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