Solution 3 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
2 2
r10=( - 4*a33*b13*m2*n3*r342 - 8*i*a33*c13*n1*n3*r342 + i*b13 *n1 *r342
2 2 2 2 2 2
+ i*b13 *n2 *r342 - 2*i*b13 *n3 *r342)/(4*a33 *b13 )
r11=( - i*a33*b13*m2*n1*r342 - a33*b13*m2*n2*r342 - 4*i*a33*c13*n1*n2*r342
2 2 2
- i*b13 *n2*n3*r342)/(2*a33 *b13 )
2
r12=( - a33*b13*m2*n1*r342 + i*a33*b13*m2*n2*r342 - 2*i*a33*c13*n1 *r342
2 2 2 2
+ 2*i*a33*c13*n2 *r342 - i*b13 *n1*n3*r342)/(2*a33 *b13 )
- i*n1*n3*r342 + n2*n3*r342
r13=------------------------------
2*a33*b13
2
- i*n1*n2*r342 + n2 *r342
r14=----------------------------
2*a33*b13
2
- i*n1 *r342 + n1*n2*r342
r15=----------------------------
2*a33*b13
2 2 2 2 2
r20=( - 8*a33 *b13*c13*m2*r342 - 8*i*a33 *c13 *n1*r342 - 8*a33 *c13 *n2*r342
2 4 4 2 3
+ 4*i*a33*b13 *c13*n3*r342 + i*b13 *n1*r342 + b13 *n2*r342)/(4*a33 *b13 )
2
- 4*a33*c13*n1*r342 - b13 *n3*r342
r21=-------------------------------------
2
2*a33 *b13
r22=0
2
4*a33*c13*n2*r342 - i*b13 *n3*r342
r23=------------------------------------
2
2*a33 *b13
r24=0
2
- 2*a33*b13*m2*r342 - 4*i*a33*c13*n1*r342 - i*b13 *n3*r342
r26=-------------------------------------------------------------
2
2*a33*b13
- n1*r342 + i*n2*r342
r27=------------------------
2*a33
i*n1*r342 + n2*r342
r28=---------------------
2*a33
- i*n2*r342
r210=--------------
a33
2
- 2*i*a33*b13*m2*r342 - 4*i*a33*c13*n2*r342 - b13 *n3*r342
r212=-------------------------------------------------------------
2
2*a33*b13
r213=0
i*n1*r342 - n2*r342
r214=---------------------
2*b13
- i*n1*r342
r215=--------------
a33
2
2*i*a33*b13*m2*r342 + 4*i*a33*c13*n2*r342 + b13 *n3*r342
r216=----------------------------------------------------------
2
2*a33*b13
r217=0
r218=0
r219=0
i*n1*r342 - n2*r342
r220=---------------------
2*b13
r30=0
r31=0
2
- 4*i*c13 *r342
r32=------------------
2
b13
c13*r342
r33=----------
a33
r34=0
r35=0
i*c13*r342
r36=------------
a33
2
- 4*i*c13 *r342
r37=------------------
2
b13
c13*r342
r38=----------
a33
i*c13*r342
r39=------------
a33
r310=0
r311=0
- 2*i*c13*r342
r312=-----------------
b13
r313=0
r314=0
- 2*i*c13*r342
r315=-----------------
b13
r316=0
r317=0
r318=0
r319=0
r320=0
r321=0
b13*r342
r322=----------
2*a33
- 4*c13*r342
r323=---------------
b13
r324=0
b13*r342
r325=----------
2*a33
r326=0
r327= - i*r342
r328= - r342
r329=0
r330=i*r342
r331=0
r332=0
r333=0
r334=0
r335=0
4*c13*r342
r336=------------
b13
i*b13*r342
r337=------------
2*a33
r338=0
r339=0
i*b13*r342
r340=------------
2*a33
r341=0
r343= - i*r342
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=i*r342
r351=0
r352=0
r353=0
r354=0
r355=0
- b13*n1 + i*b13*n2
m3=----------------------
2*a33
i*b13*m2 - 2*c13*n1 + 2*i*c13*n2
m1=----------------------------------
b13
2
b13
c33=-------
4*a33
c23= - i*c13
2
b13
c22=-------
2*a33
2
i*b13
c12=--------
4*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:
r342, m2, n3, n1, n2, 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.
