Solution 11 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r11=0
r12=0
r14=0
r15=0
r21=0
r22=0
r23=0
r24=0
r27=0
r28=0
r210=0
r212= - r216
r213=0
r215=0
r217=0
r218=0
r219=0
r220=r214
r31=0
r33=0
r34=0
r35=0
r36=0
r37=r32
r38=0
r39=0
r311=0
r313=0
r314=0
r315=r312
r317=0
r318=0
r320=0
r325=0
r326=0
r329=0
r332=0
r334=0
r335=0
r336= - r323
r337=0
r338=0
r339=0
r340=0
r341=0
r342= - r328
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=r330
r351=0
r352=0
r353=r333
r354=0
r355=0
r41=0
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
r49=r42
r410=0
r411=0
r412=0
r413=0
r416=0
r418=0
r419=0
r421=0
r422=r417
r423=0
r424=0
r426=0
r427=0
r428=0
r429=0
r432=0
r433=0
r435=0
r442=0
r444= - r420
r445=0
r450=0
r451=0
r454=0
r458=0
r483
r460=------
2
r463=0
r465=0
r467=r496
r468=0
r4119
r469=-------
2
r470=0
r471= - r439
r472=0
r473=r420
r474=0
r475=0
r476=0
r477=0
r478=2*r420
r479=0
r480=0
r481= - r448
r482=0
r484=0
r485=0
r486=0
r487= - r453
r488=0
r489=0
r490=0
r493=0
r495=0
r498=0
r499=0
r4100=0
r4102=0
r4103=0
r4104=0
r4105=r455
r4106=0
r483
r4107=------
2
r4108=0
r4109=0
r4110=r483
r4111=r461
r4112=0
r4113=0
r4114=r464
r4115=0
r4117=2*r496
r4118=0
r4120=0
r4121= - r496
r4122=0
r4123=0
r4124=0
r4119
r4125=-------
2
m3=0
m2=0
m1=0
n2=0
n1=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
a23=0
a22=0
a13=0
a12=0
a11=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:
r40, r30, r20, r10, r13, r319, r32, r42, r214, r216,
r312, r26, r415, r310, r323, r316, r431, r425, r439, r330,
r455, r417, r420, r333, r328, r453, r4119, r483, r448,
r496, r461, r464, n3, a33
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.
{a33}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11,
a12,
a13,
a22,
a23,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
b33,
c11,
c12,
c13,
c22,
c23,
c33,
n1,
n2,
m1,
m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2
HAM=u3 *a33 + u3*n3
has apart from the Hamiltonian and Casimirs the following 32 first integrals:
2 2 2 2
FI=u1 *u3 + u2 *u3
= a product of the elements of: {u3,
u3,
u1 - i*u2,
u1 + i*u2}
{HAM,FI} = 0
2 2
FI=u1 *u3*v3 + u2 *u3*v3
= a product of the elements of: {v3,
u3,
u1 - i*u2,
u1 + i*u2}
{HAM,FI} = 0
3 2 3
FI= - u1 *v2 + 2*u1 *u2*v1 + u1*u2*u3*v3 + u2 *v1
which the program can not factorize further.
{HAM,FI} = {2,
- u1 + u2,
u1 + u2,
u1*v1 + u2*v2 + u3*v3,
n3
u3*a33 + ----}
2
FI= - u1*u3*v2*v3 + u2*u3*v1*v3
= a product of the elements of: { - v3,u3,u1*v2 - u2*v1}
{HAM,FI} = 0
2 2 2 2 2 2
FI=2*u1 *v1 + u1 *v2 + 2*u1*u3*v1*v3 + u2 *v1
2 2
2 2 u1 *v2
= a product of the elements of: {2,u1 *v1 + --------- + u1*u3*v1*v3
2
2 2
u2 *v1
+ ---------}
2
{HAM,FI} = {4,
u1*v2 + u2*v1,
u1*v1 + u2*v2 + u3*v3,
n3
u3*a33 + ----}
2
4 2 2 4
FI=u1 + 2*u1 *u2 + u2
= a product of the elements of: {u1 - i*u2,
u1 - i*u2,
u1 + i*u2,
u1 + i*u2}
{HAM,FI} = 0
2 2
FI= - u1*u3 *v2 + u2*u3 *v1
= a product of the elements of: { - u3,u3,u1*v2 - u2*v1}
{HAM,FI} = 0
FI= - u1*u3*v2 + u2*u3*v1
