Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
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Equations
The following unsolved equations remain:
2 2 2 2
0=a13 + a23 + a33 + b12 *kap
Expressions
The solution is given through the following expressions:
r10=0
2 2 2
r11=( - 2*a13 *m3*n2*r333 - 2*a23 *m3*n2*r333 - 2*a33 *m3*n2*r333
2 4 2
- b12 *kap*m3*n2*r333)/(b12 *kap )
2 2 2
r12=( - 2*a13 *m3*n1*r333 - 2*a23 *m3*n1*r333 - 2*a33 *m3*n1*r333
2 4 2
- b12 *kap*m3*n1*r333)/(b12 *kap )
2 2
kap*m3 *r333 - n3 *r333
r13=-------------------------
2
b12 *kap
- n2*n3*r333
r14=---------------
2
b12 *kap
- n1*n3*r333
r15=---------------
2
b12 *kap
r20=0
r21=0
r22=0
r23=0
r24=0
2 2 2
r27=(2*a13 *n1*r333 + 2*a23 *n1*r333 + 2*a23*b12*kap*m3*r333 + 2*a33 *n1*r333
2 3 2
+ b12 *kap*n1*r333)/(b12 *kap )
2 2 2
r28=( - 2*a13 *n2*r333 + 2*a13*b12*kap*m3*r333 - 2*a23 *n2*r333 - 2*a33 *n2*r333
2 3 2
- b12 *kap*n2*r333)/(b12 *kap )
- a13*n1*r333 - a23*n2*r333 - 2*a33*n3*r333
r29=----------------------------------------------
2
b12 *kap
2 2 2 2
- 2*a13 *n1*r333 - 2*a23 *n1*r333 - 2*a33 *n1*r333 - b12 *kap*n1*r333
r210=------------------------------------------------------------------------
3 2
b12 *kap
- 2*a33*m3*r333
r211=------------------
2
b12 *kap
r212=0
- 2*a23*n3*r333
r213=------------------
2
b12 *kap
2 2 3
r214=(2*a13 *a23*n2*r333 - a13*b12 *kap*n1*r333 + 2*a23 *n2*r333
2 2 4 2
+ 2*a23*a33 *n2*r333 + a23*b12 *kap*n2*r333)/(b12 *kap )
2 2 2 2
2*a13 *n2*r333 + 2*a23 *n2*r333 + 2*a33 *n2*r333 + b12 *kap*n2*r333
r215=---------------------------------------------------------------------
3 2
b12 *kap
- 2*a33*m3*r333
r217=------------------
2
b12 *kap
- 2*a13*n3*r333
r218=------------------
2
b12 *kap
r219=0
3 2 2
r220=(2*a13 *n1*r333 + 2*a13*a23 *n1*r333 + 2*a13*a33 *n1*r333
2 2 4 2
+ a13*b12 *kap*n1*r333 - a23*b12 *kap*n2*r333)/(b12 *kap )
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
2 2 2 2
2*a13 *r333 + 2*a23 *r333 + 2*a33 *r333 + b12 *kap*r333
r312=---------------------------------------------------------
2 2
b12 *kap
r314=0
2 2 2 2
2*a13 *r333 + 2*a23 *r333 + 2*a33 *r333 + b12 *kap*r333
r315=---------------------------------------------------------
2 2
b12 *kap
- 2*a13*r333
r317=---------------
b12*kap
2*a23*r333
r318=------------
b12*kap
r319=r333
r320=0
2 2 2 2
- 2*a13 *r333 - 2*a23 *r333 - 2*a33 *r333 - b12 *kap*r333
r321=------------------------------------------------------------
2 2
b12 *kap
r322=0
r323=0
r324=0
r325=0
r327=0
- 2*a33*r333
r328=---------------
b12*kap
r329=0
r330=0
- 2*a13*r333
r331=---------------
b12*kap
r332=0
r334=0
r335=0
r336=0
r337=0
2 2 2 2
- 2*a13 *r333 - 2*a23 *r333 - 2*a33 *r333 - b12 *kap*r333
r338=------------------------------------------------------------
2 2
b12 *kap
r339=0
r340=0
2*a33*r333
r342=------------
b12*kap
r343=0
r344=0
r345=0
2*a23*r333
r346=------------
b12*kap
- 2*a13*r333
r347=---------------
b12*kap
r348=0
r349=0
r350=0
r351=0
2*a23*r333
r352=------------
b12*kap
r353=r333
r354=0
r355=0
m2=0
m1=0
a22= - a33
a11= - 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:
r333, n2, n1, m3, n3, a33, a13, a23, b12
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.
