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=a23 + a33 + b12 *kap
Expressions
The solution is given through the following expressions:
r10=0
m3*n2*r353
r11=------------
2
b12 *kap
m3*n1*r353
r12=------------
2
b12 *kap
2 2
kap*m3 *r353 - n3 *r353
r13=-------------------------
2
b12 *kap
- n2*n3*r353
r14=---------------
2
b12 *kap
- n1*n3*r353
r15=---------------
2
b12 *kap
r20=0
r21=0
r22=0
r23=0
r24=0
2*a23*m3*r353 - b12*n1*r353
r27=-----------------------------
2
b12 *kap
n2*r353
r28=---------
b12*kap
- a23*n2*r353 - 2*a33*n3*r353
r29=--------------------------------
2
b12 *kap
n1*r353
r210=---------
b12*kap
- 2*a33*m3*r353
r211=------------------
2
b12 *kap
r212=0
- 2*a23*n3*r353
r213=------------------
2
b12 *kap
- a23*n2*r353
r214=----------------
2
b12 *kap
- n2*r353
r215=------------
b12*kap
- 2*a33*m3*r353
r217=------------------
2
b12 *kap
r218=0
r219=0
- a23*n2*r353
r220=----------------
2
b12 *kap
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
- r353
r310=---------
kap
r311=0
- r353
r312=---------
kap
r313=0
r314=0
- r353
r315=---------
kap
r316=0
r317=0
2*a23*r353
r318=------------
b12*kap
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
r326=0
r327=0
- 2*a33*r353
r328=---------------
b12*kap
r329=0
r330=0
r331=0
r332=0
r333=r353
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- 2*a23*r353
r341=---------------
b12*kap
2*a33*r353
r342=------------
b12*kap
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r354=0
r355=0
m2=0
m1=0
a22= - a33
a13=0
a11= - a33
r319=r353
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:
r353, a33, m3, n3, n2, n1, 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.
{{n3,n2,n1,m3},b12,r319}
Relevance for the application:
Modulo the following equation:
2 2 2
0=a23 + a33 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2
HAM= - u1 *a33 + u1*v2*b12 + u1*n1 - u2 *a33 + 2*u2*u3*a23 - u2*v1*b12 + u2*n2
2
+ u3 *a33 + 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*a33*b12 - 2*u1*u3*v3*a23*b12
2 2 2
- 2*u1*v1*a33*m3 - u1*v3*b12*n2 - u1*n1*n3 + u2 *u3*b12 *kap - u2 *a23*n2
- 2*u2*u3*v1*a33*b12 - 2*u2*u3*a23*n3 - 2*u2*v2*a33*m3 + u2*v3*b12*n1
3 2 2 2
- u2*n2*n3 + u3 *b12 *kap + 2*u3 *v1*a23*b12 + u3 *( - a23*n2 - 2*a33*n3)
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*a33 + u1*v2*b12 + u1*n1 - u2**2*a33 + 2*u2*u3*a23 - u2*v1*b12 + u2*
n2 + u3**2*a33 + u3*n3 + v3*m3$
FI=u1**2*u3*b12**2*kap - u1**2*a23*n2 + 2*u1*u3*v2*a33*b12 - 2*u1*u3*v3*a23*b12
- 2*u1*v1*a33*m3 - u1*v3*b12*n2 - u1*n1*n3 + u2**2*u3*b12**2*kap - u2**2*a23*n2
- 2*u2*u3*v1*a33*b12 - 2*u2*u3*a23*n3 - 2*u2*v2*a33*m3 + u2*v3*b12*n1 - u2*n2*n3
+ u3**3*b12**2*kap + 2*u3**2*v1*a23*b12 + u3**2*( - a23*n2 - 2*a33*n3) - 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$