Solution 15 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
n3*r26
r10=--------
a33
- m2*r214 + n2*r26
r11=---------------------
a33
i*m2*r214 - i*n2*r26
r12=----------------------
a33
- n3*r214
r13=------------
a33
- n2*r214
r14=------------
a33
i*n2*r214
r15=-----------
a33
2
a33*m2 *r214 - a33*m2*n2*r26 + m2*n2*n3*r327
r20=----------------------------------------------
2
a33*n2
m2*r327
r21=---------
a33
r22=0
- i*m2*r327
r23=--------------
a33
r24=0
- n2*r327
r27=------------
a33
i*n2*r327
r28=-----------
a33
n2*r327
r210=---------
a33
- 2*i*a33*m2*r214 + i*a33*n2*r26 - i*n2*n3*r327
r212=--------------------------------------------------
a33*n2
r213=0
- i*n2*r327
r215=--------------
a33
2*i*a33*m2*r214 - i*a33*n2*r26 + i*n2*n3*r327
r216=-----------------------------------------------
a33*n2
r217=0
r218=0
r219=0
r220=r214
r30=0
r31=0
2
m2 *r327
r32=----------
2
n2
r33=0
r34=0
r35=0
r36=0
2
m2 *r327
r37=----------
2
n2
r38=0
r39=0
r310=0
r311=0
- m2*r327
r312=------------
n2
r313=0
r314=0
- m2*r327
r315=------------
n2
r316=0
r317=0
r318=0
r319=0
r320=0
r321=0
r322=0
2*i*m2*r327
r323=-------------
n2
r324=0
r325=0
r326=0
r328= - i*r327
r329=0
r330= - r327
r331=0
r332=0
r333=0
r334=0
r335=0
- 2*i*m2*r327
r336=----------------
n2
r337=0
r338=0
r339=0
r340=0
r341=0
r342=i*r327
r343=r327
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350= - r327
r351=0
r352=0
r353=0
r354=0
r355=0
m3=0
m1= - i*m2
n1= - i*n2
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:
r214, r26, r327, m2, n3, n2, 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.
{n2,a33,r327}
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 + i*n2,
m1 + i*m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2
HAM= - i*u1*n2 + u2*n2 + u3 *a33 + u3*n3 - i*v1*m2 + v2*m2
has apart from the Hamiltonian and Casimirs the following 3 first integrals:
2 2 2 2
FI= - u1 *v3*a33*n2 + u1*u3*v1*a33*n2 + i*u1*u3*v2*a33*n2
2 3 2 2
- 2*i*u1*v2*v3*a33*m2*n2 + i*u1*v2*n2 *n3 - i*u1*v3*n2 - u2 *v3*a33*n2
2 2
- i*u2*u3*v1*a33*n2 + u2*u3*v2*a33*n2 + 2*i*u2*v1*v3*a33*m2*n2
2 3 2 3
- i*u2*v1*n2 *n3 + u2*v3*n2 - u3*v1 *a33*m2*n2 + i*u3*v1*n2
2 3 2 2 2
- u3*v2 *a33*m2*n2 - u3*v2*n2 + v1 *v3*a33*m2 - i*v1*v3*m2*n2
2 2 2 2
+ v2 *v3*a33*m2 + v2*v3*m2*n2 + v3 *m2*n2*n3
which the program can not factorize further.
{HAM,FI} = 0
2 2
FI= - i*u1*v2*a33*n2 + i*u2*v1*a33*n2 + u3*v3*a33*n2 - i*v1*n2 + v2*n2
2
- v3 *a33*m2 + v3*n2*n3
which the program can not factorize further.
{HAM,FI} = 0
2 2 3 2 2
FI=u1 *a33*n2 + 2*i*u1*v2*a33*m2*n2 + i*u1*n2 + u2 *a33*n2
3 2 2 2
- 2*i*u2*v1*a33*m2*n2 - u2*n2 - u3*n2 *n3 + i*v1*m2*n2 - v2*m2*n2
2 2
+ v3 *a33*m2
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM= - i*u1*n2 + u2*n2 + u3**2*a33 + u3*n3 - i*v1*m2 + v2*m2$
FI= - u1**2*v3*a33*n2**2 + u1*u3*v1*a33*n2**2 + i*u1*u3*v2*a33*n2**2 - 2*i*u1*v2
*v3*a33*m2*n2 + i*u1*v2*n2**2*n3 - i*u1*v3*n2**3 - u2**2*v3*a33*n2**2 - i*u2*u3*
v1*a33*n2**2 + u2*u3*v2*a33*n2**2 + 2*i*u2*v1*v3*a33*m2*n2 - i*u2*v1*n2**2*n3 +
u2*v3*n2**3 - u3*v1**2*a33*m2*n2 + i*u3*v1*n2**3 - u3*v2**2*a33*m2*n2 - u3*v2*n2
**3 + v1**2*v3*a33*m2**2 - i*v1*v3*m2*n2**2 + v2**2*v3*a33*m2**2 + v2*v3*m2*n2**
2 + v3**2*m2*n2*n3$
FI= - i*u1*v2*a33*n2 + i*u2*v1*a33*n2 + u3*v3*a33*n2 - i*v1*n2**2 + v2*n2**2 -
v3**2*a33*m2 + v3*n2*n3$
FI=u1**2*a33*n2**2 + 2*i*u1*v2*a33*m2*n2 + i*u1*n2**3 + u2**2*a33*n2**2 - 2*i*u2
*v1*a33*m2*n2 - u2*n2**3 - u3*n2**2*n3 + i*v1*m2*n2**2 - v2*m2*n2**2 + v3**2*a33
*m2**2$