Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=a12 + a13
Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
r13=0
r14=0
r15=0
r20=0
r21=0
r23=0
r24=0
r25=0
r26=0
r27=0
r28=0
r210=0
r212=0
r213=0
r214=0
r215=0
r216=0
r217=0
r218=0
r219=0
r220=0
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=r338
r312=0
r313=r324
r314=0
r315=0
r316=r327
r317=0
r318=0
r319=0
r320=0
r321=r338
r322=r311
r323=0
r325=0
r326=r331
r328=0
r329=0
r330=0
r332=0
r333=0
r334=0
r335=0
r336=0
r337=0
r339=r311
r340=r324
r341=r352
r342=0
r343=r327
r344=0
r345=0
r346=r352
r347=r331
r348=0
r349=0
r350=0
r351=0
r353=0
r354=0
r355=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:
r352, r338, r311, r324, r331, r327, m1, m2, c33, c22,
c13, n1, n2, n3, c12, c23, b13, b23, b12, b11, b33,
a12, m3, a33, a13
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.
{{r352,r331,r327,r324,r311,r338},a13,a33,a12}
Relevance for the application:
Modulo the following equation:
2 2
0=a12 + a13
the system of equations related to the Hamiltonian HAM:
2
HAM=u1 *a33 + 2*u1*u2*a12 + 2*u1*u3*a13 + u1*v1*b11 + u1*v2*b12 + u1*v3*b13
2 2
+ u1*n1 + u2 *a33 + u2*v3*b23 + u2*n2 + u3 *a33 + u3*v3*b33 + u3*n3
2
+ 2*v1*v2*c12 + 2*v1*v3*c13 + v1*m1 + v2 *c22 + 2*v2*v3*c23 + v2*m2
2
+ v3 *c33 + v3*m3
has apart from the Hamiltonian and Casimirs the following 6 first integrals:
2
FI=u1*u3*v1 + u2*u3*v2 + u3 *v3
3 2 2 2 2
{HAM,FI} = 2*u1 *v1*a12 + 2*u1 *u2*v2*a12 + 2*u1 *u3*v3*a12 + u1 *v1 *b12
2 2 2 2
- u1 *v1*v2*b11 + u1 *v1*v3*b23 + u1 *v1*n2 - 2*u1*u2 *v1*a12
2
- 2*u1*u2*u3*v1*a13 - u1*u2*v1 *b11 - u1*u2*v1*v3*b13 - u1*u2*v1*n1
2
- u1*u2*v2 *b11 + u1*u2*v2*v3*b23 + u1*u2*v2*n2 + u1*u3*v1*v3*b12
2 3
- u1*u3*v2*v3*b11 + u1*u3*v3 *b23 + u1*u3*v3*n2 + 2*u1*v1 *c12
2 2 2 2
+ 2*u1*v1 *v2*c22 + 2*u1*v1 *v3*c23 + u1*v1 *m2 - 2*u1*v1*v2 *c12
3 2
- 2*u1*v1*v2*v3*c13 - u1*v1*v2*m1 - 2*u2 *v2*a12 - 2*u2 *u3*v2*a13
2 2 2 2 2
- 2*u2 *u3*v3*a12 - u2 *v1*v2*b11 - u2 *v2 *b12 - u2 *v2*v3*b13
2 2
- u2 *v2*n1 - 2*u2*u3 *v3*a13 - u2*u3*v1*v3*b11 - u2*u3*v2*v3*b12
2 2 2
- u2*u3*v3 *b13 - u2*u3*v3*n1 + 2*u2*v1 *v2*c12 + 2*u2*v1*v2 *c22
3 2
