Solution 10 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
2*m1*n1*n3*r214 + 2*m2*n2*n3*r214
r10=-----------------------------------
2 2
a33*n1 + a33*n2
2 2
2*m1*n1*n2*r214 - m2*n1 *r214 + m2*n2 *r214
r11=---------------------------------------------
2 2
a33*n1 + a33*n2
2 2
m1*n1 *r214 - m1*n2 *r214 + 2*m2*n1*n2*r214
r12=---------------------------------------------
2 2
a33*n1 + a33*n2
- n3*r214
r13=------------
a33
- n2*r214
r14=------------
a33
- n1*r214
r15=------------
a33
2 2
- m1 *r214 - m2 *r214
r20=------------------------
2 2
n1 + n2
r21=0
r22=0
r23=0
r24=0
2*m1*n1*r214 + 2*m2*n2*r214
r26=-----------------------------
2 2
n1 + n2
r27=0
r28=0
r210=0
r213=0
r215=0
- 2*m1*n2*r214 + 2*m2*n1*r214
r216=--------------------------------
2 2
n1 + n2
r217=0
r218=0
r219=0
r220=r214
m3=0
c33=0
c23=0
c22=0
c13=0
c12=0
c11=0
b33=0
b32=0
b31=0
b23=0
b22=0
b21=0
b13=0
b12=0
b11=0
a23=0
a22=0
a13=0
a12=0
a11=0
2*m1*n2*r214 - 2*m2*n1*r214
r212=-----------------------------
2 2
n1 + n2
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, m1, m2, n3, n2, n1, a33
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.
{a33}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11,
a12,
a13,
a22,
a23,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
b33,
c11,
c12,
c13,
c22,
c23,
c33,
m3}$
The system of equations related to the Hamiltonian HAM:
2
HAM=u1*n1 + u2*n2 + u3 *a33 + u3*n3 + v1*m1 + v2*m2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2
FI=u1 *(a33*n1 + a33*n2 ) + u1*v2*( - 2*a33*m1*n2 + 2*a33*m2*n1)
3 2 2 2 2
+ u1*( - n1 - n1*n2 ) + u2 *(a33*n1 + a33*n2 )
2 3
+ u2*v1*(2*a33*m1*n2 - 2*a33*m2*n1) + u2*( - n1 *n2 - n2 )
2 2
+ u3*v3*(2*a33*m1*n1 + 2*a33*m2*n2) + u3*( - n1 *n3 - n2 *n3)
2 2 2 2
+ v1*(m1*n1 - m1*n2 + 2*m2*n1*n2) + v2*(2*m1*n1*n2 - m2*n1 + m2*n2 )
2 2 2
+ v3 *( - a33*m1 - a33*m2 ) + v3*(2*m1*n1*n3 + 2*m2*n2*n3)
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1*n1 + u2*n2 + u3**2*a33 + u3*n3 + v1*m1 + v2*m2$
FI=u1**2*(a33*n1**2 + a33*n2**2) + u1*v2*( - 2*a33*m1*n2 + 2*a33*m2*n1) + u1*( -
n1**3 - n1*n2**2) + u2**2*(a33*n1**2 + a33*n2**2) + u2*v1*(2*a33*m1*n2 - 2*a33*
m2*n1) + u2*( - n1**2*n2 - n2**3) + u3*v3*(2*a33*m1*n1 + 2*a33*m2*n2) + u3*( -
n1**2*n3 - n2**2*n3) + v1*(m1*n1**2 - m1*n2**2 + 2*m2*n1*n2) + v2*(2*m1*n1*n2 -
m2*n1**2 + m2*n2**2) + v3**2*( - a33*m1**2 - a33*m2**2) + v3*(2*m1*n1*n3 + 2*m2*
n2*n3)$