Solution 2 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
3 3 3 3
b31 *b33*r33 - b31*b33 *r33 - b32 *b33*r39 + b32*b33 *r39
r30=-----------------------------------------------------------
3 3
b31 *b32 - b31*b32
3 2 3 2
b31 *r33 - 3*b31*b33 *r33 - b32 *r39 + 3*b32*b33 *r39
r31=-------------------------------------------------------
3 2
b31 - b31*b32
3 2 3
b31 *b33*r33 - 3*b31*b32 *b33*r33 + 2*b32 *b33*r39
r32=----------------------------------------------------
3 3
b31 *b32 - b31*b32
3 2 3 2
b31 *r33 - 3*b31*b33 *r33 - b32 *r39 + 3*b32*b33 *r39
r34=-------------------------------------------------------
2 3
b31 *b32 - b32
- 6*b31*b33*r33 + 6*b32*b33*r39
r35=----------------------------------
2 2
b31 - b32
3 2 3
b31 *r33 - 3*b31*b32 *r33 + 2*b32 *r39
r36=----------------------------------------
2 3
b31 *b32 - b32
3 2 3
- 2*b31 *b33*r33 + 3*b31 *b32*b33*r39 - b32 *b33*r39
r37=-------------------------------------------------------
3 3
b31 *b32 - b31*b32
1 2
---*b33 *r328
2
r310=---------------
a23*b31
b32*b33*r328
r311=--------------
a23*b31
1 2
---*b32 *r328
2
r312=---------------
a23*b31
b33*r328
r313=----------
a23
b32*r328
r314=----------
a23
1
---*b31*r328
2
r315=--------------
a23
1
a23*b33*r352 + ---*a33*b33*r328
2
r316=---------------------------------
a23*b31
1
a23*b32*r352 + ---*a33*b32*r328
2
r317=---------------------------------
a23*b31
1
a23*r352 + ---*a33*r328
2
r318=-------------------------
a23
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
b33*r328
r326=----------
b31
b32*r328
r327=----------
b31
r329=0
b33*r352
r330=----------
b31
b32*r352
r331=----------
b31
r332=r352
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
b33*r352
r350=----------
b31
b32*r352
r351=----------
b31
r353=0
r354=0
r355=0
c22=0
c11=0
3 2 3
- 2*b31 *r33 + 3*b31 *b32*r39 - b32 *r39
r38=-------------------------------------------
3 2
b31 - b31*b32
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:
r33, r39, r328, r352, a33, b32, b33, b31, 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.
{a23,b32}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v1*b31 + u3*v2*b32 + u3*v3*b33
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 2 2 2 2 2
FI=u1 *v1*b31 + u1 *v2*b32 + u1 *v3*b33 + u2 *v1*b31 + u2 *v2*b32 + u2 *v3*b33
2 2 2
+ u3 *v1*b31 + u3 *v2*b32 + u3 *v3*b33
2 2 2
= a product of the elements of: {u1 + u2 + u3 ,v1*b31 + v2*b32 + v3*b33}
