Solution 9 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
1
---*n3*r14
2
r00=------------
a23
1 1
---*b33*r14 + ---*n3*r210
2 2
r10=---------------------------
a23
r11=0
r12=0
1 2 1 2 1 2
---*a23 *a33*r14 + ---*a23*n3 *r333 - ---*a33*n3 *r354
2 4 8
r13=--------------------------------------------------------
3
a23
r15=0
1 1
---*b33*r210 + ---*n3*r320
2 2
r20=----------------------------
a23
r21=0
1
---*n3*r325
2
r22=-------------
a23
r23=0
r24=0
1
---*n3*r325
2
r25=-------------
a23
1 2 1 1
---*a23 *a33*r210 + ---*a23*b33*n3*r333 - ---*a33*b33*n3*r354
2 2 4
r26=---------------------------------------------------------------
3
a23
r27=0
r28=0
1 2 1 1 2
---*a23 *n3*r354 + ---*a23*a33*n3*r333 - ---*a33 *n3*r354
2 2 4
r29=-----------------------------------------------------------
3
a23
r211=0
r212=0
1
a23*n3*r333 - ---*a33*n3*r354
2
r213=-------------------------------
2
a23
1
---*n3*r354
2
r214=-------------
a23
r215=0
r216=0
r217=0
r218=0
r219=0
1
---*n3*r354
2
r220=-------------
a23
1
---*b33*r320
2
r30=--------------
a23
r31=0
1
---*b33*r325
2
r32=--------------
a23
r33=0
r34=0
r35=0
r36=0
1
---*b33*r325
2
r37=--------------
a23
r38=0
r39=0
1 2 1 2 1 2
---*a23 *a33*r320 + ---*a23*b33 *r333 - ---*a33*b33 *r354
2 4 8
r310=-----------------------------------------------------------
3
a23
r311=0
1
---*a33*r325
2
r312=--------------
a23
r313=0
r314=0
1
---*a33*r325
2
r315=--------------
a23
1 2 1 1 2
---*a23 *b33*r354 + ---*a23*a33*b33*r333 - ---*a33 *b33*r354
2 2 4
r316=--------------------------------------------------------------
3
a23
r317=0
r318=0
1 2 1 2 1 3
---*a23 *a33*r354 + ---*a23*a33 *r333 - ---*a33 *r354
2 4 8
r319=-------------------------------------------------------
3
a23
r321=0
r322=r325
r323=0
r324=0
1
a23*b33*r333 - ---*a33*b33*r354
2
r326=---------------------------------
2
a23
r327=0
r328=0
2 1 2
a23 *r354 + a23*a33*r333 - ---*a33 *r354
2
r329=------------------------------------------
2
a23
1
---*b33*r354
2
r330=--------------
a23
r331=0
r332=0
r334=r354
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
1
---*b33*r354
2
r350=--------------
a23
r351=0
r352=0
1
---*a33*r354
2
r353=--------------
a23
r355=0
m2=0
m1=0
c22=c11
b32=0
b31=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:
r14, r210, r320, r325, r333, r354, m3, n3, c11, a33,
b33, 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.
