Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
1
---*b11*n3*r214
2
r10=-------------------
a11*a33 - a22*a33
1 1
---*a11*b11*n2*r214 - ---*a33*b11*n2*r214
2 2
r11=-------------------------------------------
2 2
a11 *a33 - 2*a11*a22*a33 + a22 *a33
1 1
a11*b11*n1*r214 - ---*a22*b11*n1*r214 - ---*a33*b11*n1*r214
2 2
r12=-------------------------------------------------------------
2 2
a11 *a33 - 2*a11*a22*a33 + a22 *a33
- n3*r214
r13=------------
a11 - a22
- n2*r214
r14=------------
a11 - a22
- n1*r214
r15=------------
a11 - a22
1 2 2 1 2 1 2
r20=(---*a11 *b11 *r214 - ---*a11*a22*b11 *r214 - ---*a11*a33*b11 *r214
4 4 4
1 2 2 2 2 2 2
+ ---*a22*a33*b11 *r214)/(a11 *a33 - 2*a11*a22*a33 + a22 *a33 )
4
r21=0
1 2 2 1 2 1 2 2
---*a11 *b11 *r214 - ---*a11*a33*b11 *r214 + ---*a33 *b11 *r214
4 2 4
r22=-----------------------------------------------------------------
2 2 2 2 2
a11 *a33 - 2*a11*a22*a33 + a22 *a33
r23=0
r24=0
r26=0
r27=0
r28=0
a11*r214 - a33*r214
r29=---------------------
a11 - a22
r210=0
r212=0
r213=0
r215=0
r216=0
a11*b11*r214 - a33*b11*r214
r217=-----------------------------
a11*a33 - a22*a33
r218=0
r219=0
1 1
---*a11*b11*n3 + ---*a22*b11*n3
2 2
m3=---------------------------------
a11*a33 - a22*a33
1 1
---*a11*b11*n2 + ---*a33*b11*n2
2 2
m2=---------------------------------
a11*a33 - a22*a33
1 1
---*a22*b11*n1 + ---*a33*b11*n1
2 2
m1=---------------------------------
a11*a33 - a22*a33
c33
1 2 2 1 2 2 1 2 2 1 2 2
---*a11 *a33*b11 - ---*a11*a22 *b11 - ---*a11*a33 *b11 + ---*a22 *a33*b11
4 4 4 4
=-------------------------------------------------------------------------------
2 2 2 2 2
a11 *a33 - 2*a11*a22*a33 + a22 *a33
c23=0
1 2 1 2 2
---*a11*a22*b11 - ---*a33 *b11
4 4
c22=----------------------------------
2 2
a11*a33 - 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:
r214, n3, n1, n2, b11, a33, 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.
{a11 - a33,a22 - a33,a11 - a22,a33,a22,a11,r214}
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
1
= a product of the elements of: {---,
4
2 2 2
u1*v1*(4*a11 *a33*b11 - 4*a11*a22*a33*b11 - 4*a11*a33 *b11 + 4*a22*a33 *b11)
2 2
+ u1*( - 4*a11*a33 *n1 + 4*a22*a33 *n1)
2 2 2 2 2 2
+ u2 *(4*a11 *a33 - 8*a11*a22*a33 + 4*a22 *a33 )
2 2
+ u2*( - 4*a11*a33 *n2 + 4*a22*a33 *n2)
2 2 2 2 3 3
+ u3 *(4*a11 *a33 - 4*a11*a22*a33 - 4*a11*a33 + 4*a22*a33 )
2 2
+ u3*( - 4*a11*a33 *n3 + 4*a22*a33 *n3)
2
+ v1*(4*a11*a33*b11*n1 - 2*a22*a33*b11*n1 - 2*a33 *b11*n1)
2 2 2 2 2 2
+ v2 *(a11 *b11 - 2*a11*a33*b11 + a33 *b11 )
2
+ v2*(2*a11*a33*b11*n2 - 2*a33 *b11*n2)
2 2 2 2 2 2
+ v3 *(a11 *b11 - a11*a22*b11 - a11*a33*b11 + a22*a33*b11 )
+ v3*(2*a11*a33*b11*n3 - 2*a22*a33*b11*n3)}
{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)$