Solution 2 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
2 2
r10=( - 4*i*a33*b13*m2*n3*r330 - 8*a33*c13*n1*n3*r330 + b13 *n1 *r330
2 2 2 2 2 2
+ b13 *n2 *r330 - 2*b13 *n3 *r330)/(4*a33 *b13 )
r11=( - a33*b13*m2*n1*r330 - i*a33*b13*m2*n2*r330 - 4*a33*c13*n1*n2*r330
2 2 2
- b13 *n2*n3*r330)/(2*a33 *b13 )
2
r12=( - i*a33*b13*m2*n1*r330 + a33*b13*m2*n2*r330 - 2*a33*c13*n1 *r330
2 2 2 2
+ 2*a33*c13*n2 *r330 - b13 *n1*n3*r330)/(2*a33 *b13 )
- n1*n3*r330 + i*n2*n3*r330
r13=------------------------------
2*a33*b13
2
- n1*n2*r330 + i*n2 *r330
r14=----------------------------
2*a33*b13
2
- n1 *r330 + i*n1*n2*r330
r15=----------------------------
2*a33*b13
2 2 2 2 2
r20=( - 8*i*a33 *b13*c13*m2*r330 - 8*a33 *c13 *n1*r330 - 8*i*a33 *c13 *n2*r330
2 4 4 2 3
+ 4*a33*b13 *c13*n3*r330 + b13 *n1*r330 + i*b13 *n2*r330)/(4*a33 *b13 )
2
- 4*i*a33*c13*n1*r330 - i*b13 *n3*r330
r21=-----------------------------------------
2
2*a33 *b13
r22=0
2
4*i*a33*c13*n2*r330 - b13 *n3*r330
r23=------------------------------------
2
2*a33 *b13
r24=0
2
- 2*i*a33*b13*m2*r330 - 4*a33*c13*n1*r330 - b13 *n3*r330
r26=-----------------------------------------------------------
2
2*a33*b13
- i*n1*r330 + n2*r330
r27=------------------------
2*a33
n1*r330 + i*n2*r330
r28=---------------------
2*a33
- n2*r330
r210=------------
a33
2
- 2*a33*b13*m2*r330 - 4*a33*c13*n2*r330 - i*b13 *n3*r330
r212=-----------------------------------------------------------
2
2*a33*b13
r213=0
n1*r330 - i*n2*r330
r214=---------------------
2*b13
- n1*r330
r215=------------
a33
2
2*a33*b13*m2*r330 + 4*a33*c13*n2*r330 + i*b13 *n3*r330
r216=--------------------------------------------------------
2
2*a33*b13
r217=0
r218=0
r219=0
n1*r330 - i*n2*r330
r220=---------------------
2*b13
r30=0
r31=0
2
- 4*c13 *r330
r32=----------------
2
b13
i*c13*r330
r33=------------
a33
r34=0
r35=0
c13*r330
r36=----------
a33
2
- 4*c13 *r330
r37=----------------
2
b13
i*c13*r330
r38=------------
a33
c13*r330
r39=----------
a33
r310=0
- i*b13*r330
r311=---------------
2*a33
- 2*c13*r330
r312=---------------
b13
r313=0
r314=0
- 2*c13*r330
r315=---------------
b13
r316=r330
r317=0
r318=0
r319=0
r320=0
- 4*i*c13*r330
r323=-----------------
b13
i*b13*r330
r325=------------
2*a33
r326=0
r328= - i*r330
r329=0
r332=0
r333=0
r334=0
r335=0
4*i*c13*r330
r336=--------------
b13
b13*r330
r337=----------
2*a33
r338=0
- i*b13*r330
r339=---------------
2*a33
b13*r330
r340=----------
2*a33
r341=0
r342=i*r330
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=r330
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:
r330, m2, n1, n3, 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.