+ 2*u2*v1*v2*v3*c23 + u2*v1*v2*m2 - 2*u2*v2 *c12 - 2*u2*v2 *v3*c13
2 2 2
- u2*v2 *m1 + 2*u3*v1 *v3*c12 + 2*u3*v1*v2*v3*c22 + 2*u3*v1*v3 *c23
2 2
+ u3*v1*v3*m2 - 2*u3*v2 *v3*c12 - 2*u3*v2*v3 *c13 - u3*v2*v3*m1
2
FI=u1*u2*v1 + u2 *v2 + u2*u3*v3
3 2 2 2 2
{HAM,FI} = - 2*u1 *v1*a13 - 2*u1 *u2*v2*a13 - 2*u1 *u3*v3*a13 - u1 *v1 *b13
2 2
+ u1 *v1*v3*(b11 - b33) - u1 *v1*n3 + 2*u1*u2*u3*v1*a12
2
- u1*u2*v1 *b23 - u1*u2*v1*v2*b13 + u1*u2*v2*v3*(b11 - b33)
2 2
- u1*u2*v2*n3 + 2*u1*u3 *v1*a13 + u1*u3*v1 *(b11 - b33)
2
+ u1*u3*v1*v2*b12 + u1*u3*v1*n1 + u1*u3*v3 *(b11 - b33)
3 2 2
- u1*u3*v3*n3 - 2*u1*v1 *c13 - 2*u1*v1 *v2*c23 - 2*u1*v1 *v3*c33
2 2
- u1*v1 *m3 + 2*u1*v1*v2*v3*c12 + 2*u1*v1*v3 *c13 + u1*v1*v3*m1
2 2 2
+ 2*u2 *u3*v2*a12 - u2 *v1*v2*b23 + 2*u2*u3 *v2*a13
2
+ 2*u2*u3 *v3*a12 + u2*u3*v1*v2*(b11 - b33) - u2*u3*v1*v3*b23
2 2
+ u2*u3*v2 *b12 + u2*u3*v2*v3*b13 + u2*u3*v2*n1 - 2*u2*v1 *v2*c13
2
- 2*u2*v1*v2 *c23 - 2*u2*v1*v2*v3*c33 - u2*v1*v2*m3
2 2 3
+ 2*u2*v2 *v3*c12 + 2*u2*v2*v3 *c13 + u2*v2*v3*m1 + 2*u3 *v3*a13
2 2 2 2 2
+ u3 *v1*v3*(b11 - b33) + u3 *v2*v3*b12 + u3 *v3 *b13 + u3 *v3*n1
2 2
- 2*u3*v1 *v3*c13 - 2*u3*v1*v2*v3*c23 - 2*u3*v1*v3 *c33
2 3 2
- u3*v1*v3*m3 + 2*u3*v2*v3 *c12 + 2*u3*v3 *c13 + u3*v3 *m1
2
FI=u1*v1 + u2*v1*v2 + u3*v1*v3
2 2
{HAM,FI} = 2*u1 *v1*v2*a13 - 2*u1 *v1*v3*a12 - 2*u1*u2*v1*v3*a33
2
+ 2*u1*u2*v2 *a13 - 2*u1*u2*v2*v3*a12 + 2*u1*u3*v1*v2*a33
2
+ 2*u1*u3*v2*v3*a13 - 2*u1*u3*v3 *a12 + u1*v1*v2*v3*b33
2 2
+ u1*v1*v2*n3 - u1*v1*v3 *b23 - u1*v1*v3*n2 - 2*u2 *v2*v3*a33
2 2 2 2
+ 2*u2*u3*v2 *a33 - 2*u2*u3*v3 *a33 + u2*v2 *v3*b33 + u2*v2 *n3
2 2 2
- u2*v2*v3 *b23 - u2*v2*v3*n2 + 2*u3 *v2*v3*a33 + u3*v2*v3 *b33
3 2
+ u3*v2*v3*n3 - u3*v3 *b23 - u3*v3 *n2
2
FI=u1*v1*v2 + u2*v2 + u3*v2*v3
2 2 2
{HAM,FI} = - 2*u1 *v1 *a13 + 2*u1 *v1*v3*a33 - 2*u1*u2*v1*v2*a13
2
+ 2*u1*u2*v1*v3*a12 + 2*u1*u2*v2*v3*a33 - 2*u1*u3*v1 *a33
2 2 2
+ 2*u1*u3*v3 *a33 + u1*v1 *v3*(b11 - b33) - u1*v1 *n3
2 2
+ u1*v1*v2*v3*b12 + u1*v1*v3 *b13 + u1*v1*v3*n1 + 2*u2 *v2*v3*a12
2
- 2*u2*u3*v1*v2*a33 + 2*u2*u3*v2*v3*a13 + 2*u2*u3*v3 *a12
2
+ u2*v1*v2*v3*(b11 - b33) - u2*v1*v2*n3 + u2*v2 *v3*b12
2 2 2 2
+ u2*v2*v3 *b13 + u2*v2*v3*n1 - 2*u3 *v1*v3*a33 + 2*u3 *v3 *a13
