Solution 2 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r10=0
1
- ---*b33*n2*r29
2
r11=-------------------
2
a11
1
- ---*b33*n1*r29
2
r12=-------------------
2
a11
- n3*r29
r13=-----------
a11
- n2*r29
r14=-----------
a11
- n1*r29
r15=-----------
a11
r20=0
r21=0
r23=0
r24=0
r25=0
r26=0
r27=0
r28=0
r210=0
r212=0
r213=0
r214=0
r215=0
r216=0
r217=0
r218=0
r219=0
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r311=r339
r312=0
r313=r340
r314=0
r315=0
r317=0
r318=0
r319=0
r320=0
r321=r310
r322=r339
r323=0
r324=r340
r325=0
r327=r316
r328=0
r329=0
r330=0
r331=r326
r332=0
r333=0
r334=0
r335=0
r336=0
r337=0
r338=r310
r341=r346
r342=0
r343=r316
r344=0
r345=0
r347=r326
r348=0
r349=0
r350=0
r351=0
r352=r346
r353=0
r354=0
r355=0
- b33*n3
m3=-----------
a11
1
- ---*b33*n2
2
m2=---------------
a11
1
- ---*b33*n1
2
m1=---------------
a11
1 2
- ---*b33
4
c33=-------------
a11
c23=0
c22=0
c13=0
c12=0
b32=0
b31=0
b11=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:
r310, r316, r340, r339, r326, r346, r29, n3, b33, n1,
n2, a11
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.
{{r346,r316,r340,r339,r326,r310},r29,n1,a11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2
HAM=(u1 *a11 + u1*a11*n1 + u2 *a11 + u2*a11*n2 + u3*v3*a11*b33 + u3*a11*n3
1 1 1 2 2
- ---*v1*b33*n1 - ---*v2*b33*n2 - ---*v3 *b33 - v3*b33*n3)/a11
2 2 4
has apart from the Hamiltonian and Casimirs the following 7 first integrals:
2 2 1
FI= - u1*a11*n1 - u2*a11*n2 + u3 *a11 - u3*a11*n3 - ---*v1*b33*n1
2
1
- ---*v2*b33*n2
2
{HAM,FI} = 0
2
FI=u1 *v1 + u1*u2*v2 + u1*u3*v3
2
{HAM,FI} = ( - 2*u1*u2*u3*v1*a11 + u1*u2*v1*v3*a11*b33 + u1*u2*v1*a11*n3
1 2
+ u1*u3*v1*v2*a11*b33 - u1*u3*v1*a11*n2 - ---*u1*v1*v2*v3*b33
2
1 2 2
- u1*v1*v2*b33*n3 + ---*u1*v1*v3*b33*n2 - 2*u2 *u3*v2*a11
2
2 2 2 2
+ u2 *v2*v3*a11*b33 + u2 *v2*a11*n3 - 2*u2*u3 *v3*a11
2 2
+ u2*u3*v2 *a11*b33 - u2*u3*v2*a11*n2 + u2*u3*v3 *a11*b33
1 2 2 2
+ u2*u3*v3*a11*n3 - ---*u2*v2 *v3*b33 - u2*v2 *b33*n3
2
1 2 2
+ ---*u2*v2*v3*b33*n2 + u3 *v2*v3*a11*b33 - u3 *v3*a11*n2
2
1 2 2 1 2
- ---*u3*v2*v3 *b33 - u3*v2*v3*b33*n3 + ---*u3*v3 *b33*n2)/a11
