Solution 6 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
2 2
- ---*b33 *r32
3
r30=-----------------
2 2
b32 - b33
r31=0
1 3 2
---*b32 *r32 - b32*b33 *r32
3
r33=-----------------------------
2 3
b32 *b33 - b33
r34=0
r35=0
r36=0
2
- b33 *r32
r37=-------------
2 2
b32 - b33
- b32*b33*r32
r38=----------------
2 2
b32 - b33
r39=0
1 2
---*b33 *r327
2
r310=---------------
a23*b32
b33*r327
r311=----------
a23
1
---*b32*r327
2
r312=--------------
a23
r313=0
r314=0
r315=0
1
---*a33*b33*r327
2
r316=------------------
a23*b32
1
---*a33*r327
2
r317=--------------
a23
r318=0
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
b33*r327
r326=----------
b32
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
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
c22=0
c11=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:
r327, r32, a33, b33, b32, 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.
{b33,b32 + b33,b32 - b33,b32,r32,a23}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v2*b32 + u3*v3*b33
has apart from the Hamiltonian and Casimirs the following 2 first integrals:
2 2 2 3 3 1 3 2
FI= - v1 *v2*b32*b33 - v1 *v3*b33 + v2 *(---*b32 - b32*b33 )
3
2 2 3 2 3 3
+ v2 *v3*(b32 *b33 - b33 ) - ---*v3 *b33
3
1
= a product of the elements of: {---,
3
v2*b32 + v3*b33,
2 2 2 2 2 2 2
- 3*v1 *b33 + v2 *(b32 - 3*b33 ) + 2*v2*v3*b32*b33 - 2*v3 *b33 }
{HAM,FI} = 0
1 2 1 2
FI=u2*u3*v2*a23*b32 + u2*u3*v3*a23*b33 + ---*u3 *v2*a33*b32 + ---*u3 *v3*a33*b33
2 2
1 2 2 1 2 2
+ ---*u3*v2 *b32 + u3*v2*v3*b32*b33 + ---*u3*v3 *b33
2 2
1
= a product of the elements of: {---,
2
u3,
v2*b32 + v3*b33,
2*u2*a23 + u3*a33 + v2*b32 + v3*b33}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v2*b32 + u3*v3*b33$
FI= - v1**2*v2*b32*b33**2 - v1**2*v3*b33**3 + v2**3*(1/3*b32**3 - b32*b33**2) +
v2**2*v3*(b32**2*b33 - b33**3) - 2/3*v3**3*b33**3$
FI=u2*u3*v2*a23*b32 + u2*u3*v3*a23*b33 + 1/2*u3**2*v2*a33*b32 + 1/2*u3**2*v3*a33
*b33 + 1/2*u3*v2**2*b32**2 + u3*v2*v3*b32*b33 + 1/2*u3*v3**2*b33**2$