Solution 9 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r10=0
1 2 1 2
r11=( - ---*a22 *b33*n2*r29 + a22*a33*b33*n2*r29 - ---*a33 *b33*n2*r29
2 2
1 2 4 3 2 2 3
+ ---*b33*n2*n3 *r434)/(a22 - 3*a22 *a33 + 3*a22 *a33 - a22*a33 )
2
r12=0
2 2 3
- a22 *n3*r29 + 2*a22*a33*n3*r29 - a33 *n3*r29 + n3 *r434
r13=------------------------------------------------------------
3 2 2 3
a22 - 3*a22 *a33 + 3*a22*a33 - a33
2 2 2
- a22 *n2*r29 + 2*a22*a33*n2*r29 - a33 *n2*r29 + n2*n3 *r434
r14=---------------------------------------------------------------
3 2 2 3
a22 - 3*a22 *a33 + 3*a22*a33 - a33
r15=0
1 2 2
- ---*b33 *n2 *r434
4
r20=-------------------------------
4 3 2 2
a22 - 2*a22 *a33 + a22 *a33
r21=0
r23=0
r24=0
1 2 2
- ---*b33 *n2 *r434
4
r25=-------------------------------
4 3 2 2
a22 - 2*a22 *a33 + a22 *a33
2
- b33*n2 *r434
r26=------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
b33*n2*n3*r434
r27=------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
r28=0
r210=0
r212=0
2*n2*n3*r434
r213=-------------------------
2 2
a22 - 2*a22*a33 + a33
2
n2 *r434
r214=-------------------------
2 2
a22 - 2*a22*a33 + a33
r215=0
r216=0
2
- b33*n2 *r434
r217=------------------------------
3 2 2
a22 - 2*a22 *a33 + a22*a33
r218=0
r219=0
r30=0
1
- ---*b33*n2*r425
2
r31=--------------------
2
a22 - a22*a33
r32=0
1
- ---*b33*n2*r425
2
r33=--------------------
2
a22 - a22*a33
r34=0
r35=0
r36=0
r37=0
1
- ---*b33*n2*r425
2
r38=--------------------
2
a22 - a22*a33
r39=0
- n3*r425
r310=------------
a22 - a33
n2*r425
r311=-----------
a22 - a33
- n3*r425
r312=------------
a22 - a33
r313=0
r314=0
- n3*r425
r315=------------
a22 - a33
r316=0
- b33*n2*r434
r317=----------------
2
a22 - a22*a33
r318=0
- 2*n3*r434
r319=--------------
a22 - a33
- n2*r425
r320=------------
a22 - a33
r323=0
- n2*r425
r325=------------
a22 - a33
r326=0
r328=0
- 2*n2*r434
r329=--------------
a22 - a33
r330=0
r332=0
r333=0
r334=0
r335=0
r336=0
r337=0
r338=0
n2*r425
r339=-----------
a22 - a33
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
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
r427=r425
r428=0
r429=0
r430=r425
r431=0
r432=0
r433=0
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
r464=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
r4114=0
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
- b33*n3
m3=-----------
a22 - a33
1 1
- ---*a22*b33*n2 - ---*a33*b33*n2
2 2
m2=------------------------------------
2
a22 - a22*a33
m1=0
n1=0
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
a23=0
a13=0
a12=0
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, r425, r434, n3, b33, n2, a33, 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.
