Solution 6 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r10=0
b32*r12
r11=---------
b31
r13=0
r14=0
r15=0
1
- ---*b31*r24
2
r20=----------------
b32
r21=0
1 2 1 2
- ---*b31 *r24 + ---*b32 *r24
2 2
r22=--------------------------------
b31*b32
r23=0
r26=0
1
---*b32*n3*r328
2
r27=-----------------
a23*b31
1
---*n3*r328
2
r28=-------------
a23
r29=0
r210=0
r211=0
r212=0
r214=0
r215=0
r216=0
r217=0
r218=0
r219=0
r30=0
r31=0
r32=0
3
b32 *r39
r33=----------
3
b31
r34=0
r35=0
2
3*b32 *r39
r36=------------
2
b31
r37=0
3*b32*r39
r38=-----------
b31
r310=0
r311=0
1 2
---*b32 *r328
2
r312=---------------
a23*b31
r313=0
b32*r328
r314=----------
a23
1
---*b31*r328
2
r315=--------------
a23
r316=0
1
---*a33*b32*r328
2
r317=------------------
a23*b31
1
---*a33*r328
2
r318=--------------
a23
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
r326=0
b32*r328
r327=----------
b31
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
m3=0
c22=0
c11=0
b33=0
b32*m1
m2=--------
b31
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:
b32, r00, r12, r24, r328, r39, n3, b31, a33, m1, 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.
{r39,a23}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2
HAM=(2*u2*u3*a23*b31 + u3 *a33*b31 + u3*v1*b31 + u3*v2*b31*b32 + u3*b31*n3
+ v1*b31*m1 + v2*b32*m1)/b31
has apart from the Hamiltonian and Casimirs the following 5 first integrals:
3 3 2 2 2 2 3 3
FI=v1 *b31 + 3*v1 *v2*b31 *b32 + 3*v1*v2 *b31*b32 + v2 *b32
= a product of the elements of: {v1*b31 + v2*b32,
v1*b31 + v2*b32,
v1*b31 + v2*b32}
{HAM,FI} = 0
1 2 1 2
FI=u2*u3*v1*a23*b31 + u2*u3*v2*a23*b32 + ---*u3 *v1*a33*b31 + ---*u3 *v2*a33*b32
2 2
1 2 2 1 1 2 2
+ ---*u3*v1 *b31 + u3*v1*v2*b31*b32 + ---*u3*v1*b31*n3 + ---*u3*v2 *b32
2 2 2
1
+ ---*u3*v2*b32*n3
2
1
= a product of the elements of: {---,
2
u3,
v1*b31 + v2*b32,
2*u2*a23 + u3*a33 + v1*b31 + v2*b32 + n3}
{HAM,FI} = 0
2 1 2 1 2 1 2 2
FI=v1*v2*b31*b32 + v2 *( - ---*b31 + ---*b32 ) - ---*v3 *b31
2 2 2
1
= a product of the elements of: { - ---,
2
2 2 2 2 2
- 2*v1*v2*b31*b32 + v2 *(b31 - b32 ) + v3 *b31 }
{HAM,FI} = 0
FI=v1*b31 + v2*b32
which the program can not factorize further.
{HAM,FI} = 0
FI=1
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=(2*u2*u3*a23*b31 + u3**2*a33*b31 + u3*v1*b31**2 + u3*v2*b31*b32 + u3*b31*n3
+ v1*b31*m1 + v2*b32*m1)/b31$
FI=v1**3*b31**3 + 3*v1**2*v2*b31**2*b32 + 3*v1*v2**2*b31*b32**2 + v2**3*b32**3$
FI=u2*u3*v1*a23*b31 + u2*u3*v2*a23*b32 + 1/2*u3**2*v1*a33*b31 + 1/2*u3**2*v2*a33
*b32 + 1/2*u3*v1**2*b31**2 + u3*v1*v2*b31*b32 + 1/2*u3*v1*b31*n3 + 1/2*u3*v2**2*
b32**2 + 1/2*u3*v2*b32*n3$
FI=v1*v2*b31*b32 + v2**2*( - 1/2*b31**2 + 1/2*b32**2) - 1/2*v3**2*b31**2$
FI=v1*b31 + v2*b32$
FI=1$