Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
1 1
- ---*a11*b33*n3*r29 + ---*a22*b33*n3*r29
2 2
r10=--------------------------------------------
2
a11 *a22
1
- ---*b33*n2*r29
2
r11=-------------------
a11*a22
1
- a11*b33*n1*r29 + ---*a22*b33*n1*r29
2
r12=----------------------------------------
2
a11 *a22
- n3*r29
r13=-----------
a11
- n2*r29
r14=-----------
a11
- n1*r29
r15=-----------
a11
1 2 2 1 2 1 2 2
---*a11 *b33 *r29 - ---*a11*a22*b33 *r29 + ---*a22 *b33 *r29
4 2 4
r20=--------------------------------------------------------------
2 2
a11 *a22
r21=0
1 2 1 2
---*a11*b33 *r29 - ---*a22*b33 *r29
4 4
r22=-------------------------------------
2
a11*a22
r23=0
r24=0
r26=0
r27=0
r28=0
r210=0
r212=0
r213=0
a11*r29 - a22*r29
r214=-------------------
a11
r215=0
r216=0
- a11*b33*r29 + a22*b33*r29
r217=------------------------------
a11*a22
r218=0
r219=0
1 1
- ---*a11*b33*n3 - ---*a22*b33*n3
2 2
m3=------------------------------------
a11*a22
1
- ---*b33*n2
2
m2=---------------
a22
1
- ---*b33*n1
2
m1=---------------
a11
1 2
- ---*b33
4
c33=-------------
a11
c23=0
1 2 1 2
---*a11*b33 - ---*a22*b33
4 4
c22=-----------------------------
a11*a22
c13=0
c12=0
c11=0
b32=0
b31=0
b23=0
b22=0
b21=0
b13=0
b12=0
b11=0
a33=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, n2, n1, b33, n3, a11, a22
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.
{r29,a11,a22,a11 - a22}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a12,
a13,
a23,
a33,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
c11,
c12,
c13,
a11*a22*c22 - 1/4*a11*b33**2 + 1/4*a22*b33**2,
c23,
a11*c33 + 1/4*b33**2,
a11*m1 + 1/2*b33*n1,
a22*m2 + 1/2*b33*n2,
a11*a22*m3 + 1/2*a11*b33*n3 + 1/2*a22*b33*n3}$
The system of equations related to the Hamiltonian HAM:
1
- ---*v1*b33*n1
2 2 2
HAM=u1 *a11 + u1*n1 + u2 *a22 + u2*n2 + u3*v3*b33 + u3*n3 + ------------------
a11
1 2 1 2 1
---*a11*b33 - ---*a22*b33 - ---*v2*b33*n2
2 4 4 2
+ v2 *----------------------------- + ------------------
a11*a22 a22
1 2 2 1 1
- ---*v3 *b33 - ---*a11*b33*n3 - ---*a22*b33*n3
4 2 2
+ ----------------- + v3*------------------------------------
a11 a11*a22
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2
FI=u1*v1*( - a11 *a22*b33 + a11*a22 *b33) - u1*a11*a22 *n1
2 2 2 3 2 2 2 2
+ u2 *(a11 *a22 - a11*a22 ) - u2*a11*a22 *n2 + u3 *a11 *a22
2 1 2
- u3*a11*a22 *n3 + v1*( - a11*a22*b33*n1 + ---*a22 *b33*n1)
2
2 1 2 2 1 2 1
+ v2 *(---*a11 *b33 - ---*a11*a22*b33 ) - ---*v2*a11*a22*b33*n2
4 4 2
2 1 2 2 1 2 1 2 2
+ v3 *(---*a11 *b33 - ---*a11*a22*b33 + ---*a22 *b33 )
4 2 4
1 1 2
+ v3*( - ---*a11*a22*b33*n3 + ---*a22 *b33*n3)
2 2
1
= a product of the elements of: {---,
4
2 2 2
u1*v1*( - 4*a11 *a22*b33 + 4*a11*a22 *b33) - 4*u1*a11*a22 *n1
2 2 2 3 2 2 2 2
+ u2 *(4*a11 *a22 - 4*a11*a22 ) - 4*u2*a11*a22 *n2 + 4*u3 *a11 *a22
2 2
- 4*u3*a11*a22 *n3 + v1*( - 4*a11*a22*b33*n1 + 2*a22 *b33*n1)
2 2 2 2
+ v2 *(a11 *b33 - a11*a22*b33 ) - 2*v2*a11*a22*b33*n2
2 2 2 2 2 2
+ v3 *(a11 *b33 - 2*a11*a22*b33 + a22 *b33 )
2
+ v3*( - 2*a11*a22*b33*n3 + 2*a22 *b33*n3)}
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*n1 + u2**2*a22 + u2*n2 + u3*v3*b33 + u3*n3 + ( - 1/2*v1*b33*
n1)/a11 + v2**2*(1/4*a11*b33**2 - 1/4*a22*b33**2)/(a11*a22) + ( - 1/2*v2*b33*n2)
/a22 + ( - 1/4*v3**2*b33**2)/a11 + v3*( - 1/2*a11*b33*n3 - 1/2*a22*b33*n3)/(a11*
a22)$
FI=u1*v1*( - a11**2*a22*b33 + a11*a22**2*b33) - u1*a11*a22**2*n1 + u2**2*(a11**2
*a22**2 - a11*a22**3) - u2*a11*a22**2*n2 + u3**2*a11**2*a22**2 - u3*a11*a22**2*
n3 + v1*( - a11*a22*b33*n1 + 1/2*a22**2*b33*n1) + v2**2*(1/4*a11**2*b33**2 - 1/4
*a11*a22*b33**2) - 1/2*v2*a11*a22*b33*n2 + v3**2*(1/4*a11**2*b33**2 - 1/2*a11*
a22*b33**2 + 1/4*a22**2*b33**2) + v3*( - 1/2*a11*a22*b33*n3 + 1/2*a22**2*b33*n3)
$