{{m3,n1,n2,n3},r333,a13,b12}
Relevance for the application:
Modulo the following equation:
2 2 2 2
0=a13 + a23 + a33 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2
HAM= - u1 *a33 + 2*u1*u3*a13 + u1*v2*b12 + u1*n1 - u2 *a33 + 2*u2*u3*a23
2
- u2*v1*b12 + u2*n2 + u3 *a33 + u3*n3 + v3*m3
has apart from the Hamiltonian and Casimirs only the following first integral:
2 4 2 2 3 2 3 2
FI=u1 *u3*b12 *kap + 2*u1 *v1*a23*b12 *kap + u1 *(2*a13 *n1 + 2*a13*a23 *n1
2 2 2
+ 2*a13*a33 *n1 + a13*b12 *kap*n1 - a23*b12 *kap*n2)
3 3
- 2*u1*u2*v1*a13*b12 *kap + 2*u1*u2*v2*a23*b12 *kap
3 2
+ 2*u1*u3*v2*a33*b12 *kap - 2*u1*u3*a13*b12 *kap*n3
2 2 2 2 2 2 4
+ u1*v1*v3*( - 2*a13 *b12 - 2*a23 *b12 - 2*a33 *b12 - b12 *kap)
2
- 2*u1*v1*a33*b12 *kap*m3
2 2 2 3
+ u1*v3*(2*a13 *b12*n2 + 2*a23 *b12*n2 + 2*a33 *b12*n2 + b12 *kap*n2)
2 2 4 2 2 3 2
- u1*b12 *kap*n1*n3 + u2 *u3*b12 *kap - 2*u2 *v2*a13*b12 *kap + u2 *(
2 2 3 2
2*a13 *a23*n2 - a13*b12 *kap*n1 + 2*a23 *n2 + 2*a23*a33 *n2
2 3 2
+ a23*b12 *kap*n2) - 2*u2*u3*v1*a33*b12 *kap - 2*u2*u3*a23*b12 *kap*n3
2 2 2 2 2 2 4
+ u2*v2*v3*( - 2*a13 *b12 - 2*a23 *b12 - 2*a33 *b12 - b12 *kap)
2
- 2*u2*v2*a33*b12 *kap*m3
2 2 2 3
+ u2*v3*( - 2*a13 *b12*n1 - 2*a23 *b12*n1 - 2*a33 *b12*n1 - b12 *kap*n1)
2 3 4 2 2 3
- u2*b12 *kap*n2*n3 + u3 *b12 *kap + 2*u3 *v1*a23*b12 *kap
2 3
- 2*u3 *v2*a13*b12 *kap
2 2 2 2
+ u3 *( - a13*b12 *kap*n1 - a23*b12 *kap*n2 - 2*a33*b12 *kap*n3)
2 2 2 2 2 2 2 4
+ u3*v1 *(2*a13 *b12 + 2*a23 *b12 + 2*a33 *b12 + b12 *kap) + u3*v1*(
2 2 2 2
- 2*a13 *b12*n2 + 2*a13*b12 *kap*m3 - 2*a23 *b12*n2 - 2*a33 *b12*n2
3
- b12 *kap*n2)
2 2 2 2 2 2 2 4
+ u3*v2 *(2*a13 *b12 + 2*a23 *b12 + 2*a33 *b12 + b12 *kap) + u3*v2*(
2 2 2 2
2*a13 *b12*n1 + 2*a23 *b12*n1 + 2*a23*b12 *kap*m3 + 2*a33 *b12*n1
3 2 2 2 2 2
+ b12 *kap*n1) + u3*(b12 *kap *m3 - b12 *kap*n3 )
2 2 2 2
+ v1*( - 2*a13 *m3*n1 - 2*a23 *m3*n1 - 2*a33 *m3*n1 - b12 *kap*m3*n1)
2 2 2 2
+ v2*( - 2*a13 *m3*n2 - 2*a23 *m3*n2 - 2*a33 *m3*n2 - b12 *kap*m3*n2)
3 2 3 5 2 2 3
