Solution 1 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
1 1
- ---*a12*b12*n2*r218 - ---*a13*b12*n3*r218
4 4
r12=----------------------------------------------
a12*a13*a33
1
---*n3*r218
2
r13=-------------
a13
1
---*n2*r218
2
r14=-------------
a13
1
---*n1*r218
2
r15=-------------
a13
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
1 2 1 2
- ---*a12 *b12*r218 - ---*a13 *b12*r218
2 2
r217=------------------------------------------
a12*a13*a33
a12*r218
r219=----------
a13
r220=0
1
---*b12*n3
2
m3=------------
a12
1
---*b12*n2
2
m2=------------
a12
1 1 1
- ---*a12*b12*n2 - ---*a13*b12*n3 + ---*a33*b12*n1
2 2 2
m1=-----------------------------------------------------
a12*a33
1 2 2 1 2 2
---*a12 *b12 + ---*a13 *b12
4 4
c33=-------------------------------
2
a12 *a33
c23=0
1 2 2 1 2 2
---*a12 *b12 + ---*a13 *b12
4 4
c22=-------------------------------
2
a12 *a33
c13=0
c12=0
b33=0
b23=0
a13*b12
b13=---------
a12
2 2
- a12 *b12 - a13 *b12
b11=------------------------
a12*a33
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:
r218, n1, n3, n2, b12, a12, 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.
{a13,
a33,
r218,
a12,
{n3,
n2,
n1,
b12*n3,
b12*n2,
1 1 1
---*a12*b12*n2 + ---*a13*b12*n3 - ---*a33*b12*n1}}
2 2 2
Relevance for the application:
Modulo the following equation:
2 2
0=a12 + a13
the system of equations related to the Hamiltonian HAM:
2 2 2 3 2
HAM=(u1 *a12 *a33 + 2*u1*u2*a12 *a33 + 2*u1*u3*a12 *a13*a33
3 2 2
+ u1*v1*( - a12 *b12 - a12*a13 *b12) + u1*v2*a12 *a33*b12
2 2 2 2 2
+ u1*v3*a12*a13*a33*b12 + u1*a12 *a33*n1 + u2 *a12 *a33 + u2*a12 *a33*n2
2 2 2 2
+ u3 *a12 *a33 + u3*a12 *a33*n3
1 2 1 1
+ v1*( - ---*a12 *b12*n2 - ---*a12*a13*b12*n3 + ---*a12*a33*b12*n1)
2 2 2
2 1 2 2 1 2 2 1
+ v2 *(---*a12 *b12 + ---*a13 *b12 ) + ---*v2*a12*a33*b12*n2
4 4 2
2 1 2 2 1 2 2 1 2
+ v3 *(---*a12 *b12 + ---*a13 *b12 ) + ---*v3*a12*a33*b12*n3)/(a12 *a33)
4 4 2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 1 2 1 2
FI=u1*u2*a12 *a33 + u1*u3*a12*a13*a33 + u1*v1*( - ---*a12 *b12 - ---*a13 *b12)
2 2
1 1 1
+ ---*u1*a12*a33*n1 + ---*u2*a12*a33*n2 + ---*u3*a12*a33*n3
2 2 2
1 1
+ v1*( - ---*a12*b12*n2 - ---*a13*b12*n3)
4 4
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a12**2*a33**2 + 2*u1*u2*a12**3*a33 + 2*u1*u3*a12**2*a13*a33 + u1*v1*(
- a12**3*b12 - a12*a13**2*b12) + u1*v2*a12**2*a33*b12 + u1*v3*a12*a13*a33*b12 +
u1*a12**2*a33*n1 + u2**2*a12**2*a33**2 + u2*a12**2*a33*n2 + u3**2*a12**2*a33**2
+ u3*a12**2*a33*n3 + v1*( - 1/2*a12**2*b12*n2 - 1/2*a12*a13*b12*n3 + 1/2*a12*
a33*b12*n1) + v2**2*(1/4*a12**2*b12**2 + 1/4*a13**2*b12**2) + 1/2*v2*a12*a33*b12
*n2 + v3**2*(1/4*a12**2*b12**2 + 1/4*a13**2*b12**2) + 1/2*v3*a12*a33*b12*n3)/(
a12**2*a33)$
FI=u1*u2*a12**2*a33 + u1*u3*a12*a13*a33 + u1*v1*( - 1/2*a12**2*b12 - 1/2*a13**2*
b12) + 1/2*u1*a12*a33*n1 + 1/2*u2*a12*a33*n2 + 1/2*u3*a12*a33*n3 + v1*( - 1/4*
a12*b12*n2 - 1/4*a13*b12*n3)$