22*n2 - a22**2*n2) + u2*v3**2*(
- a11**2*n2 + 2*a11*a22*n2 - a22**2*n2) + u3**2*v1**2*(1/2*a11**3 - 1/2*a11**2*
a22 - 1/2*a11*a22**2 + 1/2*a22**3) + u3**2*v2**2*(1/2*a11**3 - 1/2*a11**2*a22 -
1/2*a11*a22**2 + 1/2*a22**3) + u3**2*v3**2*(a11**3 - 2*a11**2*a22 + a11*a22**2)
+ u3*v1**2*( - a11**2*n3 + 2*a11*a22*n3 - a22**2*n3) + u3*v2**2*( - a11**2*n3 +
2*a11*a22*n3 - a22**2*n3) + u3*v2*v3*(a11**2*n2 - 2*a11*a22*n2 + a22**2*n2) + u3
*v3**2*( - a11**2*n3 + 2*a11*a22*n3 - a22**2*n3)$
FI=u1*( - a11**2*n1 + 2*a11*a22*n1 - a22**2*n1) + u2**2*(a11**3 - 3*a11**2*a22 +
3*a11*a22**2 - a22**3) + u2*( - a11**2*n2 + 2*a11*a22*n2 - a22**2*n2) + u3**2*(
a11**3 - 2*a11**2*a22 + a11*a22**2) + u3*( - a11**2*n3 + 2*a11*a22*n3 - a22**2*
n3)$