Solution 5 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
r13=0
r14=0
r15=0
- i*a33*m1*r323 + a33*m2*r323 + 2*i*b13*n3*r323
r20=--------------------------------------------------
2*a33*b13
r21=0
r22=0
r23=0
r24=0
r26=0
r27=0
r28=0
r210=0
r212=0
r213=0
r214=0
r215=0
r216=0
r217=0
r218=0
r219=0
r220=0
r30=0
2*b13*r323
r31=------------
a33
2*i*c13*r323
r32=--------------
b13
r33=0
2*i*b13*r323
r34=--------------
a33
r35=0
r36=0
2*i*c13*r323
r37=--------------
b13
r38=0
r39=0
r310=i*r323
r311=0
r312=0
r313=0
r314=0
r315=0
r316=0
r317=0
r318=0
r319=0
r320=0
r325=0
r326=0
r328=0
r329=0
r330=0
r332=0
r333=0
r334=0
r335=0
r336= - r323
r337=0
r338=0
r339=0
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
m3=0
n2=0
n1=0
2
i*a33*c12 + 2*b13
c33=--------------------
a33
c23= - i*c13
2
2*i*a33*c12 + 4*b13
c22=----------------------
a33
c11=0
b33=0
b32=0
b31=0
b23= - i*b13
b22=0
b21=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:
r323, m1, m2, n3, c12, c13, b13, 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.
{r34,r323,a33,b13}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11,
a12,
a13,
a22,
a23,
b11,
b12,
b21,
b22,
i*b13 + b23,
b31,
b32,
b33,
c11,
- 2*i*a33*c12 + a33*c22 - 4*b13**2,
i*c13 + c23,
- i*a33*c12 + a33*c33 - 2*b13**2,
n1,
n2,
m3}$
The system of equations related to the Hamiltonian HAM:
2
HAM=u1*v3*b13 - i*u2*v3*b13 + u3 *a33 + u3*n3 + 2*v1*v2*c12 + 2*v1*v3*c13
2
2 2*i*a33*c12 + 4*b13
+ v1*m1 + v2 *---------------------- - 2*i*v2*v3*c13 + v2*m2
a33
2
2 i*a33*c12 + 2*b13
+ v3 *--------------------
a33
has apart from the Hamiltonian and Casimirs only the following first integral:
2
FI= - 2*u1*v2*v3*a33*b13 + 2*u2*v1*v3*a33*b13 + 2*i*u3*v3 *a33*b13
2 2 2 2 2 2
+ 4*i*v1 *v3*a33*c13 + 4*i*v1*v3 *b13 + 4*i*v2 *v3*a33*c13 + 4*v2*v3 *b13
2
+ v3 *( - i*a33*m1 + a33*m2 + 2*i*b13*n3)
= a product of the elements of: {2,
- v3,
2
u1*v2*a33*b13 - u2*v1*a33*b13 - i*u3*v3*a33*b13 - 2*i*v1 *a33*c13
2 2 2
- 2*i*v1*v3*b13 - 2*i*v2 *a33*c13 - 2*v2*v3*b13
i*a33*m1 - a33*m2 - 2*i*b13*n3
+ v3*--------------------------------}
2
{HAM,FI} = {2,
- b13,
b13,
v3,
v1 - i*v2,
u1*v1 + u2*v2 + u3*v3,
a33}
And again in machine readable form:
HAM=u1*v3*b13 - i*u2*v3*b13 + u3**2*a33 + u3*n3 + 2*v1*v2*c12 + 2*v1*v3*c13 + v1
*m1 + v2**2*(2*i*a33*c12 + 4*b13**2)/a33 - 2*i*v2*v3*c13 + v2*m2 + v3**2*(i*a33*
c12 + 2*b13**2)/a33$
FI= - 2*u1*v2*v3*a33*b13 + 2*u2*v1*v3*a33*b13 + 2*i*u3*v3**2*a33*b13 + 4*i*v1**2
*v3*a33*c13 + 4*i*v1*v3**2*b13**2 + 4*i*v2**2*v3*a33*c13 + 4*v2*v3**2*b13**2 +
v3**2*( - i*a33*m1 + a33*m2 + 2*i*b13*n3)$