{b33,a22,a33,a22 - a33,a11,r434,n2}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11 - a22,
a12,
a13,
a23,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
c11,
c12,
c13,
c22,
c23,
a22*c33 + 1/4*b33**2,
n1,
m1,
a22**2*m2 - a22*a33*m2 + 1/2*a22*b33*n2 + 1/2*a33*b33*n2,
a22*m3 - a33*m3 + b33*n3}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a22 + u2 *a22 + u2*n2 + u3 *a33 + u3*v3*b33 + u3*n3
1 1 1 2 2
- ---*a22*b33*n2 - ---*a33*b33*n2 - ---*v3 *b33
2 2 4
+ v2*------------------------------------ + -----------------
2 a22
a22 - a22*a33
- v3*b33*n3
+ --------------
a22 - a33
has apart from the Hamiltonian and Casimirs the following 3 first integrals:
2 2 2 2 3 2 2 2
FI=u1*v1*( - a22 *b33*n2 + a22*a33*b33*n2 ) + u2 *(a22 *n2 - a22 *a33*n2 )
2 4 3 2 2
+ u2*u3 *( - 2*a22 *n2 + 4*a22 *a33*n2 - 2*a22 *a33 *n2)
3 2 2 2
+ u2*u3*(2*a22 *n2*n3 - 2*a22 *a33*n2*n3) + u2*a22 *n2*n3
4 5 4 3 2 2 3
+ u3 *(a22 - 3*a22 *a33 + 3*a22 *a33 - a22 *a33 )
3 4 3 2 2
+ u3 *( - 2*a22 *n3 + 4*a22 *a33*n3 - 2*a22 *a33 *n3)
2 3 2 2
+ u3 *v2*( - a22 *b33*n2 + 2*a22 *a33*b33*n2 - a22*a33 *b33*n2)
2
+ u3*v2*(a22 *b33*n2*n3 - a22*a33*b33*n2*n3)
2 2 2 2 3
+ u3*v3*( - a22 *b33*n2 + a22*a33*b33*n2 ) + u3*a22 *n3
2 1 2 2 1 2 2 1 2
+ v1 *( - ---*a22*b33 *n2 + ---*a33*b33 *n2 ) + ---*v2*a22*b33*n2*n3
4 4 2
2 1 2 2 1 2 2
+ v3 *( - ---*a22*b33 *n2 + ---*a33*b33 *n2 )
4 4
which the program can not factorize further.
{HAM,FI} = 0
3 2 2
FI=u1*v1*v2*(a22 *n2 - 2*a22 *a33*n2 + a22*a33 *n2)
2 3 2 2
+ u2*v1 *( - a22 *n2 + 2*a22 *a33*n2 - a22*a33 *n2)
2 3 2 2
+ u2*v3 *( - a22 *n2 + 2*a22 *a33*n2 - a22*a33 *n2)
2 2 4 3 2 2 3
+ u3 *v1 *(a22 - 3*a22 *a33 + 3*a22 *a33 - a22*a33 )
2 2 4 3 2 2 3
+ u3 *v2 *(a22 - 3*a22 *a33 + 3*a22 *a33 - a22*a33 )
2 2 4 3 2 2 3
+ u3 *v3 *(a22 - 3*a22 *a33 + 3*a22 *a33 - a22*a33 )
2 3 2 2
+ u3*v1 *( - a22 *n3 + 2*a22 *a33*n3 - a22*a33 *n3)
2 3 2 2
+ u3*v2 *( - a22 *n3 + 2*a22 *a33*n3 - a22*a33 *n3)
3 2 2
+ u3*v2*v3*(a22 *n2 - 2*a22 *a33*n2 + a22*a33 *n2)
2 3 2 2
+ u3*v3 *( - a22 *n3 + 2*a22 *a33*n3 - a22*a33 *n3)
2 1 2 1 2
+ v1 *v2*( - ---*a22 *b33*n2 + a22*a33*b33*n2 - ---*a33 *b33*n2)
2 2
3 1 2 1 2
+ v2 *( - ---*a22 *b33*n2 + a22*a33*b33*n2 - ---*a33 *b33*n2)
2 2
2 1 2 1 2
+ v2*v3 *( - ---*a22 *b33*n2 + a22*a33*b33*n2 - ---*a33 *b33*n2)
2 2
= a product of the elements of: {a22 - a33,
a22 - a33,
2 2 2 2 2
u1*v1*v2*a22*n2 - u2*v1 *a22*n2 - u2*v3 *a22*n2 + u3 *v1 *(a22 - a22*a33)
2 2 2 2 2 2 2
+ u3 *v2 *(a22 - a22*a33) + u3 *v3 *(a22 - a22*a33) - u3*v1 *a22*n3
2 2 1 2
- u3*v2 *a22*n3 + u3*v2*v3*a22*n2 - u3*v3 *a22*n3 - ---*v1 *v2*b33*n2