{b23,a33,b13,r330}
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
HAM=u1*v3*b13 + u1*n1 + i*u2*v3*b13 + u2*n2 + u3 *a33 + u3*n3
2
- i*v1*v2*b13
+ ----------------- + 2*v1*v3*c13
2*a33
2 2
- i*b13*m2 - 2*c13*n1 - 2*i*c13*n2 v2 *b13
+ v1*------------------------------------- + ---------- + 2*i*v2*v3*c13
b13 2*a33
2 2
v3 *b13 - b13*n1 - i*b13*n2
+ 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*u1 *v3*a33 *b13 + u1 *(2*a33 *b13 *n1 - 2*i*a33 *b13 *n2)
2 3 2 4 4
+ 4*i*u1*u3*v2*a33 *b13 + 2*u1*v1 *a33*b13 - 2*i*u1*v1*v2*a33*b13
2 4 2 2
+ 2*u1*v2 *a33*b13 + 16*i*u1*v2*v3*a33 *b13 *c13
2 2 2 3
+ u1*v2*(4*a33 *b13 *m2 + 8*a33 *b13*c13*n2 + 2*i*a33*b13 *n3)
3 2 2 2
- 4*u1*v3*a33*b13 *n1 + u1*( - 2*a33*b13 *n1 + 2*i*a33*b13 *n1*n2)
2 2 3 2 2 2 2 2
+ 4*u2 *v3*a33 *b13 + u2 *(2*a33 *b13 *n1 - 2*i*a33 *b13 *n2)
2 3 2 4 2 2
- 4*i*u2*u3*v1*a33 *b13 + 2*i*u2*v1 *a33*b13 - 16*i*u2*v1*v3*a33 *b13 *c13
2 2 2 3
+ u2*v1*( - 4*a33 *b13 *m2 - 8*a33 *b13*c13*n2 - 2*i*a33*b13 *n3)
3 2 2 2
- 4*u2*v3*a33*b13 *n2 + u2*( - 2*a33*b13 *n1*n2 + 2*i*a33*b13 *n2 )
2 2 3 2 2 2
+ 4*u3 *v3*a33 *b13 - 8*u3*v1 *a33 *b13 *c13
3 3 2 2 2
+ u3*v1*(2*a33*b13 *n1 + 2*i*a33*b13 *n2) - 8*u3*v2 *a33 *b13 *c13
4 3 3
- 2*i*u3*v2*v3*a33*b13 + u3*v2*( - 2*i*a33*b13 *n1 + 2*a33*b13 *n2)
2 2 2 3
+ u3*v3*( - 4*i*a33 *b13 *m2 - 8*a33 *b13*c13*n1 - 2*a33*b13 *n3)
2 2 3 3
+ u3*( - 2*a33*b13 *n1*n3 + 2*i*a33*b13 *n2*n3) + 4*v1 *a33*b13 *c13
2 3 2 2 2 2 3
+ 4*i*v1 *v2*a33*b13 *c13 - 16*v1 *v3*a33 *b13*c13 + 4*v1*v2 *a33*b13 *c13
2 4 2
+ v1*v3*(8*i*a33*b13 *c13*n2 - 2*b13 *n3) + v1*( - 2*i*a33*b13 *m2*n1
2 2 2 3
+ 2*a33*b13 *m2*n2 - 4*a33*b13*c13*n1 + 4*a33*b13*c13*n2 - 2*b13 *n1*n3
3 3 2 2 2
) + 4*i*v2 *a33*b13 *c13 - 16*v2 *v3*a33 *b13*c13
2 4 2
+ v2*v3*( - 8*i*a33*b13 *c13*n1 - 2*i*b13 *n3) + v2*( - 2*a33*b13 *m2*n1
2 3 2
- 2*i*a33*b13 *m2*n2 - 8*a33*b13*c13*n1*n2 - 2*b13 *n2*n3) + v3 *(
2 2 2 2 2
- 8*i*a33 *b13*c13*m2 - 8*a33 *c13 *n1 - 8*i*a33 *c13 *n2
2 4 4 2
+ 4*a33*b13 *c13*n3 + b13 *n1 + i*b13 *n2) + v3*( - 4*i*a33*b13 *m2*n3
3 2 3 2 3 2
- 8*a33*b13*c13*n1*n3 + b13 *n1 + b13 *n2 - 2*b13 *n3 )
= a product of the elements of: {4,
2 2 2 2
2 2 3 2 a33 *b13 *n1 - i*a33 *b13 *n2 2 3
