Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
b33*r12
r10=---------
b31
b32*r12
r11=---------
b31
r13=0
r14=0
r15=0
2 2
b31 *r22 - b33 *r22
r20=---------------------
2 2
b31 - b32
- 2*b32*b33*r22
r21=------------------
2 2
b31 - b32
- 2*b31*b33*r22
r23=------------------
2 2
b31 - b32
1
---*b33*n3*r328
2
r26=-----------------
a23*b31
1
---*b32*n3*r328
2
r27=-----------------
a23*b31
1
---*n3*r328
2
r28=-------------
a23
r29=0
r210=0
r211=0
r212=0
r214=0
r215=0
r216=0
r217=0
r218=0
r219=0
1 2 1 2
- ---*b31 *r35 + b31*b32*r37 + ---*b33 *r35
2 6
r30=----------------------------------------------
b31*b32
1 2 1 2
- ---*b31 *r35 + b31*b32*r37 + ---*b33 *r35
2 2
r31=----------------------------------------------
b31*b33
1 2 1 2
- ---*b31 *r35 + b31*b32*r37 + ---*b32 *r35
2 2
r32=----------------------------------------------
b31*b32
1 2 1 2
- ---*b31 *r35 + b31*b32*r37 + ---*b32 *r35
2 6
r33=----------------------------------------------
b31*b33
1 2 1 2
- ---*b31 *r35 + b31*b32*r37 + ---*b33 *r35
2 2
r34=----------------------------------------------
b32*b33
1 2 1 2
- ---*b31 *r35 + b31*b32*r37 + ---*b32 *r35
2 2
r36=----------------------------------------------
b32*b33
b32*r37
r38=---------
b33
1 2
- ---*b31 *r35 + b31*b32*r37
3
r39=-------------------------------
b32*b33
1 2
---*b33 *r328
2
r310=---------------
a23*b31
b32*b33*r328
r311=--------------
a23*b31
1 2
---*b32 *r328
2
r312=---------------
a23*b31
b33*r328
r313=----------
a23
b32*r328
r314=----------
a23
1
---*b31*r328
2
r315=--------------
a23
1
a23*b31*b33*r351 + ---*a33*b32*b33*r328
2
r316=-----------------------------------------
a23*b31*b32
1
a23*b31*r351 + ---*a33*b32*r328
2
r317=---------------------------------
a23*b31
1
a23*b31*r351 + ---*a33*b32*r328
2
r318=---------------------------------
a23*b32
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
b33*r328
r326=----------
b31
b32*r328
r327=----------
b31
r329=0
b33*r351
r330=----------
b32
r331=r351
b31*r351
r332=----------
b32
r333=0
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
b33*r351
r350=----------
b32
b31*r351
r352=----------
b32
r353=0
r354=0
r355=0
b33*m1
m3=--------
b31
b32*m1
m2=--------
b31
c22=0
c11=0
- 2*b31*b32*r22
r24=------------------
2 2
b31 - b32
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:
r22, r00, r12, r35, r37, r328, r351, m1, n3, a33, b33,
b31, b32, a23
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.
{b31,a23,b33}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2
HAM=(2*u2*u3*a23*b31 + u3 *a33*b31 + u3*v1*b31 + u3*v2*b31*b32 + u3*v3*b31*b33
+ u3*b31*n3 + v1*b31*m1 + v2*b32*m1 + v3*b33*m1)/b31
has apart from the Hamiltonian and Casimirs the following 7 first integrals:
2 2 2 2 2 2
FI=u1 *v1*b31 + u1 *v2*b32 + u1 *v3*b33 + u2 *v1*b31 + u2 *v2*b32 + u2 *v3*b33
2 2 2
+ u3 *v1*b31 + u3 *v2*b32 + u3 *v3*b33