2
1 3 1 2
- ---*v2 *b33*n2 - ---*v2*v3 *b33*n2}
2 2
{HAM,FI} = {2,
a22 - a33,
a22 - a33,
u1*v1 + u2*v2 + u3*v3,
n2,
a22,
1 1
u1*v3*a22 - u3*v1*a33 - ---*v1*v3*b33 - ---*v1*n3}
2 2
3 2 2
FI=u2*( - a22 *n2 + 2*a22 *a33*n2 - a22*a33 *n2)
2 4 3 2 2 3
+ u3 *(a22 - 3*a22 *a33 + 3*a22 *a33 - a22*a33 )
3 2 2
+ u3*( - a22 *n3 + 2*a22 *a33*n3 - a22*a33 *n3)
1 2 1 2
+ v2*( - ---*a22 *b33*n2 + a22*a33*b33*n2 - ---*a33 *b33*n2)
2 2
= a product of the elements of: {a22 - a33,
a22 - a33,
2 2 1
- u2*a22*n2 + u3 *(a22 - a22*a33) - u3*a22*n3 - ---*v2*b33*n2}
2
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a22 + u2**2*a22 + u2*n2 + u3**2*a33 + u3*v3*b33 + u3*n3 + v2*( - 1/2*
a22*b33*n2 - 1/2*a33*b33*n2)/(a22**2 - a22*a33) + ( - 1/4*v3**2*b33**2)/a22 + (
- v3*b33*n3)/(a22 - a33)$
FI=u1*v1*( - a22**2*b33*n2**2 + a22*a33*b33*n2**2) + u2**2*(a22**3*n2**2 - a22**
2*a33*n2**2) + u2*u3**2*( - 2*a22**4*n2 + 4*a22**3*a33*n2 - 2*a22**2*a33**2*n2)
+ u2*u3*(2*a22**3*n2*n3 - 2*a22**2*a33*n2*n3) + u2*a22**2*n2*n3**2 + u3**4*(a22
**5 - 3*a22**4*a33 + 3*a22**3*a33**2 - a22**2*a33**3) + u3**3*( - 2*a22**4*n3 +
4*a22**3*a33*n3 - 2*a22**2*a33**2*n3) + u3**2*v2*( - a22**3*b33*n2 + 2*a22**2*
a33*b33*n2 - a22*a33**2*b33*n2) + u3*v2*(a22**2*b33*n2*n3 - a22*a33*b33*n2*n3) +
u3*v3*( - a22**2*b33*n2**2 + a22*a33*b33*n2**2) + u3*a22**2*n3**3 + v1**2*( - 1
/4*a22*b33**2*n2**2 + 1/4*a33*b33**2*n2**2) + 1/2*v2*a22*b33*n2*n3**2 + v3**2*(
- 1/4*a22*b33**2*n2**2 + 1/4*a33*b33**2*n2**2)$
FI=u1*v1*v2*(a22**3*n2 - 2*a22**2*a33*n2 + a22*a33**2*n2) + u2*v1**2*( - a22**3*
n2 + 2*a22**2*a33*n2 - a22*a33**2*n2) + u2*v3**2*( - a22**3*n2 + 2*a22**2*a33*n2
- a22*a33**2*n2) + u3**2*v1**2*(a22**4 - 3*a22**3*a33 + 3*a22**2*a33**2 - a22*
a33**3) + u3**2*v2**2*(a22**4 - 3*a22**3*a33 + 3*a22**2*a33**2 - a22*a33**3) +
u3**2*v3**2*(a22**4 - 3*a22**3*a33 + 3*a22**2*a33**2 - a22*a33**3) + u3*v1**2*(
- a22**3*n3 + 2*a22**2*a33*n3 - a22*a33**2*n3) + u3*v2**2*( - a22**3*n3 + 2*a22
**2*a33*n3 - a22*a33**2*n3) + u3*v2*v3*(a22**3*n2 - 2*a22**2*a33*n2 + a22*a33**2
*n2) + u3*v3**2*( - a22**3*n3 + 2*a22**2*a33*n3 - a22*a33**2*n3) + v1**2*v2*( -
1/2*a22**2*b33*n2 + a22*a33*b33*n2 - 1/2*a33**2*b33*n2) + v2**3*( - 1/2*a22**2*
b33*n2 + a22*a33*b33*n2 - 1/2*a33**2*b33*n2) + v2*v3**2*( - 1/2*a22**2*b33*n2 +
a22*a33*b33*n2 - 1/2*a33**2*b33*n2)$
FI=u2*( - a22**3*n2 + 2*a22**2*a33*n2 - a22*a33**2*n2) + u3**2*(a22**4 - 3*a22**
3*a33 + 3*a22**2*a33**2 - a22*a33**3) + u3*( - a22**3*n3 + 2*a22**2*a33*n3 - a22
*a33**2*n3) + v2*( - 1/2*a22**2*b33*n2 + a22*a33*b33*n2 - 1/2*a33**2*b33*n2)$