u1 *v3*a33 *b13 + u1 *------------------------------- + i*u1*u3*v2*a33 *b13
2
2 4 4 2 4
u1*v1 *a33*b13 - i*u1*v1*v2*a33*b13 u1*v2 *a33*b13
+ ----------------- + ------------------------ + -----------------
2 2 2
2 2
+ 4*i*u1*v2*v3*a33 *b13 *c13
2 2 2 3
2*a33 *b13 *m2 + 4*a33 *b13*c13*n2 + i*a33*b13 *n3
+ u1*v2*----------------------------------------------------
2
2 2 2
3 - a33*b13 *n1 + i*a33*b13 *n1*n2
- u1*v3*a33*b13 *n1 + u1*------------------------------------
2
2 2 2 2
2 2 3 2 a33 *b13 *n1 - i*a33 *b13 *n2
+ u2 *v3*a33 *b13 + u2 *-------------------------------
2
2 4
2 3 i*u2*v1 *a33*b13 2 2
- i*u2*u3*v1*a33 *b13 + ------------------- - 4*i*u2*v1*v3*a33 *b13 *c13
2
2 2 2 3
- 2*a33 *b13 *m2 - 4*a33 *b13*c13*n2 - i*a33*b13 *n3
+ u2*v1*-------------------------------------------------------
2
2 2 2
3 - a33*b13 *n1*n2 + i*a33*b13 *n2
- u2*v3*a33*b13 *n2 + u2*------------------------------------
2
2 2 3 2 2 2
+ u3 *v3*a33 *b13 - 2*u3*v1 *a33 *b13 *c13
3 3
a33*b13 *n1 + i*a33*b13 *n2 2 2 2
+ u3*v1*----------------------------- - 2*u3*v2 *a33 *b13 *c13
2
4 3 3
- i*u3*v2*v3*a33*b13 - i*a33*b13 *n1 + a33*b13 *n2
+ ------------------------ + u3*v2*--------------------------------
2 2
2 2 2 3
- 2*i*a33 *b13 *m2 - 4*a33 *b13*c13*n1 - a33*b13 *n3
+ u3*v3*-------------------------------------------------------
2
2 2
- a33*b13 *n1*n3 + i*a33*b13 *n2*n3 3 3
+ u3*-------------------------------------- + v1 *a33*b13 *c13
2
2 3 2 2 2 2 3
+ i*v1 *v2*a33*b13 *c13 - 4*v1 *v3*a33 *b13*c13 + v1*v2 *a33*b13 *c13
2 4
4*i*a33*b13 *c13*n2 - b13 *n3 2
+ v1*v3*------------------------------- + v1*( - i*a33*b13 *m2*n1
2
2 2 2 3
+ a33*b13 *m2*n2 - 2*a33*b13*c13*n1 + 2*a33*b13*c13*n2 - b13 *n1*n3)/2
3 3 2 2 2
+ i*v2 *a33*b13 *c13 - 4*v2 *v3*a33 *b13*c13
2 4
- 4*i*a33*b13 *c13*n1 - i*b13 *n3
+ v2*v3*------------------------------------ + v2
2
2 2 3
- a33*b13 *m2*n1 - i*a33*b13 *m2*n2 - 4*a33*b13*c13*n1*n2 - b13 *n2*n3
*------------------------------------------------------------------------- +
2
2 2 2 2 2 2
v3 *( - 8*i*a33 *b13*c13*m2 - 8*a33 *c13 *n1 - 8*i*a33 *c13 *n2
2 4 4
+ 4*a33*b13 *c13*n3 + b13 *n1 + i*b13 *n2)/4 + v3*(
2 3 2 3 2
- 4*i*a33*b13 *m2*n3 - 8*a33*b13*c13*n1*n3 + b13 *n1 + b13 *n2
3 2
- 2*b13 *n3 )/4}
{HAM,FI} = {4*i,
