Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
b11*n3*r29
r10=--------------------
2
2*a11*a33 - 2*a33
b11*n2*r29
r11=-----------------------
2*a11*a33 - 2*a22*a33
2*a11*b11*n1*r29 - a22*b11*n1*r29 - a33*b11*n1*r29
r12=------------------------------------------------------
2 2 2
2*a11 *a33 - 2*a11*a22*a33 - 2*a11*a33 + 2*a22*a33
- n3*r29
r13=-----------
a11 - a33
- n2*r29
r14=-----------
a11 - a33
- n1*r29
r15=-----------
a11 - a33
2
b11 *r29
r20=----------
2
4*a33
r21=0
2 2
a11*b11 *r29 - a33*b11 *r29
r22=-----------------------------
2 2
4*a11*a33 - 4*a22*a33
r23=0
r24=0
r26=0
r27=0
r28=0
r210=0
r212=0
r213=0
a11*r29 - a22*r29
r214=-------------------
a11 - a33
r215=0
r216=0
b11*r29
r217=---------
a33
r218=0
r219=0
a11*b11*n3 + a22*b11*n3
m3=-------------------------
2*a11*a33 - 2*a22*a33
a11*b11*n2 + a33*b11*n2
m2=-------------------------
2*a11*a33 - 2*a22*a33
a22*b11*n1 + a33*b11*n1
m1=-------------------------
2*a11*a33 - 2*a22*a33
2 2 2 2 2 2 2 2
a11 *a33*b11 - a11*a22 *b11 - a11*a33 *b11 + a22 *a33*b11
c33=---------------------------------------------------------------
2 2 2 2 2
4*a11 *a33 - 8*a11*a22*a33 + 4*a22 *a33
c23=0
2 2 2
a11*a22*b11 - a33 *b11
c22=--------------------------
2 2
4*a11*a33 - 4*a22*a33
c13=0
c12=0
c11=0
- a11*a22*b11 + a11*a33*b11
b33=------------------------------
a11*a33 - a22*a33
b32=0
b31=0
b23=0
b22=0
b21=0
b13=0
b12=0
a23=0
a13=0
a12=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:
r29, n3, n2, n1, b11, a33, a22, 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 - a33,a22 - a33,a11 - a22,a33,a22,a11,r29}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a12,
a13,
a23,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
a11*a22*b11 - a11*a33*b11 + a11*a33*b33 - a22*a33*b33,
c11,
c12,
c13,
- 1/4*a11*a22*b11**2 + a11*a33**2*c22 - a22*a33**2*c22 + 1/4*a33**2*b11**2,
c23,
a11**2*a33**2*c33 - 1/4*a11**2*a33*b11**2 + 1/4*a11*a22**2*b11**2 - 2*a11*a22*
a33**2*c33 + 1/4*a11*a33**2*b11**2 + a22**2*a33**2*c33 - 1/4*a22**2*a33*b11**2,
a11*a33*m1 - a22*a33*m1 - 1/2*a22*b11*n1 - 1/2*a33*b11*n1,
a11*a33*m2 - 1/2*a11*b11*n2 - a22*a33*m2 - 1/2*a33*b11*n2,
a11*a33*m3 - 1/2*a11*b11*n3 - a22*a33*m3 - 1/2*a22*b11*n3}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a11 + u1*v1*b11 + u1*n1 + u2 *a22 + u2*n2 + u3 *a33
- a11*a22*b11 + a11*a33*b11
+ u3*v3*------------------------------ + u3*n3
a11*a33 - a22*a33
1 1
---*a22*b11*n1 + ---*a33*b11*n1
2 2
+ v1*---------------------------------
a11*a33 - a22*a33
1 2 1 2 2
---*a11*a22*b11 - ---*a33 *b11
2 4 4