{r342,b13,a33}
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,
4*a33*c12 - i*b13**2,
2*a33*c22 - b13**2,
i*c13 + c23,
4*a33*c33 - b13**2,
b13*m1 - i*b13*m2 + 2*c13*n1 - 2*i*c13*n2,
2*a33*m3 + b13*n1 - i*b13*n2}$
The system of equations related to the Hamiltonian HAM:
2
2 i*v1*v2*b13
HAM=u1*v3*b13 + u1*n1 - i*u2*v3*b13 + u2*n2 + u3 *a33 + u3*n3 + --------------
2*a33
2 2
i*b13*m2 - 2*c13*n1 + 2*i*c13*n2 v2 *b13
+ 2*v1*v3*c13 + v1*---------------------------------- + ----------
b13 2*a33
2 2
v3 *b13 - b13*n1 + i*b13*n2
- 2*i*v2*v3*c13 + v2*m2 + ---------- + v3*----------------------
4*a33 2*a33
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 3 2 2 2 2 2
FI=4*i*u1 *v3*a33 *b13 + u1 *(2*i*a33 *b13 *n1 - 2*a33 *b13 *n2)
2 3 2 3 2 4
- 4*i*u1*u3*v1*a33 *b13 + 4*u1*u3*v2*a33 *b13 + 2*i*u1*v1 *a33*b13
2 4 2 2
+ 2*i*u1*v2 *a33*b13 + 16*u1*v2*v3*a33 *b13 *c13
2 2 2 3
+ u1*v2*(4*i*a33 *b13 *m2 + 8*i*a33 *b13*c13*n2 + 2*a33*b13 *n3)
3 2 2 2
- 4*i*u1*v3*a33*b13 *n1 + u1*( - 2*i*a33*b13 *n1 + 2*a33*b13 *n1*n2)
2 2 3 2 2 2 2 2
+ 4*i*u2 *v3*a33 *b13 + u2 *(2*i*a33 *b13 *n1 - 2*a33 *b13 *n2)
2 3 2 3 2 4
- 4*u2*u3*v1*a33 *b13 - 4*i*u2*u3*v2*a33 *b13 + 2*u2*v1 *a33*b13
2 2
- 16*u2*v1*v3*a33 *b13 *c13
2 2 2 3
+ u2*v1*( - 4*i*a33 *b13 *m2 - 8*i*a33 *b13*c13*n2 - 2*a33*b13 *n3)
2 4 3
+ 2*u2*v2 *a33*b13 - 4*i*u2*v3*a33*b13 *n2
2 2 2 2 2 2
+ u2*( - 2*i*a33*b13 *n1*n2 + 2*a33*b13 *n2 ) - 8*i*u3*v1 *a33 *b13 *c13
3 3 2 2 2
+ u3*v1*(2*i*a33*b13 *n1 + 2*a33*b13 *n2) - 8*i*u3*v2 *a33 *b13 *c13
3 3
+ u3*v2*( - 2*a33*b13 *n1 + 2*i*a33*b13 *n2)
2 2 2 3
+ u3*v3*( - 4*a33 *b13 *m2 - 8*i*a33 *b13*c13*n1 - 2*i*a33*b13 *n3)
2 2 3 3
+ u3*( - 2*i*a33*b13 *n1*n3 + 2*a33*b13 *n2*n3) + 4*i*v1 *a33*b13 *c13
2 3 2 2 2
+ 4*v1 *v2*a33*b13 *c13 - 16*i*v1 *v3*a33 *b13*c13
2 3 2 4
+ 4*i*v1*v2 *a33*b13 *c13 + v1*v3*(8*a33*b13 *c13*n2 - 2*i*b13 *n3) + v1*(
2 2 2
- 2*a33*b13 *m2*n1 + 2*i*a33*b13 *m2*n2 - 4*i*a33*b13*c13*n1
2 3 3 3
+ 4*i*a33*b13*c13*n2 - 2*i*b13 *n1*n3) + 4*v2 *a33*b13 *c13
2 2 2 2 4
- 16*i*v2 *v3*a33 *b13*c13 + v2*v3*( - 8*a33*b13 *c13*n1 - 2*b13 *n3) + v2*
2 2
( - 2*i*a33*b13 *m2*n1 - 2*a33*b13 *m2*n2 - 8*i*a33*b13*c13*n1*n2
3 2 2 2 2
- 2*i*b13 *n2*n3) + v3 *( - 8*a33 *b13*c13*m2 - 8*i*a33 *c13 *n1
2 2 2 4 4
- 8*a33 *c13 *n2 + 4*i*a33*b13 *c13*n3 + i*b13 *n1 + b13 *n2) + v3*(
2 3 2 3 2
- 4*a33*b13 *m2*n3 - 8*i*a33*b13*c13*n1*n3 + i*b13 *n1 + i*b13 *n2
3 2
- 2*i*b13 *n3 )
Stop of a subroutine.