{HAM,FI} = 0
1 2
FI=u2*u3*v1*a23*b31 + u2*u3*v2*a23*b32 + u2*u3*v3*a23*b33 + ---*u3 *v1*a33*b31
2
1 2 1 2 1 2 2
+ ---*u3 *v2*a33*b32 + ---*u3 *v3*a33*b33 + ---*u3*v1 *b31
2 2 2
1 2 2
+ u3*v1*v2*b31*b32 + u3*v1*v3*b31*b33 + ---*u3*v2 *b32 + u3*v2*v3*b32*b33
2
1 2 2
+ ---*u3*v3 *b33
2
1
= a product of the elements of: {---,
2
u3,
v1*b31 + v2*b32 + v3*b33,
2*u2*a23 + u3*a33 + v1*b31 + v2*b32 + v3*b33}
{HAM,FI} = 0
3 3 2 2 2 3
FI=v1 *(b31 - b31*b32 ) + v1 *v2*(3*b31 *b32 - b32 )
2 2 2 2 2
+ v1 *v3*(3*b31 *b33 - b32 *b33) + 2*v1*v2 *b31*b32
2 2 2
+ 6*v1*v2*v3*b31*b32*b33 + v1*v3 *( - b31*b32 + 3*b31*b33 )
2 2 2 3 2
+ 2*v2 *v3*b32 *b33 + v2*v3 *( - b32 + 3*b32*b33 )
3 2 3
+ v3 *( - b32 *b33 + b33 )
= a product of the elements of: {v1*b31 + v2*b32 + v3*b33,
2 2 2
v1 *(b31 - b32 ) + 2*v1*v2*b31*b32 + 2*v1*v3*b31*b33 + 2*v2*v3*b32*b33
2 2 2
+ v3 *( - b32 + b33 )}
{HAM,FI} = 0
2 2 2 2 2 3 2
FI= - 2*v1 *v2*b31 *b32 - 2*v1 *v3*b31 *b33 + v1*v2 *(b31 - 3*b31*b32 )
2 3 2
- 6*v1*v2*v3*b31*b32*b33 + v1*v3 *(b31 - 3*b31*b33 )
3 2 3 2 2 2
+ v2 *(b31 *b32 - b32 ) + v2 *v3*(b31 *b33 - 3*b32 *b33)
2 2 2 3 2 3
+ v2*v3 *(b31 *b32 - 3*b32*b33 ) + v3 *(b31 *b33 - b33 )
= a product of the elements of: { - v1*b31 - v2*b32 - v3*b33,
2 2 2
2*v1*v2*b31*b32 + 2*v1*v3*b31*b33 + v2 *( - b31 + b32 ) + 2*v2*v3*b32*b33
2 2 2
+ v3 *( - b31 + b33 )}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v1*b31 + u3*v2*b32 + u3*v3*b33$
FI=u1**2*v1*b31 + u1**2*v2*b32 + u1**2*v3*b33 + u2**2*v1*b31 + u2**2*v2*b32 + u2
**2*v3*b33 + u3**2*v1*b31 + u3**2*v2*b32 + u3**2*v3*b33$
FI=u2*u3*v1*a23*b31 + u2*u3*v2*a23*b32 + u2*u3*v3*a23*b33 + 1/2*u3**2*v1*a33*b31
+ 1/2*u3**2*v2*a33*b32 + 1/2*u3**2*v3*a33*b33 + 1/2*u3*v1**2*b31**2 + u3*v1*v2*
b31*b32 + u3*v1*v3*b31*b33 + 1/2*u3*v2**2*b32**2 + u3*v2*v3*b32*b33 + 1/2*u3*v3
**2*b33**2$
FI=v1**3*(b31**3 - b31*b32**2) + v1**2*v2*(3*b31**2*b32 - b32**3) + v1**2*v3*(3*
b31**2*b33 - b32**2*b33) + 2*v1*v2**2*b31*b32**2 + 6*v1*v2*v3*b31*b32*b33 + v1*
v3**2*( - b31*b32**2 + 3*b31*b33**2) + 2*v2**2*v3*b32**2*b33 + v2*v3**2*( - b32
**3 + 3*b32*b33**2) + v3**3*( - b32**2*b33 + b33**3)$
FI= - 2*v1**2*v2*b31**2*b32 - 2*v1**2*v3*b31**2*b33 + v1*v2**2*(b31**3 - 3*b31*
b32**2) - 6*v1*v2*v3*b31*b32*b33 + v1*v3**2*(b31**3 - 3*b31*b33**2) + v2**3*(b31
**2*b32 - b32**3) + v2**2*v3*(b31**2*b33 - 3*b32**2*b33) + v2*v3**2*(b31**2*b32
- 3*b32*b33**2) + v3**3*(b31**2*b33 - b33**3)$