+ v2 *----------------------------------
2 2
a11*a33 - a22*a33
1 1
---*a11*b11*n2 + ---*a33*b11*n2
2 2 2 1 2 2
+ v2*--------------------------------- + v3 *(---*a11 *a33*b11
a11*a33 - a22*a33 4
1 2 2 1 2 2 1 2 2 2 2
- ---*a11*a22 *b11 - ---*a11*a33 *b11 + ---*a22 *a33*b11 )/(a11 *a33
4 4 4
1 1
---*a11*b11*n3 + ---*a22*b11*n3
2 2 2 2 2
- 2*a11*a22*a33 + a22 *a33 ) + v3*---------------------------------
a11*a33 - a22*a33
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2
FI=u1*v1*(a11 *a33*b11 - a11*a22*a33*b11 - a11*a33 *b11 + a22*a33 *b11)
2 2
+ u1*( - a11*a33 *n1 + a22*a33 *n1)
2 2 2 2 2 2
+ u2 *(a11 *a33 - 2*a11*a22*a33 + a22 *a33 )
2 2
+ u2*( - a11*a33 *n2 + a22*a33 *n2)
2 2 2 2 3 3
+ u3 *(a11 *a33 - a11*a22*a33 - a11*a33 + a22*a33 )
2 2
+ u3*( - a11*a33 *n3 + a22*a33 *n3)
1 1 2
+ v1*(a11*a33*b11*n1 - ---*a22*a33*b11*n1 - ---*a33 *b11*n1)
2 2
2 1 2 2 1 2 1 2 2
+ v2 *(---*a11 *b11 - ---*a11*a33*b11 + ---*a33 *b11 )
4 2 4
1 1 2 2
+ v2*(---*a11*a33*b11*n2 - ---*a33 *b11*n2) + v3
2 2
1 2 2 1 2 1 2 1 2
*(---*a11 *b11 - ---*a11*a22*b11 - ---*a11*a33*b11 + ---*a22*a33*b11 )
4 4 4 4
1 1
+ v3*(---*a11*a33*b11*n3 - ---*a22*a33*b11*n3)
2 2
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*v1*b11 + u1*n1 + u2**2*a22 + u2*n2 + u3**2*a33 + u3*v3*( -
a11*a22*b11 + a11*a33*b11)/(a11*a33 - a22*a33) + u3*n3 + v1*(1/2*a22*b11*n1 + 1/
2*a33*b11*n1)/(a11*a33 - a22*a33) + v2**2*(1/4*a11**2*a22*b11**2 - 1/4*a11*a22**
2*b11**2 - 1/4*a11*a33**2*b11**2 + 1/4*a22*a33**2*b11**2)/(a11**2*a33**2 - 2*a11
*a22*a33**2 + a22**2*a33**2) + v2*(1/2*a11**2*b11*n2 - 1/2*a11*a22*b11*n2 + 1/2*
a11*a33*b11*n2 - 1/2*a22*a33*b11*n2)/(a11**2*a33 - 2*a11*a22*a33 + a22**2*a33) +
v3**2*(1/4*a11**2*a33*b11**2 - 1/4*a11*a22**2*b11**2 - 1/4*a11*a33**2*b11**2 +
1/4*a22**2*a33*b11**2)/(a11**2*a33**2 - 2*a11*a22*a33**2 + a22**2*a33**2) + v3*(
1/2*a11**2*b11*n3 - 1/2*a22**2*b11*n3)/(a11**2*a33 - 2*a11*a22*a33 + a22**2*a33)
$
FI=u1*v1*(a11**2*a33*b11 - a11*a22*a33*b11 - a11*a33**2*b11 + a22*a33**2*b11) +
u1*( - a11*a33**2*n1 + a22*a33**2*n1) + u2**2*(a11**2*a33**2 - 2*a11*a22*a33**2
+ a22**2*a33**2) + u2*( - a11*a33**2*n2 + a22*a33**2*n2) + u3**2*(a11**2*a33**2
- a11*a22*a33**2 - a11*a33**3 + a22*a33**3) + u3*( - a11*a33**2*n3 + a22*a33**2*
n3) + v1*(a11*a33*b11*n1 - 1/2*a22*a33*b11*n1 - 1/2*a33**2*b11*n1) + v2**2*(1/4*
a11**2*b11**2 - 1/2*a11*a33*b11**2 + 1/4*a33**2*b11**2) + v2*(1/2*a11*a33*b11*n2
- 1/2*a33**2*b11*n2) + v3**2*(1/4*a11**2*b11**2 - 1/4*a11*a22*b11**2 - 1/4*a11*
a33*b11**2 + 1/4*a22*a33*b11**2) + v3*(1/2*a11*a33*b11*n3 - 1/2*a22*a33*b11*n3)$