Solution 16 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 - i*n1*r26
r11=-----------------------
a33
- i*m2*r214 + n1*r26
r12=-----------------------
a33
- n3*r214
r13=------------
a33
i*n1*r214
r14=-----------
a33
- n1*r214
r15=------------
a33
2
- a33*m2 *r214 - i*a33*m2*n1*r26 - i*m2*n1*n3*r316
r20=-----------------------------------------------------
2
a33*n1
- m2*r316
r21=------------
a33
r22=0
- i*m2*r316
r23=--------------
a33
r24=0
- i*n1*r316
r27=--------------
a33
n1*r316
r28=---------
a33
i*n1*r316
r210=-----------
a33
- 2*a33*m2*r214 - i*a33*n1*r26 - i*n1*n3*r316
r212=------------------------------------------------
a33*n1
r213=0
- n1*r316
r215=------------
a33
2*a33*m2*r214 + i*a33*n1*r26 + i*n1*n3*r316
r216=---------------------------------------------
a33*n1
r217=0
r218=0
r219=0
r220=r214
r30=0
r31=0
2
m2 *r316
r32=----------
2
n1
r33=0
r34=0
r35=0
r36=0
2
m2 *r316
r37=----------
2
n1
r38=0
r39=0
r310=0
r311=0
i*m2*r316
r312=-----------
n1
r313=0
r314=0
i*m2*r316
r315=-----------
n1
r317=0
r318=0
r319=0
r320=0
- 2*m2*r316
r323=--------------
n1
r325=0
r326=0
r328= - i*r316
r329=0
r330=r316
r332=0
r333=0
r334=0
r335=0
2*m2*r316
r336=-----------
n1
r337=0
r338=0
r339=0
r340=0
r341=0
r342=i*r316
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=r316
r351=0
r352=0
r353=0
r354=0
r355=0
m3=0
m1=i*m2
n2= - i*n1
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, r316, m2, n3, n1, 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.
{n1,r316,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,
i*n1 + n2,
m1 - i*m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2
HAM=u1*n1 - i*u2*n1 + 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*n1 + i*u1*u3*v2*a33*n1 + 2*u1*v2*v3*a33*m2*n1 + i*u1*v2*n1 *n3
3 2 2 2
- u1*v3*n1 + u2 *v3*a33*n1 - i*u2*u3*v1*a33*n1 - 2*u2*v1*v3*a33*m2*n1
2 3 2 2 2
- i*u2*v1*n1 *n3 + i*u2*v3*n1 + u3 *v3*a33*n1 + i*u3*v1 *a33*m2*n1
3 2 3 2 2
+ u3*v1*n1 + i*u3*v2 *a33*m2*n1 - i*u3*v2*n1 + v1 *v3*a33*m2
2 2 2 2 2
- i*v1*v3*m2*n1 + v2 *v3*a33*m2 - v2*v3*m2*n1 - i*v3 *m2*n1*n3
which the program can not factorize further.
{HAM,FI} = {n1,
n1,
u1*v1 + u2*v2 + u3*v3,
- i*u1*n1 - u2*n1 + v1*m2 - i*v2*m2,
a33}
2 2
FI=i*u1*v2*a33*n1 - i*u2*v1*a33*n1 + u3*v3*a33*n1 + v1*n1 - i*v2*n1
2
- i*v3 *a33*m2 + v3*n1*n3
= a product of the elements of: { - i,
2 2
- u1*v2*a33*n1 + u2*v1*a33*n1 + i*u3*v3*a33*n1 + i*v1*n1 + v2*n1
2
+ v3 *a33*m2 + i*v3*n1*n3}
{HAM,FI} = 0
2 2 3 2 2
FI=u1 *a33*n1 + 2*u1*v2*a33*m2*n1 - u1*n1 + u2 *a33*n1 - 2*u2*v1*a33*m2*n1
3 2 2 2 2 2
+ i*u2*n1 - u3*n1 *n3 - i*v1*m2*n1 - v2*m2*n1 - v3 *a33*m2
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1*n1 - i*u2*n1 + u3**2*a33 + u3*n3 + i*v1*m2 + v2*m2$
FI=u1**2*v3*a33*n1**2 + i*u1*u3*v2*a33*n1**2 + 2*u1*v2*v3*a33*m2*n1 + i*u1*v2*n1
**2*n3 - u1*v3*n1**3 + u2**2*v3*a33*n1**2 - i*u2*u3*v1*a33*n1**2 - 2*u2*v1*v3*
a33*m2*n1 - i*u2*v1*n1**2*n3 + i*u2*v3*n1**3 + u3**2*v3*a33*n1**2 + i*u3*v1**2*
a33*m2*n1 + u3*v1*n1**3 + i*u3*v2**2*a33*m2*n1 - i*u3*v2*n1**3 + v1**2*v3*a33*m2
**2 - i*v1*v3*m2*n1**2 + v2**2*v3*a33*m2**2 - v2*v3*m2*n1**2 - i*v3**2*m2*n1*n3$
FI=i*u1*v2*a33*n1 - i*u2*v1*a33*n1 + u3*v3*a33*n1 + v1*n1**2 - i*v2*n1**2 - i*v3
**2*a33*m2 + v3*n1*n3$
FI=u1**2*a33*n1**2 + 2*u1*v2*a33*m2*n1 - u1*n1**3 + u2**2*a33*n1**2 - 2*u2*v1*
a33*m2*n1 + i*u2*n1**3 - u3*n1**2*n3 - i*v1*m2*n1**2 - v2*m2*n1**2 - v3**2*a33*
m2**2$