Solution 1 to problem h
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem h
Equations
The following unsolved equations remain:
2 2 2
0=a11 + a23 + b12 *kap
Expressions
The solution is given through the following expressions:
r10=0
m3*n2*r319
r11=------------
2
b12 *kap
m3*n1*r319
r12=------------
2
b12 *kap
2 2
kap*m3 *r319 - n3 *r319
r13=-------------------------
2
b12 *kap
- n2*n3*r319
r14=---------------
2
b12 *kap
- n1*n3*r319
r15=---------------
2
b12 *kap
r20=0
r21=0
r22=0
r23=0
r24=0
2*a23*m3*r319 - b12*n1*r319
r27=-----------------------------
2
b12 *kap
n2*r319
r28=---------
b12*kap
2*a11*n3*r319 - a23*n2*r319
r29=-----------------------------
2
b12 *kap
n1*r319
r210=---------
b12*kap
2*a11*m3*r319
r211=---------------
2
b12 *kap
r212=0
- 2*a23*n3*r319
r213=------------------
2
b12 *kap
- a23*n2*r319
r214=----------------
2
b12 *kap
- n2*r319
r215=------------
b12*kap
2*a11*m3*r319
r217=---------------
2
b12 *kap
r218=0
r219=0
- a23*n2*r319
r220=----------------
2
b12 *kap
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
- r319
r310=---------
kap
r311=0
- r319
r312=---------
kap
r313=0
r314=0
- r319
r315=---------
kap
r316=0
r317=0
2*a23*r319
r318=------------
b12*kap
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
r326=0
r327=0
2*a11*r319
r328=------------
b12*kap
r329=0
r330=0
r331=0
r332=0
r333=r319
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- 2*a23*r319
r341=---------------
b12*kap
- 2*a11*r319
r342=---------------
b12*kap
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=r319
r354=0
r355=0
m2=0
m1=0
a33= - a11
a22=a11
a13=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:
r319, m3, n3, n2, n1, a11, a23, b12
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.
{r319,b12,a23,{n3,n2,n1,m3}}
Relevance for the application:
Modulo the following equation:
2 2 2
0=a11 + a23 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2
HAM=u1 *a11 + u1*v2*b12 + u1*n1 + u2 *a11 + 2*u2*u3*a23 - u2*v1*b12 + u2*n2
2
- u3 *a11 + u3*n3 + v3*m3
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2
FI=u1 *u3*b12 *kap - u1 *a23*n2 - 2*u1*u3*v2*a11*b12 - 2*u1*u3*v3*a23*b12
2 2 2
+ 2*u1*v1*a11*m3 - u1*v3*b12*n2 - u1*n1*n3 + u2 *u3*b12 *kap - u2 *a23*n2
+ 2*u2*u3*v1*a11*b12 - 2*u2*u3*a23*n3 + 2*u2*v2*a11*m3 + u2*v3*b12*n1
3 2 2 2
- u2*n2*n3 + u3 *b12 *kap + 2*u3 *v1*a23*b12 + u3 *(2*a11*n3 - a23*n2)
2 2 2 2
- u3*v1 *b12 + u3*v1*b12*n2 - u3*v2 *b12 + u3*v2*(2*a23*m3 - b12*n1)
2 2 2 2
- u3*v3 *b12 + u3*(kap*m3 - n3 ) + v1*m3*n1 + v2*m3*n2
which is not further factorizable.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*v2*b12 + u1*n1 + u2**2*a11 + 2*u2*u3*a23 - u2*v1*b12 + u2*n2
- u3**2*a11 + u3*n3 + v3*m3$
FI=u1**2*u3*b12**2*kap - u1**2*a23*n2 - 2*u1*u3*v2*a11*b12 - 2*u1*u3*v3*a23*b12
+ 2*u1*v1*a11*m3 - u1*v3*b12*n2 - u1*n1*n3 + u2**2*u3*b12**2*kap - u2**2*a23*n2
+ 2*u2*u3*v1*a11*b12 - 2*u2*u3*a23*n3 + 2*u2*v2*a11*m3 + u2*v3*b12*n1 - u2*n2*n3
+ u3**3*b12**2*kap + 2*u3**2*v1*a23*b12 + u3**2*(2*a11*n3 - a23*n2) - u3*v1**2*
b12**2 + u3*v1*b12*n2 - u3*v2**2*b12**2 + u3*v2*(2*a23*m3 - b12*n1) - u3*v3**2*
b12**2 + u3*(kap*m3**2 - n3**2) + v1*m3*n1 + v2*m3*n2$