{HAM,FI} = u1 *v1*(4*a13 *b12 *kap + b12 *kap ) + 8*u1 *u2*v1*a13*a23*b12 *kap
2 2 3 5 2
+ u1 *u2*v2*(4*a13 *b12 *kap + b12 *kap )
2 3
+ 8*u1 *u3*v1*a13*a33*b12 *kap
2 2 3 5 2 2 4
+ u1 *u3*v3*(4*a13 *b12 *kap + b12 *kap ) + 2*u1 *v1*v2*a33*b12 *kap
2 4 2 3
- 2*u1 *v1*v3*a23*b12 *kap + 2*u1 *v1*a13*b12 *kap*n3
2 2 3 5 2
+ u1*u2 *v1*(4*a23 *b12 *kap + b12 *kap )
2 3 3
+ 8*u1*u2 *v2*a13*a23*b12 *kap + 8*u1*u2*u3*v1*a23*a33*b12 *kap
3 3
+ 8*u1*u2*u3*v2*a13*a33*b12 *kap + 8*u1*u2*u3*v3*a13*a23*b12 *kap
2 4 4
- 2*u1*u2*v1 *a33*b12 *kap + 2*u1*u2*v1*v3*a13*b12 *kap
3 2 4
+ 2*u1*u2*v1*a23*b12 *kap*n3 + 2*u1*u2*v2 *a33*b12 *kap
4 3
- 2*u1*u2*v2*v3*a23*b12 *kap + 2*u1*u2*v2*a13*b12 *kap*n3
2 2 3 2 3
+ u1*u3 *v1*( - 4*a13 *b12 *kap - 4*a23 *b12 *kap)
2 3 2 4
+ 8*u1*u3 *v3*a13*a33*b12 *kap + 4*u1*u3*v1 *a23*b12 *kap
4
- 4*u1*u3*v1*v2*a13*b12 *kap
3 3
+ u1*u3*v1*( - 2*a13*b12 *kap*n1 - 2*a23*b12 *kap*n2)
4 2 4
+ 2*u1*u3*v2*v3*a33*b12 *kap - 2*u1*u3*v3 *a23*b12 *kap
3 3 5
+ 2*u1*u3*v3*a13*b12 *kap*n3 - u1*v1 *b12 *kap
2 3 4 2 5
+ u1*v1 *(2*a13*b12 *kap*m3 + b12 *kap*n2) - u1*v1*v2 *b12 *kap
3 4
+ u1*v1*v2*(2*a23*b12 *kap*m3 - b12 *kap*n1)
3 2 3 5 2
+ u2 *v2*(4*a23 *b12 *kap + b12 *kap )
2 3
+ 8*u2 *u3*v2*a23*a33*b12 *kap
2 2 3 5 2 2 4
+ u2 *u3*v3*(4*a23 *b12 *kap + b12 *kap ) - 2*u2 *v1*v2*a33*b12 *kap
2 4 2 3
+ 2*u2 *v2*v3*a13*b12 *kap + 2*u2 *v2*a23*b12 *kap*n3
2 2 3 2 3
+ u2*u3 *v2*( - 4*a13 *b12 *kap - 4*a23 *b12 *kap)
2 3 4
+ 8*u2*u3 *v3*a23*a33*b12 *kap + 4*u2*u3*v1*v2*a23*b12 *kap
4 2 4
- 2*u2*u3*v1*v3*a33*b12 *kap - 4*u2*u3*v2 *a13*b12 *kap
3 3
+ u2*u3*v2*( - 2*a13*b12 *kap*n1 - 2*a23*b12 *kap*n2)
2 4 3
+ 2*u2*u3*v3 *a13*b12 *kap + 2*u2*u3*v3*a23*b12 *kap*n3
2 5 3 4
- u2*v1 *v2*b12 *kap + u2*v1*v2*(2*a13*b12 *kap*m3 + b12 *kap*n2)
3 5 2 3 4
- u2*v2 *b12 *kap + u2*v2 *(2*a23*b12 *kap*m3 - b12 *kap*n1)
3 2 3 2 3
+ u3 *v3*( - 4*a13 *b12 *kap - 4*a23 *b12 *kap)
2 4 2 4
+ 4*u3 *v1*v3*a23*b12 *kap - 4*u3 *v2*v3*a13*b12 *kap
2 3 3
