Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=b12 + 4*c23
Expressions
The solution is given through the following expressions:
1 2 2 1 2
----*b12 *m3 *r338 - ----*b12 *m3*n3*r338
16 16
r10=-------------------------------------------
4
c23
r11=0
1 3 2 1 3 1 2 2
r12=( - ----*b12 *m3 *r338 + ----*b12 *m3*n3*r338 - ---*b12*c23 *m3 *r338
32 32 8
1 2 5
+ ---*b12*c23 *m3*n3*r338)/c23
8
r13=0
r14=0
r15=0
r20=0
1 2 1 2 2 2
- ---*b12 *m3*r338 + ---*b12 *n3*r338 - c23 *m3*r338 + c23 *n3*r338
8 8
r21=----------------------------------------------------------------------
3
c23
r22=0
r23=0
r24=0
r26=0
1 2 1 2
---*b12 *m3*r338 - ---*b12 *n3*r338
8 8
r27=-------------------------------------
3
c23
r28=0
r29=0
r210=0
r211=0
1
---*b12*m3*r338
4
r212=-----------------
2
c23
r214=0
r215=0
1
- ---*b12*m3*r338
4
r216=--------------------
2
c23
r217=0
r218=0
r219=0
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=r338
r311=0
r312=0
1
- ---*b12*r338
2
r313=-----------------
c23
r314=0
1 2 2
- ---*b12 *r338 - c23 *r338
4
r315=------------------------------
2
c23
1 2
- ---*b12 *r338
4
r316=------------------
2
c23
r317=0
1
- ---*b12*r338
2
r318=-----------------
c23
r319=0
r320=0
r321=r338
r322=0
r323=0
r324=0
r325=0
r326=0
1 2
- ---*b12 *r338
4
r327=------------------
2
c23
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
1
---*b12*r338
2
r335=--------------
c23
r336=0
r337=0
r339=0
r340=0
1
---*b12*r338
2
r341=--------------
c23
r342=0
1 2
- ---*b12 *r338
4
r343=------------------
2
c23
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
c33=0
b33=0
b22=0
a33=0
a23=c23
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, r338, m3, n3, b12, c23
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.
{b12,r338,c23}
Relevance for the application:
Modulo the following equation:
2 2
0=b12 + 4*c23
the system of equations related to the Hamiltonian HAM:
HAM=u1*v2*b12 + 2*u2*u3*c23 - u2*v1*b12 + 2*u2*v3*c23 + 2*u3*v2*c23 + u3*n3
+ 2*v2*v3*c23 + v3*m3
has apart from the Hamiltonian and Casimirs the following 2 first integrals:
1 2 3 1 4 5
FI= - ---*u1*u3*v1*b12 *c23 + ---*u1*u3*v3*b12*c23 + u1*v1*v3*c23
4 2
1 3 1 2 4 1 2 3
- ---*u1*v2*b12*c23 *m3 + ---*u1*v3 *b12*c23 - ---*u2*u3*v2*b12 *c23
4 2 4
1 3 5 1 2 4
+ ---*u2*v1*b12*c23 *m3 + u2*v2*v3*c23 - ---*u3 *v1*b12*c23
4 2
1 2 2 3 2 1 2 3 5
- ---*u3 *v3*b12 *c23 + u3*v1 *( - ---*b12 *c23 - c23 )
4 4
1 4 1 2 2 1 2 2
- ---*u3*v1*v3*b12*c23 + u3*v2*(---*b12 *c23 *m3 - ---*b12 *c23 *n3)
2 8 8
2 5
+ u3*v3 *c23 + v1
1 3 2 1 3 1 2 2 1 2
*( - ----*b12 *m3 + ----*b12 *m3*n3 - ---*b12*c23 *m3 + ---*b12*c23 *m3*n3)
32 32 8 8
1 2 2 1 2 2 4 4
+ v2*v3*( - ---*b12 *c23 *m3 + ---*b12 *c23 *n3 - c23 *m3 + c23 *n3)
8 8
1 2 2 1 2
+ v3*(----*b12 *c23*m3 - ----*b12 *c23*m3*n3)
16 16
1
= a product of the elements of: { - ----,
32
2 3 4 5
8*u1*u3*v1*b12 *c23 - 16*u1*u3*v3*b12*c23 - 32*u1*v1*v3*c23
3 2 4 2 3
+ 8*u1*v2*b12*c23 *m3 - 16*u1*v3 *b12*c23 + 8*u2*u3*v2*b12 *c23
3 5 2 4
- 8*u2*v1*b12*c23 *m3 - 32*u2*v2*v3*c23 + 16*u3 *v1*b12*c23
2 2 3 2 2 3 5 4
+ 8*u3 *v3*b12 *c23 + u3*v1 *(8*b12 *c23 + 32*c23 ) + 16*u3*v1*v3*b12*c23
2 2 2 2 2 5
+ u3*v2*( - 4*b12 *c23 *m3 + 4*b12 *c23 *n3) - 32*u3*v3 *c23
3 2 3 2 2 2
+ v1*(b12 *m3 - b12 *m3*n3 + 4*b12*c23 *m3 - 4*b12*c23 *m3*n3)
2 2 2 2 4 4
+ v2*v3*(4*b12 *c23 *m3 - 4*b12 *c23 *n3 + 32*c23 *m3 - 32*c23 *n3)
2 2 2
+ v3*( - 2*b12 *c23*m3 + 2*b12 *c23*m3*n3)}
{HAM,FI} = 0
FI=1
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1*v2*b12 + 2*u2*u3*c23 - u2*v1*b12 + 2*u2*v3*c23 + 2*u3*v2*c23 + u3*n3 + 2*
v2*v3*c23 + v3*m3$
FI= - 1/4*u1*u3*v1*b12**2*c23**3 + 1/2*u1*u3*v3*b12*c23**4 + u1*v1*v3*c23**5 - 1
/4*u1*v2*b12*c23**3*m3 + 1/2*u1*v3**2*b12*c23**4 - 1/4*u2*u3*v2*b12**2*c23**3 +
1/4*u2*v1*b12*c23**3*m3 + u2*v2*v3*c23**5 - 1/2*u3**2*v1*b12*c23**4 - 1/4*u3**2*
v3*b12**2*c23**3 + u3*v1**2*( - 1/4*b12**2*c23**3 - c23**5) - 1/2*u3*v1*v3*b12*
c23**4 + u3*v2*(1/8*b12**2*c23**2*m3 - 1/8*b12**2*c23**2*n3) + u3*v3**2*c23**5 +
v1*( - 1/32*b12**3*m3**2 + 1/32*b12**3*m3*n3 - 1/8*b12*c23**2*m3**2 + 1/8*b12*
c23**2*m3*n3) + v2*v3*( - 1/8*b12**2*c23**2*m3 + 1/8*b12**2*c23**2*n3 - c23**4*
m3 + c23**4*n3) + v3*(1/16*b12**2*c23*m3**2 - 1/16*b12**2*c23*m3*n3)$
FI=1$