Number of garbage collections exceeds max_gc_fac.
The factorization crashed!
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1*v3*b13 + u1*n1 - i*u2*v3*b13 + u2*n2 + u3**2*a33 + u3*n3 + (i*v1*v2*b13**
2)/(2*a33) + 2*v1*v3*c13 + v1*(i*b13*m2 - 2*c13*n1 + 2*i*c13*n2)/b13 + (v2**2*
b13**2)/(2*a33) - 2*i*v2*v3*c13 + v2*m2 + (v3**2*b13**2)/(4*a33) + v3*( - b13*n1
+ i*b13*n2)/(2*a33)$
FI=4*i*u1**2*v3*a33**2*b13**3 + u1**2*(2*i*a33**2*b13**2*n1 - 2*a33**2*b13**2*n2
) - 4*i*u1*u3*v1*a33**2*b13**3 + 4*u1*u3*v2*a33**2*b13**3 + 2*i*u1*v1**2*a33*b13
**4 + 2*i*u1*v2**2*a33*b13**4 + 16*u1*v2*v3*a33**2*b13**2*c13 + u1*v2*(4*i*a33**
2*b13**2*m2 + 8*i*a33**2*b13*c13*n2 + 2*a33*b13**3*n3) - 4*i*u1*v3*a33*b13**3*n1
+ u1*( - 2*i*a33*b13**2*n1**2 + 2*a33*b13**2*n1*n2) + 4*i*u2**2*v3*a33**2*b13**
3 + u2**2*(2*i*a33**2*b13**2*n1 - 2*a33**2*b13**2*n2) - 4*u2*u3*v1*a33**2*b13**3
- 4*i*u2*u3*v2*a33**2*b13**3 + 2*u2*v1**2*a33*b13**4 - 16*u2*v1*v3*a33**2*b13**
2*c13 + u2*v1*( - 4*i*a33**2*b13**2*m2 - 8*i*a33**2*b13*c13*n2 - 2*a33*b13**3*n3
) + 2*u2*v2**2*a33*b13**4 - 4*i*u2*v3*a33*b13**3*n2 + u2*( - 2*i*a33*b13**2*n1*
n2 + 2*a33*b13**2*n2**2) - 8*i*u3*v1**2*a33**2*b13**2*c13 + u3*v1*(2*i*a33*b13**
3*n1 + 2*a33*b13**3*n2) - 8*i*u3*v2**2*a33**2*b13**2*c13 + u3*v2*( - 2*a33*b13**
3*n1 + 2*i*a33*b13**3*n2) + u3*v3*( - 4*a33**2*b13**2*m2 - 8*i*a33**2*b13*c13*n1
- 2*i*a33*b13**3*n3) + u3*( - 2*i*a33*b13**2*n1*n3 + 2*a33*b13**2*n2*n3) + 4*i*
v1**3*a33*b13**3*c13 + 4*v1**2*v2*a33*b13**3*c13 - 16*i*v1**2*v3*a33**2*b13*c13
**2 + 4*i*v1*v2**2*a33*b13**3*c13 + v1*v3*(8*a33*b13**2*c13*n2 - 2*i*b13**4*n3)
+ v1*( - 2*a33*b13**2*m2*n1 + 2*i*a33*b13**2*m2*n2 - 4*i*a33*b13*c13*n1**2 + 4*i
*a33*b13*c13*n2**2 - 2*i*b13**3*n1*n3) + 4*v2**3*a33*b13**3*c13 - 16*i*v2**2*v3*
a33**2*b13*c13**2 + v2*v3*( - 8*a33*b13**2*c13*n1 - 2*b13**4*n3) + v2*( - 2*i*
a33*b13**2*m2*n1 - 2*a33*b13**2*m2*n2 - 8*i*a33*b13*c13*n1*n2 - 2*i*b13**3*n2*n3
) + v3**2*( - 8*a33**2*b13*c13*m2 - 8*i*a33**2*c13**2*n1 - 8*a33**2*c13**2*n2 +
4*i*a33*b13**2*c13*n3 + i*b13**4*n1 + b13**4*n2) + v3*( - 4*a33*b13**2*m2*n3 - 8
*i*a33*b13*c13*n1*n3 + i*b13**3*n1**2 + i*b13**3*n2**2 - 2*i*b13**3*n3**2)$