Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=a12 + a13
Expressions
The solution is given through the following expressions:
1 4 2 31 2 2 2 1 4
r10=(----*a12 *n2 *r350 - ----*a12 *a13 *n2 *r350 - ---*a12*a13 *n2*r215
64 64 2
1 4 2 6
- ---*a13 *n2 *r350)/a13
4
1 3 2 9 2 2 1 4
----*a12 *n2 *r350 + ----*a12*a13 *n2 *r350 + ---*a13 *n2*r215
32 32 2
r11=----------------------------------------------------------------
5
a13
1 5 9 3 2 33 4
r12=(----*a12 *n1*n2*r350 + ----*a12 *a13 *n1*n2*r350 + ----*a12*a13 *n1*n2*r350
64 32 64
1 6 7
+ ---*a13 *n1*r215)/a13
2
r13=0
r14=0
r15=0
r20=0
r21=0
r23=0
r24=0
r25=0
r26=0
1 3 17 2
- ----*a12 *n1*r350 - ----*a12*a13 *n1*r350
32 32
r27=----------------------------------------------
4
a13
1 3 9 2
----*a12 *n2*r350 + ----*a12*a13 *n2*r350
16 16
r28=-------------------------------------------
4
a13
r29=r214
1 3 17 2
----*a12 *n1*r350 + ----*a12*a13 *n1*r350
32 32
r210=-------------------------------------------
4
a13
1 2 9 2
- ----*a12 *n2*r350 - ----*a13 *n2*r350
16 16
r212=------------------------------------------
3
a13
r213=0
3 4 13 2 2 4
r216=( - ----*a12 *n2*r350 + ----*a12 *a13 *n2*r350 + a12*a13 *r215
32 32
1 4 5
+ ---*a13 *n2*r350)/a13
2
1 4 17 2 2 1 4
----*a12 *n1*r350 + ----*a12 *a13 *n1*r350 + ---*a13 *n1*r350
32 32 2
r217=---------------------------------------------------------------
5
a13
r219=0
r220=r214
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=0
r311=0
r312=0
r313=0
r314=0
r315=0
r316=0
r317=0
r318=0
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
r326=0
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
1 2 9 2
- ---*a12 *r350 - ---*a13 *r350
8 8
r343=----------------------------------
2
a13
r344=0
r345=0
r346=0
1 3 9 2
- ---*a12 *r350 - ---*a12*a13 *r350
8 8
r347=--------------------------------------
3
a13
r348=0
r349=0
a12*r350
r351=----------
a13
r352=0
r353=0
r354=0
r355=0
m3=0
m2=0
m1=0
- a12*n2
n3=-----------
a13
c33=0
c23=0
c22=0
c13=0
c12=0
b33=0
b31=0
b21=0
b13=0
b11=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, r215, r350, n2, n1, a12, a13
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.
{a12,r350,a13}
Relevance for the application:
Modulo the following equation:
2 2
0=a12 + a13
the system of equations related to the Hamiltonian HAM:
2
2*u1*u2*a12*a13 + 2*u1*u3*a13 + u1*a13*n1 + u2*a13*n2 - u3*a12*n2
HAM=--------------------------------------------------------------------
a13
has apart from the Hamiltonian and Casimirs the following 3 first integrals:
2 6 2 7 1 3 4 9 6
FI=u1 *v2*a12*a13 + u1 *v3*a13 + u1*u2*v1*( - ---*a12 *a13 - ---*a12*a13 )
8 8
1 2 5 9 7
+ u1*u3*v1*( - ---*a12 *a13 - ---*a13 )
8 8
1 4 2 17 2 4 1 6
+ u1*v1*(----*a12 *a13 *n1 + ----*a12 *a13 *n1 + ---*a13 *n1)
32 32 2
3 4 2 13 2 4 1 6
+ u1*v2*( - ----*a12 *a13 *n2 + ----*a12 *a13 *n2 + ---*a13 *n2)
32 32 2
1 2 4 9 6
+ u2*v1*( - ----*a12 *a13 *n2 - ----*a13 *n2)
16 16
1 3 3 17 5
+ u2*v3*(----*a12 *a13 *n1 + ----*a12*a13 *n1)
32 32
1 3 3 9 5
+ u3*v1*(----*a12 *a13 *n2 + ----*a12*a13 *n2)
16 16
1 3 3 17 5
+ u3*v2*( - ----*a12 *a13 *n1 - ----*a12*a13 *n1)
32 32
1 5 9 3 2 33 4
+ v1*(----*a12 *n1*n2 + ----*a12 *a13 *n1*n2 + ----*a12*a13 *n1*n2)
64 32 64
1 3 2 2 9 4 2
+ v2*(----*a12 *a13 *n2 + ----*a12*a13 *n2 )
32 32
1 4 2 31 2 3 2 1 5 2
+ v3*(----*a12 *a13*n2 - ----*a12 *a13 *n2 - ---*a13 *n2 )
64 64 4
{HAM,FI} = 0
2 1 1 1
FI=u1*v2*a12*a13 + u1*v3*a13 + ---*v1*a13*n1 + ---*v2*a13*n2 - ---*v3*a12*n2
2 2 2
{HAM,FI} = 0
2 2 2
FI=u1 + u2 + u3
{HAM,FI} = 0
And again in machine readable form:
HAM=(2*u1*u2*a12*a13 + 2*u1*u3*a13**2 + u1*a13*n1 + u2*a13*n2 - u3*a12*n2)/a13$
FI=u1**2*v2*a12*a13**6 + u1**2*v3*a13**7 + u1*u2*v1*( - 1/8*a12**3*a13**4 - 9/8*
a12*a13**6) + u1*u3*v1*( - 1/8*a12**2*a13**5 - 9/8*a13**7) + u1*v1*(1/32*a12**4*
a13**2*n1 + 17/32*a12**2*a13**4*n1 + 1/2*a13**6*n1) + u1*v2*( - 3/32*a12**4*a13
**2*n2 + 13/32*a12**2*a13**4*n2 + 1/2*a13**6*n2) + u2*v1*( - 1/16*a12**2*a13**4*
n2 - 9/16*a13**6*n2) + u2*v3*(1/32*a12**3*a13**3*n1 + 17/32*a12*a13**5*n1) + u3*
v1*(1/16*a12**3*a13**3*n2 + 9/16*a12*a13**5*n2) + u3*v2*( - 1/32*a12**3*a13**3*
n1 - 17/32*a12*a13**5*n1) + v1*(1/64*a12**5*n1*n2 + 9/32*a12**3*a13**2*n1*n2 +
33/64*a12*a13**4*n1*n2) + v2*(1/32*a12**3*a13**2*n2**2 + 9/32*a12*a13**4*n2**2)
+ v3*(1/64*a12**4*a13*n2**2 - 31/64*a12**2*a13**3*n2**2 - 1/4*a13**5*n2**2)$
FI=u1*v2*a12*a13 + u1*v3*a13**2 + 1/2*v1*a13*n1 + 1/2*v2*a13*n2 - 1/2*v3*a12*n2$
FI=u1**2 + u2**2 + u3**2$