2 2 2
= a product of the elements of: {u1 + u2 + u3 ,v1*b31 + v2*b32 + v3*b33}
{HAM,FI} = 0
1 2
FI=u2*u3*v1*a23*b31 + u2*u3*v2*a23*b32 + u2*u3*v3*a23*b33 + ---*u3 *v1*a33*b31
2
1 2 1 2 1 2 2
+ ---*u3 *v2*a33*b32 + ---*u3 *v3*a33*b33 + ---*u3*v1 *b31
2 2 2
1 1 2 2
+ u3*v1*v2*b31*b32 + u3*v1*v3*b31*b33 + ---*u3*v1*b31*n3 + ---*u3*v2 *b32
2 2
1 1 2 2 1
+ u3*v2*v3*b32*b33 + ---*u3*v2*b32*n3 + ---*u3*v3 *b33 + ---*u3*v3*b33*n3
2 2 2
1
= a product of the elements of: {---,
2
u3,
v1*b31 + v2*b32 + v3*b33,
2*u2*a23 + u3*a33 + v1*b31 + v2*b32 + v3*b33 + n3}
{HAM,FI} = 0
3 2 2 2 2 3
FI=v1 *b31 + v1 *v2*b32 + v1 *v3*b33 + v1*v2 *b31 + v1*v3 *b31 + v2 *b32
2 2 3
+ v2 *v3*b33 + v2*v3 *b32 + v3 *b33
2 2 2
= a product of the elements of: {v1 + v2 + v3 ,v1*b31 + v2*b32 + v3*b33}
{HAM,FI} = 0
1 3 3 2 1 3 1 2
FI= - ---*v1 *b31 + v1*v2 *( - ---*b31 + ---*b31*b32 ) + v1*v2*v3*b31*b32*b33
3 2 2
2 1 3 1 2 3 1 2 1 3
+ v1*v3 *( - ---*b31 + ---*b31*b33 ) + v2 *( - ---*b31 *b32 + ---*b32 )
2 2 2 6
2 1 2 1 2
+ v2 *v3*( - ---*b31 *b33 + ---*b32 *b33)
2 2
2 1 2 1 2 3 1 2 1 3
+ v2*v3 *( - ---*b31 *b32 + ---*b32*b33 ) + v3 *( - ---*b31 *b33 + ---*b33 )
2 2 2 6
1
= a product of the elements of: { - ---,
6
v1*b31 + v2*b32 + v3*b33,
2 2 2 2 2
2*v1 *b31 - 2*v1*v2*b31*b32 - 2*v1*v3*b31*b33 + v2 *(3*b31 - b32 )
2 2 2
- 2*v2*v3*b32*b33 + v3 *(3*b31 - b33 )}
{HAM,FI} = 0
FI=v1*b31 + v2*b32 + v3*b33
which the program can not factorize further.
{HAM,FI} = 0
FI=1
which the program can not factorize further.
{HAM,FI} = 0
2 2 2
FI= - 2*v1*v2*b31*b32 - 2*v1*v3*b31*b33 + v2 *(b31 - b32 ) - 2*v2*v3*b32*b33
2 2 2
+ v3 *(b31 - b33 )
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=(2*u2*u3*a23*b31 + u3**2*a33*b31 + u3*v1*b31**2 + u3*v2*b31*b32 + u3*v3*b31*
b33 + u3*b31*n3 + v1*b31*m1 + v2*b32*m1 + v3*b33*m1)/b31$
FI=u1**2*v1*b31 + u1**2*v2*b32 + u1**2*v3*b33 + u2**2*v1*b31 + u2**2*v2*b32 + u2
**2*v3*b33 + u3**2*v1*b31 + u3**2*v2*b32 + u3**2*v3*b33$
FI=u2*u3*v1*a23*b31 + u2*u3*v2*a23*b32 + u2*u3*v3*a23*b33 + 1/2*u3**2*v1*a33*b31
+ 1/2*u3**2*v2*a33*b32 + 1/2*u3**2*v3*a33*b33 + 1/2*u3*v1**2*b31**2 + u3*v1*v2*
b31*b32 + u3*v1*v3*b31*b33 + 1/2*u3*v1*b31*n3 + 1/2*u3*v2**2*b32**2 + u3*v2*v3*
b32*b33 + 1/2*u3*v2*b32*n3 + 1/2*u3*v3**2*b33**2 + 1/2*u3*v3*b33*n3$
FI=v1**3*b31 + v1**2*v2*b32 + v1**2*v3*b33 + v1*v2**2*b31 + v1*v3**2*b31 + v2**3
*b32 + v2**2*v3*b33 + v2*v3**2*b32 + v3**3*b33$
FI= - 1/3*v1**3*b31**3 + v1*v2**2*( - 1/2*b31**3 + 1/2*b31*b32**2) + v1*v2*v3*
b31*b32*b33 + v1*v3**2*( - 1/2*b31**3 + 1/2*b31*b33**2) + v2**3*( - 1/2*b31**2*
b32 + 1/6*b32**3) + v2**2*v3*( - 1/2*b31**2*b33 + 1/2*b32**2*b33) + v2*v3**2*( -
1/2*b31**2*b32 + 1/2*b32*b33**2) + v3**3*( - 1/2*b31**2*b33 + 1/6*b33**3)$
FI=v1*b31 + v2*b32 + v3*b33$
FI=1$
FI= - 2*v1*v2*b31*b32 - 2*v1*v3*b31*b33 + v2**2*(b31**2 - b32**2) - 2*v2*v3*b32*
b33 + v3**2*(b31**2 - b33**2)$