b13,
b13,
u1*v1 + u2*v2 + u3*v3,
a33,
2 2
u1*v3*a33*b13 - i*u1*a33*b13*n2 + i*u2*v3*a33*b13
2 3
2 - v1 *b13 3
+ i*u2*a33*b13*n1 + u3*v1*a33*b13 + ------------- - i*v1*v2*b13
2
2
- 2*i*a33*b13*m2 + b13 *n3
+ 2*v1*v3*a33*b13*c13 + v1*-----------------------------
2
2 3
v2 *b13
+ ---------- + 2*i*v2*v3*a33*b13*c13
2
2 3
- v3 *b13
+ v2*(a33*b13*m2 - 2*i*a33*c13*n1 + 2*a33*c13*n2) + -------------
2
2
- v3*b13 *n1
+ ---------------}
2
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*u1**2*v3*a33**2*b13**3 + u1**2*(2*a33**2*b13**2*n1 - 2*i*a33**2*b13**2*n2)
+ 4*i*u1*u3*v2*a33**2*b13**3 + 2*u1*v1**2*a33*b13**4 - 2*i*u1*v1*v2*a33*b13**4 +
2*u1*v2**2*a33*b13**4 + 16*i*u1*v2*v3*a33**2*b13**2*c13 + u1*v2*(4*a33**2*b13**
2*m2 + 8*a33**2*b13*c13*n2 + 2*i*a33*b13**3*n3) - 4*u1*v3*a33*b13**3*n1 + u1*( -
2*a33*b13**2*n1**2 + 2*i*a33*b13**2*n1*n2) + 4*u2**2*v3*a33**2*b13**3 + u2**2*(
2*a33**2*b13**2*n1 - 2*i*a33**2*b13**2*n2) - 4*i*u2*u3*v1*a33**2*b13**3 + 2*i*u2
*v1**2*a33*b13**4 - 16*i*u2*v1*v3*a33**2*b13**2*c13 + u2*v1*( - 4*a33**2*b13**2*
m2 - 8*a33**2*b13*c13*n2 - 2*i*a33*b13**3*n3) - 4*u2*v3*a33*b13**3*n2 + u2*( - 2
*a33*b13**2*n1*n2 + 2*i*a33*b13**2*n2**2) + 4*u3**2*v3*a33**2*b13**3 - 8*u3*v1**
2*a33**2*b13**2*c13 + u3*v1*(2*a33*b13**3*n1 + 2*i*a33*b13**3*n2) - 8*u3*v2**2*
a33**2*b13**2*c13 - 2*i*u3*v2*v3*a33*b13**4 + u3*v2*( - 2*i*a33*b13**3*n1 + 2*
a33*b13**3*n2) + u3*v3*( - 4*i*a33**2*b13**2*m2 - 8*a33**2*b13*c13*n1 - 2*a33*
b13**3*n3) + u3*( - 2*a33*b13**2*n1*n3 + 2*i*a33*b13**2*n2*n3) + 4*v1**3*a33*b13
**3*c13 + 4*i*v1**2*v2*a33*b13**3*c13 - 16*v1**2*v3*a33**2*b13*c13**2 + 4*v1*v2
**2*a33*b13**3*c13 + v1*v3*(8*i*a33*b13**2*c13*n2 - 2*b13**4*n3) + v1*( - 2*i*
a33*b13**2*m2*n1 + 2*a33*b13**2*m2*n2 - 4*a33*b13*c13*n1**2 + 4*a33*b13*c13*n2**
2 - 2*b13**3*n1*n3) + 4*i*v2**3*a33*b13**3*c13 - 16*v2**2*v3*a33**2*b13*c13**2 +
v2*v3*( - 8*i*a33*b13**2*c13*n1 - 2*i*b13**4*n3) + v2*( - 2*a33*b13**2*m2*n1 -
2*i*a33*b13**2*m2*n2 - 8*a33*b13*c13*n1*n2 - 2*b13**3*n2*n3) + v3**2*( - 8*i*a33
**2*b13*c13*m2 - 8*a33**2*c13**2*n1 - 8*i*a33**2*c13**2*n2 + 4*a33*b13**2*c13*n3
+ b13**4*n1 + i*b13**4*n2) + v3*( - 4*i*a33*b13**2*m2*n3 - 8*a33*b13*c13*n1*n3
+ b13**3*n1**2 + b13**3*n2**2 - 2*b13**3*n3**2)$