Solution 3 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
r14=0
r15=0
r20=0
r21=0
r23=0
r24=0
r25=0
m3*r333
r26=---------
a11
r27=0
r28=0
r210=0
r212=0
r213=0
r214=0
r215=0
r216=0
r217=0
r218=0
r219=0
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r311=r339
r313=r340
r314=0
r315=r312
r317=0
r318=0
r320=0
a11*r310 - a11*r312 - c33*r333
r321=--------------------------------
a11
r322=r339
r323=0
r324=r340
r325=0
a11*r316 - b33*r333
r327=---------------------
a11
r328=0
r329=0
r330=0
r331=r326
r332=0
r334=0
r335=0
r336=0
r337=0
a11*r310 - a11*r312 - c33*r333
r338=--------------------------------
a11
r341=r346
r342=0
a11*r316 - b33*r333
r343=---------------------
a11
r344=0
r345=0
r347=r326
r348=0
r349=0
r350=0
r351=0
r352=r346
r353=r333
r354=0
r355=0
m2=0
m1=0
n2=0
n1=0
c23=0
c22=0
c13=0
c12=0
b32=0
b31=0
b11=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, r29, r13, r310, r312, r316, r340, r339, r326, r346,
r333, m3, c33, b33, n3, a11
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.
{a11,n3}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a11 + u2 *a11 + u3*v3*b33 + u3*n3 + v3 *c33 + v3*m3
has apart from the Hamiltonian and Casimirs the following 11 first integrals:
2 2
FI=u1 *u3*a11 - u1*u3*v1*b33 - u1*v1*v3*c33 + u2 *u3*a11 - u2*u3*v2*b33
- u2*v2*v3*c33 + u3*v3*m3
2 2 2
{HAM,FI} = 2*u1 *v1*v2*a11*c33 - 2*u1*u2*v1 *a11*c33 + 2*u1*u2*v2 *a11*c33
2
+ 2*u1*u3*v2*v3*a11*c33 - 2*u2 *v1*v2*a11*c33
- 2*u2*u3*v1*v3*a11*c33
2
FI=u1 *v1 + u1*u2*v2 + u1*u3*v3
{HAM,FI} = - 2*u1*u2*u3*v1*a11 + u1*u2*v1*v3*b33 + u1*u2*v1*n3
+ u1*u3*v1*v2*b33 + 2*u1*v1*v2*v3*c33 + u1*v1*v2*m3
2 2 2 2
- 2*u2 *u3*v2*a11 + u2 *v2*v3*b33 + u2 *v2*n3 - 2*u2*u3 *v3*a11
2 2 2
+ u2*u3*v2 *b33 + u2*u3*v3 *b33 + u2*u3*v3*n3 + 2*u2*v2 *v3*c33
2 2 2
+ u2*v2 *m3 + u3 *v2*v3*b33 + 2*u3*v2*v3 *c33 + u3*v2*v3*m3
2
FI=u1*u2*v1 + u2 *v2 + u2*u3*v3
2 2 2
{HAM,FI} = 2*u1 *u3*v1*a11 - u1 *v1*v3*b33 - u1 *v1*n3 + 2*u1*u2*u3*v2*a11
2 2
- u1*u2*v2*v3*b33 - u1*u2*v2*n3 + 2*u1*u3 *v3*a11 - u1*u3*v1 *b33
2 2 2
- u1*u3*v3 *b33 - u1*u3*v3*n3 - 2*u1*v1 *v3*c33 - u1*v1 *m3
2
- u2*u3*v1*v2*b33 - 2*u2*v1*v2*v3*c33 - u2*v1*v2*m3 - u3 *v1*v3*b33
2
- 2*u3*v1*v3 *c33 - u3*v1*v3*m3
2
FI=u1*v1*v2 + u2*v2 + u3*v2*v3
2 2 2
{HAM,FI} = 2*u1 *v1*v3*a11 + 2*u1*u2*v2*v3*a11 + 2*u1*u3*v3 *a11 - u1*v1 *v3*b33
2 2
- u1*v1 *n3 - u2*v1*v2*v3*b33 - u2*v1*v2*n3 - u3*v1*v3 *b33
- u3*v1*v3*n3
2
FI=u1*v1 + u2*v1*v2 + u3*v1*v3
{HAM,FI} = - 2*u1*u2*v1*v3*a11 + u1*v1*v2*v3*b33 + u1*v1*v2*n3
2 2 2 2
- 2*u2 *v2*v3*a11 - 2*u2*u3*v3 *a11 + u2*v2 *v3*b33 + u2*v2 *n3
2
+ u3*v2*v3 *b33 + u3*v2*v3*n3
2
FI=u1*u3*v1 + u2*u3*v2 + u3 *v3
{HAM,FI} = 0
2 2
FI= - u1*v1*v3 - u2*v2*v3 + u3*v1 + u3*v2
2 2 2
{HAM,FI} = 2*u1 *v1*v2*a11 - 2*u1*u2*v1 *a11 + 2*u1*u2*v2 *a11
2
+ 2*u1*u3*v2*v3*a11 - 2*u2 *v1*v2*a11 - 2*u2*u3*v1*v3*a11
2
FI=u1*v1*v3 + u2*v2*v3 + u3*v3
2 2 2
{HAM,FI} = - 2*u1 *v1*v2*a11 + 2*u1*u2*v1 *a11 - 2*u1*u2*v2 *a11
2
- 2*u1*u3*v2*v3*a11 + 2*u2 *v1*v2*a11 + 2*u2*u3*v1*v3*a11
FI=u3
{HAM,FI} = 0
2
FI=u3
{HAM,FI} = 0
3
FI=u3
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u2**2*a11 + u3*v3*b33 + u3*n3 + v3**2*c33 + v3*m3$
FI=u1**2*u3*a11 - u1*u3*v1*b33 - u1*v1*v3*c33 + u2**2*u3*a11 - u2*u3*v2*b33 - u2
*v2*v3*c33 + u3*v3*m3$
FI=u1**2*v1 + u1*u2*v2 + u1*u3*v3$
FI=u1*u2*v1 + u2**2*v2 + u2*u3*v3$
FI=u1*v1*v2 + u2*v2**2 + u3*v2*v3$
FI=u1*v1**2 + u2*v1*v2 + u3*v1*v3$
FI=u1*u3*v1 + u2*u3*v2 + u3**2*v3$
FI= - u1*v1*v3 - u2*v2*v3 + u3*v1**2 + u3*v2**2$
FI=u1*v1*v3 + u2*v2*v3 + u3*v3**2$
FI=u3$
FI=u3**2$
FI=u3**3$