Solution 4 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
m3*r11
r10=--------
m2
r12=0
r13=0
r14=0
r15=0
2
m3 *r22
r20=---------
2
m2
2*m3*r22
r21=----------
m2
r23=0
r24=0
1
---*m3*n3*r327
2
r26=----------------
a23*m2
1
---*n3*r327
2
r27=-------------
a23
r28=0
r29=0
r210=0
r211=0
r212=0
r214=0
r215=0
r216=0
r217=0
r218=0
r219=0
2 3 3
m2 *m3*r38 + m3 *r33 - m3 *r38
r30=--------------------------------
3
m2
2 2 2
m2 *r38 + 3*m3 *r33 - 3*m3 *r38
r31=---------------------------------
2
m2
3*m3*r33 - 2*m3*r38
r32=---------------------
m2
r34=0
r35=0
r36=0
m3*r38
r37=--------
m2
r39=0
r310=0
r311=0
r312=0
r313=0
r314=0
r315=0
m2*r316
r317=---------
m3
r318=0
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
m3*r327
r326=---------
m2
r328=0
r329=0
1
a23*m2*r316 - ---*a33*m3*r327
2
r330=-------------------------------
a23*m2
1
a23*m2*r316 - ---*a33*m3*r327
2
r331=-------------------------------
a23*m3
r332=0
r333=0
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
1
a23*m2*r316 - ---*a33*m3*r327
2
r350=-------------------------------
a23*m2
1
a23*m2*r316 - ---*a33*m3*r327
2
r351=-------------------------------
a23*m3
r352=0
r353=0
r354=0
r355=0
m1=0
c22=0
c11=0
b33=0
b32=0
b31=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:
r00, r11, r22, r33, r38, r327, r316, n3, m2, a33, m3,
a23
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.
{a33,a23,m2}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2
HAM=2*u2*u3*a23 + u3 *a33 + u3*n3 + v2*m2 + v3*m3
has apart from the Hamiltonian and Casimirs the following 7 first integrals:
2 2 2 2 2 2
FI=u1 *v2*m2 + u1 *v3*m3 + u2 *v2*m2 + u2 *v3*m3 + u3 *v2*m2 + u3 *v3*m3
2 2 2
= a product of the elements of: {u1 + u2 + u3 ,v2*m2 + v3*m3}
{HAM,FI} = 0
1 2 1 2 1 2
FI= - ---*u1 *v2*a33*m2 - ---*u1 *v3*a33*m3 - ---*u2 *v2*a33*m2
2 2 2
1 2 1
- ---*u2 *v3*a33*m3 + u2*u3*v2*a23*m2 + u2*u3*v3*a23*m3 + ---*u3*v2*m2*n3
2 2
1
+ ---*u3*v3*m3*n3
2
1
= a product of the elements of: {---,
2
v2*m2 + v3*m3,
2 2
- u1 *a33 - u2 *a33 + 2*u2*u3*a23 + u3*n3}
{HAM,FI} = 0
2 3 2 2 2 2 2 3 2
FI=v1 *v2*m2 + v1 *v3*m2 *m3 - 2*v2 *v3*m2 *m3 + v2*v3 *(m2 - 3*m2*m3 )
3 2 3
+ v3 *(m2 *m3 - m3 )
= a product of the elements of: {v2*m2 + v3*m3,
2 2 2 2 2
v1 *m2 - 2*v2*v3*m2*m3 + v3 *(m2 - m3 )}
{HAM,FI} = 0
3 3 2 2 2 2 3 3
FI=v2 *m2 + 3*v2 *v3*m2 *m3 + 3*v2*v3 *m2*m3 + v3 *m3
= a product of the elements of: {v2*m2 + v3*m3,
v2*m2 + v3*m3,
v2*m2 + v3*m3}
{HAM,FI} = 0
2 2 2 2
FI=v2 *m2 + 2*v2*v3*m2*m3 + v3 *m3
= a product of the elements of: {v2*m2 + v3*m3,v2*m2 + v3*m3}
{HAM,FI} = 0
FI=v2*m2 + v3*m3
which the program can not factorize further.
{HAM,FI} = 0
FI=1
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*n3 + v2*m2 + v3*m3$
FI=u1**2*v2*m2 + u1**2*v3*m3 + u2**2*v2*m2 + u2**2*v3*m3 + u3**2*v2*m2 + u3**2*
v3*m3$
FI= - 1/2*u1**2*v2*a33*m2 - 1/2*u1**2*v3*a33*m3 - 1/2*u2**2*v2*a33*m2 - 1/2*u2**
2*v3*a33*m3 + u2*u3*v2*a23*m2 + u2*u3*v3*a23*m3 + 1/2*u3*v2*m2*n3 + 1/2*u3*v3*m3
*n3$
FI=v1**2*v2*m2**3 + v1**2*v3*m2**2*m3 - 2*v2**2*v3*m2**2*m3 + v2*v3**2*(m2**3 -
3*m2*m3**2) + v3**3*(m2**2*m3 - m3**3)$
FI=v2**3*m2**3 + 3*v2**2*v3*m2**2*m3 + 3*v2*v3**2*m2*m3**2 + v3**3*m3**3$
FI=v2**2*m2**2 + 2*v2*v3*m2*m3 + v3**2*m3**2$
FI=v2*m2 + v3*m3$
FI=1$