= a product of the elements of: { - u3,u1*v2 - u2*v1}
{HAM,FI} = 0
2 2
FI=u1 *u3 + u2 *u3
= a product of the elements of: {u3,u1 - i*u2,u1 + i*u2}
{HAM,FI} = 0
2 3 3
FI=2*u1*v1 *v2 + u1*v2 - u2*v1 + u3*v1*v2*v3
3 3
2 u1*v2 - u2*v1
= a product of the elements of: {2,u1*v1 *v2 + -------- + -----------
2 2
u3*v1*v2*v3
+ -------------}
2
{HAM,FI} = {2,
- v1 + v2,
v1 + v2,
u1*v1 + u2*v2 + u3*v3,
n3
u3*a33 + ----}
2
2 2
FI=u3*v1 *v3 + u3*v2 *v3
= a product of the elements of: {v3,
v1 - i*v2,
v1 + i*v2,
u3}
{HAM,FI} = 0
2 2 2 2
FI=u1 *v3 + u2 *v3
= a product of the elements of: {v3,
v3,
u1 - i*u2,
u1 + i*u2}
{HAM,FI} = 0
2 2
FI=u1 *v3 + u2 *v3
= a product of the elements of: {v3,u1 - i*u2,u1 + i*u2}
{HAM,FI} = 0
2 2
FI= - u1*v2*v3 + u2*v1*v3
= a product of the elements of: { - v3,v3,u1*v2 - u2*v1}
{HAM,FI} = 0
2 2
FI=u3 *v3
= a product of the elements of: {v3,v3,u3,u3}
{HAM,FI} = 0
3
FI=u3 *v3
= a product of the elements of: {u3,u3,u3,v3}
{HAM,FI} = 0
2
FI=u3 *v3
= a product of the elements of: {u3,u3,v3}
{HAM,FI} = 0
FI= - u1*v2*v3 + u2*v1*v3
= a product of the elements of: { - v3,u1*v2 - u2*v1}
{HAM,FI} = 0
2
FI=u3*v3
= a product of the elements of: {v3,v3,u3}
{HAM,FI} = 0
3
FI=u3*v3
= a product of the elements of: {v3,v3,v3,u3}
{HAM,FI} = 0
FI=u3*v3
= a product of the elements of: {v3,u3}
{HAM,FI} = 0
2 2
FI=u3*v1 + u3*v2
= a product of the elements of: {v1 - i*v2,v1 + i*v2,u3}
{HAM,FI} = 0
FI=u1*v2 - u2*v1
which the program can not factorize further.
{HAM,FI} = 0
2 2
FI=u1 + u2
= a product of the elements of: {u1 - i*u2,u1 + i*u2}
{HAM,FI} = 0
2 2 2 2
FI=v1 *v3 + v2 *v3
= a product of the elements of: {v3,
v3,
v1 - i*v2,
v1 + i*v2}
{HAM,FI} = 0
2 2
FI=v1 *v3 + v2 *v3
= a product of the elements of: {v3,v1 - i*v2,v1 + i*v2}
{HAM,FI} = 0
3
FI=u3
= a product of the elements of: {u3,u3,u3}
{HAM,FI} = 0
FI=u3
which the program can not factorize further.
{HAM,FI} = 0
FI=v3
which the program can not factorize further.
{HAM,FI} = 0
2
FI=v3
= a product of the elements of: {v3,v3}
{HAM,FI} = 0
3
FI=v3
= a product of the elements of: {v3,v3,v3}
{HAM,FI} = 0
4
FI=v3
= a product of the elements of: {v3,v3,v3,v3}
{HAM,FI} = 0
And again in machine readable form:
HAM=u3**2*a33 + u3*n3$
FI=u1**2*u3**2 + u2**2*u3**2$
FI=u1**2*u3*v3 + u2**2*u3*v3$
FI= - u1**3*v2 + 2*u1**2*u2*v1 + u1*u2*u3*v3 + u2**3*v1$
FI= - u1*u3*v2*v3 + u2*u3*v1*v3$
FI=2*u1**2*v1**2 + u1**2*v2**2 + 2*u1*u3*v1*v3 + u2**2*v1**2$
FI=u1**4 + 2*u1**2*u2**2 + u2**4$
FI= - u1*u3**2*v2 + u2*u3**2*v1$
FI= - u1*u3*v2 + u2*u3*v1$
FI=u1**2*u3 + u2**2*u3$
FI=2*u1*v1**2*v2 + u1*v2**3 - u2*v1**3 + u3*v1*v2*v3$
FI=u3*v1**2*v3 + u3*v2**2*v3$
FI=u1**2*v3**2 + u2**2*v3**2$
FI=u1**2*v3 + u2**2*v3$
FI= - u1*v2*v3**2 + u2*v1*v3**2$
FI=u3**2*v3**2$
FI=u3**3*v3$
FI=u3**2*v3$
FI= - u1*v2*v3 + u2*v1*v3$
FI=u3*v3**2$
FI=u3*v3**3$
FI=u3*v3$
FI=u3*v1**2 + u3*v2**2$
FI=u1*v2 - u2*v1$
FI=u1**2 + u2**2$
FI=v1**2*v3**2 + v2**2*v3**2$
FI=v1**2*v3 + v2**2*v3$
FI=u3**3$
FI=u3$
FI=v3$
FI=v3**2$
FI=v3**3$
FI=v3**4$