+ u3 *v3*( - 2*a13*b12 *kap*n1 - 2*a23*b12 *kap*n2)
2 5 3 4
- u3*v1 *v3*b12 *kap + u3*v1*v3*(2*a13*b12 *kap*m3 + b12 *kap*n2)
2 5 3 4
- u3*v2 *v3*b12 *kap + u3*v2*v3*(2*a23*b12 *kap*m3 - b12 *kap*n1)
And again in machine readable form:
HAM= - u1**2*a33 + 2*u1*u3*a13 + u1*v2*b12 + u1*n1 - u2**2*a33 + 2*u2*u3*a23 -
u2*v1*b12 + u2*n2 + u3**2*a33 + u3*n3 + v3*m3$
FI=u1**2*u3*b12**4*kap**2 + 2*u1**2*v1*a23*b12**3*kap + u1**2*(2*a13**3*n1 + 2*
a13*a23**2*n1 + 2*a13*a33**2*n1 + a13*b12**2*kap*n1 - a23*b12**2*kap*n2) - 2*u1*
u2*v1*a13*b12**3*kap + 2*u1*u2*v2*a23*b12**3*kap + 2*u1*u3*v2*a33*b12**3*kap - 2
*u1*u3*a13*b12**2*kap*n3 + u1*v1*v3*( - 2*a13**2*b12**2 - 2*a23**2*b12**2 - 2*
a33**2*b12**2 - b12**4*kap) - 2*u1*v1*a33*b12**2*kap*m3 + u1*v3*(2*a13**2*b12*n2
+ 2*a23**2*b12*n2 + 2*a33**2*b12*n2 + b12**3*kap*n2) - u1*b12**2*kap*n1*n3 + u2
**2*u3*b12**4*kap**2 - 2*u2**2*v2*a13*b12**3*kap + u2**2*(2*a13**2*a23*n2 - a13*
b12**2*kap*n1 + 2*a23**3*n2 + 2*a23*a33**2*n2 + a23*b12**2*kap*n2) - 2*u2*u3*v1*
a33*b12**3*kap - 2*u2*u3*a23*b12**2*kap*n3 + u2*v2*v3*( - 2*a13**2*b12**2 - 2*
a23**2*b12**2 - 2*a33**2*b12**2 - b12**4*kap) - 2*u2*v2*a33*b12**2*kap*m3 + u2*
v3*( - 2*a13**2*b12*n1 - 2*a23**2*b12*n1 - 2*a33**2*b12*n1 - b12**3*kap*n1) - u2
*b12**2*kap*n2*n3 + u3**3*b12**4*kap**2 + 2*u3**2*v1*a23*b12**3*kap - 2*u3**2*v2
*a13*b12**3*kap + u3**2*( - a13*b12**2*kap*n1 - a23*b12**2*kap*n2 - 2*a33*b12**2
*kap*n3) + u3*v1**2*(2*a13**2*b12**2 + 2*a23**2*b12**2 + 2*a33**2*b12**2 + b12**
4*kap) + u3*v1*( - 2*a13**2*b12*n2 + 2*a13*b12**2*kap*m3 - 2*a23**2*b12*n2 - 2*
a33**2*b12*n2 - b12**3*kap*n2) + u3*v2**2*(2*a13**2*b12**2 + 2*a23**2*b12**2 + 2
*a33**2*b12**2 + b12**4*kap) + u3*v2*(2*a13**2*b12*n1 + 2*a23**2*b12*n1 + 2*a23*
b12**2*kap*m3 + 2*a33**2*b12*n1 + b12**3*kap*n1) + u3*(b12**2*kap**2*m3**2 - b12
**2*kap*n3**2) + v1*( - 2*a13**2*m3*n1 - 2*a23**2*m3*n1 - 2*a33**2*m3*n1 - b12**
2*kap*m3*n1) + v2*( - 2*a13**2*m3*n2 - 2*a23**2*m3*n2 - 2*a33**2*m3*n2 - b12**2*
kap*m3*n2)$