2 2 3
+ u3*v1*v3 *(b11 - b33) - u3*v1*v3*n3 + u3*v2*v3 *b12 + u3*v3 *b13
2
+ u3*v3 *n1
2
FI=u1*v1*v3 + u2*v2*v3 + u3*v3
2 2 2 2 2
{HAM,FI} = 2*u1 *v1 *a12 - 2*u1 *v1*v2*a33 + 2*u1*u2*v1 *a33 - 2*u1*u2*v2 *a33
- 2*u1*u3*v1*v2*a13 + 2*u1*u3*v1*v3*a12 - 2*u1*u3*v2*v3*a33
2 2 2 2
- u1*v1 *v2*b11 + u1*v1 *v3*b23 + u1*v1 *n2 - u1*v1*v2 *b12
2 2 2
- u1*v1*v2*v3*b13 - u1*v1*v2*n1 + 2*u2 *v1*v2*a33 - 2*u2 *v2 *a12
2
+ 2*u2*u3*v1*v3*a33 - 2*u2*u3*v2 *a13 - 2*u2*u3*v2*v3*a12
2 3
- u2*v1*v2 *b11 + u2*v1*v2*v3*b23 + u2*v1*v2*n2 - u2*v2 *b12
2 2 2
- u2*v2 *v3*b13 - u2*v2 *n1 - 2*u3 *v2*v3*a13 - u3*v1*v2*v3*b11
2 2 2
+ u3*v1*v3 *b23 + u3*v1*v3*n2 - u3*v2 *v3*b12 - u3*v2*v3 *b13
- u3*v2*v3*n1
2
FI=u1 *v1 + u1*u2*v2 + u1*u3*v3
2 2 2 2
{HAM,FI} = 2*u1 *u2*v1*a13 - 2*u1 *u3*v1*a12 + u1 *v1*v2*b13 - u1 *v1*v3*b12
2
+ 2*u1*u2 *v2*a13 - 2*u1*u2*u3*v2*a12 + 2*u1*u2*u3*v3*a13
2
+ u1*u2*v1*v2*b23 + u1*u2*v1*v3*b33 + u1*u2*v1*n3 + u1*u2*v2 *b13
2
- u1*u2*v2*v3*b12 - 2*u1*u3 *v3*a12 + u1*u3*v1*v2*b33
2
- u1*u3*v1*v3*b23 - u1*u3*v1*n2 + u1*u3*v2*v3*b13 - u1*u3*v3 *b12
2 2 2
+ 2*u1*v1 *v2*c13 - 2*u1*v1 *v3*c12 + 2*u1*v1*v2 *c23
2
+ u1*v1*v2*v3*( - 2*c22 + 2*c33) + u1*v1*v2*m3 - 2*u1*v1*v3 *c23
2 2 2 2
- u1*v1*v3*m2 + u2 *v2 *b23 + u2 *v2*v3*b33 + u2 *v2*n3
2 2
+ u2*u3*v2 *b33 - u2*u3*v2*n2 + u2*u3*v3 *b33 + u2*u3*v3*n3
2 3
+ 2*u2*v1*v2 *c13 - 2*u2*v1*v2*v3*c12 + 2*u2*v2 *c23
2 2 2
+ u2*v2 *v3*( - 2*c22 + 2*c33) + u2*v2 *m3 - 2*u2*v2*v3 *c23
2 2 2 2
- u2*v2*v3*m2 + u3 *v2*v3*b33 - u3 *v3 *b23 - u3 *v3*n2
2 2
+ 2*u3*v1*v2*v3*c13 - 2*u3*v1*v3 *c12 + 2*u3*v2 *v3*c23
2 3
+ u3*v2*v3 *( - 2*c22 + 2*c33) + u3*v2*v3*m3 - 2*u3*v3 *c23
2
- u3*v3 *m2
And again in machine readable form:
HAM=u1**2*a33 + 2*u1*u2*a12 + 2*u1*u3*a13 + u1*v1*b11 + u1*v2*b12 + u1*v3*b13 +
u1*n1 + u2**2*a33 + u2*v3*b23 + u2*n2 + u3**2*a33 + u3*v3*b33 + u3*n3 + 2*v1*v2*
c12 + 2*v1*v3*c13 + v1*m1 + v2**2*c22 + 2*v2*v3*c23 + v2*m2 + v3**2*c33 + v3*m3$
FI=u1*u3*v1 + u2*u3*v2 + u3**2*v3$
FI=u1*u2*v1 + u2**2*v2 + u2*u3*v3$
FI=u1*v1**2 + u2*v1*v2 + u3*v1*v3$
FI=u1*v1*v2 + u2*v2**2 + u3*v2*v3$
FI=u1*v1*v3 + u2*v2*v3 + u3*v3**2$
FI=u1**2*v1 + u1*u2*v2 + u1*u3*v3$