2 2
2
FI=u1*u2*v1 + u2 *v2 + u2*u3*v3
2 2 2 2
{HAM,FI} = (2*u1 *u3*v1*a11 - u1 *v1*v3*a11*b33 - u1 *v1*a11*n3
2
+ 2*u1*u2*u3*v2*a11 - u1*u2*v2*v3*a11*b33 - u1*u2*v2*a11*n3
2 2 2
+ 2*u1*u3 *v3*a11 - u1*u3*v1 *a11*b33 + u1*u3*v1*a11*n1
2 1 2 2
- u1*u3*v3 *a11*b33 - u1*u3*v3*a11*n3 + ---*u1*v1 *v3*b33
2
2 1
+ u1*v1 *b33*n3 - ---*u1*v1*v3*b33*n1 - u2*u3*v1*v2*a11*b33
2
1 2
+ u2*u3*v2*a11*n1 + ---*u2*v1*v2*v3*b33 + u2*v1*v2*b33*n3
2
1 2 2
- ---*u2*v2*v3*b33*n1 - u3 *v1*v3*a11*b33 + u3 *v3*a11*n1
2
1 2 2 1 2
+ ---*u3*v1*v3 *b33 + u3*v1*v3*b33*n3 - ---*u3*v3 *b33*n1)/a11
2 2
2
FI=u1*v1*v2 + u2*v2 + u3*v2*v3
2 2 2
{HAM,FI} = 2*u1 *v1*v3*a11 + 2*u1*u2*v2*v3*a11 + 2*u1*u3*v3 *a11 - u1*v1 *v3*b33
2
- u1*v1 *n3 + u1*v1*v3*n1 - u2*v1*v2*v3*b33 - u2*v1*v2*n3
2 2
+ u2*v2*v3*n1 - u3*v1*v3 *b33 - u3*v1*v3*n3 + u3*v3 *n1
2
FI=u1*v1 + u2*v1*v2 + u3*v1*v3
{HAM,FI} = - 2*u1*u2*v1*v3*a11 + u1*v1*v2*v3*b33 + u1*v1*v2*n3 - u1*v1*v3*n2
2 2 2 2
- 2*u2 *v2*v3*a11 - 2*u2*u3*v3 *a11 + u2*v2 *v3*b33 + u2*v2 *n3
2 2
- u2*v2*v3*n2 + u3*v2*v3 *b33 + u3*v2*v3*n3 - u3*v3 *n2
2
FI=u1*u3*v1 + u2*u3*v2 + u3 *v3
2
{HAM,FI} = (u1 *v1*a11*n2 - u1*u2*v1*a11*n1 + u1*u2*v2*a11*n2 + u1*u3*v3*a11*n2
1 2 1 2
- ---*u1*v1 *b33*n2 + ---*u1*v1*v2*b33*n1 - u2 *v2*a11*n1
2 2
1 1 2
- u2*u3*v3*a11*n1 - ---*u2*v1*v2*b33*n2 + ---*u2*v2 *b33*n1
2 2
1 1
- ---*u3*v1*v3*b33*n2 + ---*u3*v2*v3*b33*n1)/a11
2 2
2
FI=u1*v1*v3 + u2*v2*v3 + u3*v3
2 2 2
{HAM,FI} = - 2*u1 *v1*v2*a11 + 2*u1*u2*v1 *a11 - 2*u1*u2*v2 *a11
2 2
- 2*u1*u3*v2*v3*a11 + u1*v1 *n2 - u1*v1*v2*n1 + 2*u2 *v1*v2*a11
2
+ 2*u2*u3*v1*v3*a11 + u2*v1*v2*n2 - u2*v2 *n1 + u3*v1*v3*n2
- u3*v2*v3*n1
And again in machine readable form:
HAM=(u1**2*a11**2 + u1*a11*n1 + u2**2*a11**2 + u2*a11*n2 + u3*v3*a11*b33 + u3*
a11*n3 - 1/2*v1*b33*n1 - 1/2*v2*b33*n2 - 1/4*v3**2*b33**2 - v3*b33*n3)/a11$
FI= - u1*a11*n1 - u2*a11*n2 + u3**2*a11**2 - u3*a11*n3 - 1/2*v1*b33*n1 - 1/2*v2*
b33*n2$
FI=u1**2*v1 + u1*u2*v2 + u1*u3*v3$
FI=u1*u2*v1 + u2**2*v2 + u2*u3*v3$
FI=u1*v1*v2 + u2*v2**2 + u3*v2*v3$
FI=u1*v1**2 + u2*v1*v2 + u3*v1*v3$
FI=u1*u3*v1 + u2*u3*v2 + u3**2*v3$
FI=u1*v1*v3 + u2*v2*v3 + u3*v3**2$