{a23,b33}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v3*b33 + u3*n3 + v1 *c11 + v2 *c11 + v3*m3
has apart from the Hamiltonian and Casimirs the following 6 first integrals:
2 3 2 2 2 2 2
FI=8*u1 *u2*u3*a23 + 4*u1 *u3 *a23 *a33 + 4*u1 *u3*v3*a23 *b33
2 2 3 3 2 2 2 2
+ 4*u1 *u3*a23 *n3 + 8*u2 *u3*a23 + 4*u2 *u3*v3*a23 *b33 + 4*u2 *u3*a23 *n3
3 3 2 2
+ u2*u3 *(8*a23 - 4*a23*a33 ) - 4*u2*u3 *v3*a23*a33*b33
2 4 2 3
- 4*u2*u3 *a23*a33*n3 + u3 *(4*a23 *a33 - a33 )
3 2 2 3 2 2
+ u3 *v3*(4*a23 *b33 - 2*a33 *b33) + u3 *(4*a23 *n3 - 2*a33 *n3)
2 2 2 2 2 2
- u3 *v3 *a33*b33 - 2*u3 *v3*a33*b33*n3 - u3 *a33*n3
= a product of the elements of: {u3,
2*u2*a23 + u3*a33 + v3*b33 + n3,
2 2 2 2 2 2 2
4*u1 *a23 + 4*u2 *a23 - 2*u2*u3*a23*a33 + u3 *(4*a23 - a33 )
- u3*v3*a33*b33 - u3*a33*n3}
{HAM,FI} = 0
2 2 2 3 2 2
FI=4*u2 *u3 *a23 + 4*u2*u3 *a23*a33 + 4*u2*u3 *v3*a23*b33 + 4*u2*u3 *a23*n3
4 2 3 3 2 2 2
+ u3 *a33 + 2*u3 *v3*a33*b33 + 2*u3 *a33*n3 + u3 *v3 *b33
2 2 2
+ 2*u3 *v3*b33*n3 + u3 *n3
= a product of the elements of: {u3,
u3,
2*u2*a23 + u3*a33 + v3*b33 + n3,
2*u2*a23 + u3*a33 + v3*b33 + n3}
{HAM,FI} = 0
2 2 2 2 2 2 2
FI=2*u2*u3*v1 *a23 + 2*u2*u3*v2 *a23 + u3 *v1 *a33 + u3 *v2 *a33 + u3*v1 *v3*b33
2 2 2
+ u3*v1 *n3 + u3*v2 *v3*b33 + u3*v2 *n3
2 2
= a product of the elements of: {v1 + v2 ,
u3,
2*u2*a23 + u3*a33 + v3*b33 + n3}
{HAM,FI} = 0
2 2 2 3 2
FI=2*u2*u3*v3 *a23 + u3 *v3 *a33 + u3*v3 *b33 + u3*v3 *n3
= a product of the elements of: {v3,
v3,
u3,
2*u2*a23 + u3*a33 + v3*b33 + n3}
{HAM,FI} = 0
2 2
FI=2*u2*u3*v3*a23 + u3 *v3*a33 + u3*v3 *b33 + u3*v3*n3
= a product of the elements of: {v3,
u3,
2*u2*a23 + u3*a33 + v3*b33 + n3}
{HAM,FI} = 0
2
FI=2*u2*u3*a23 + u3 *a33 + u3*v3*b33 + u3*n3
= a product of the elements of: {u3,2*u2*a23 + u3*a33 + v3*b33 + n3}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v3*b33 + u3*n3 + v1**2*c11 + v2**2*c11 + v3*m3$
FI=8*u1**2*u2*u3*a23**3 + 4*u1**2*u3**2*a23**2*a33 + 4*u1**2*u3*v3*a23**2*b33 +
4*u1**2*u3*a23**2*n3 + 8*u2**3*u3*a23**3 + 4*u2**2*u3*v3*a23**2*b33 + 4*u2**2*u3
*a23**2*n3 + u2*u3**3*(8*a23**3 - 4*a23*a33**2) - 4*u2*u3**2*v3*a23*a33*b33 - 4*
u2*u3**2*a23*a33*n3 + u3**4*(4*a23**2*a33 - a33**3) + u3**3*v3*(4*a23**2*b33 - 2
*a33**2*b33) + u3**3*(4*a23**2*n3 - 2*a33**2*n3) - u3**2*v3**2*a33*b33**2 - 2*u3
**2*v3*a33*b33*n3 - u3**2*a33*n3**2$
FI=4*u2**2*u3**2*a23**2 + 4*u2*u3**3*a23*a33 + 4*u2*u3**2*v3*a23*b33 + 4*u2*u3**
2*a23*n3 + u3**4*a33**2 + 2*u3**3*v3*a33*b33 + 2*u3**3*a33*n3 + u3**2*v3**2*b33
**2 + 2*u3**2*v3*b33*n3 + u3**2*n3**2$
FI=2*u2*u3*v1**2*a23 + 2*u2*u3*v2**2*a23 + u3**2*v1**2*a33 + u3**2*v2**2*a33 +
u3*v1**2*v3*b33 + u3*v1**2*n3 + u3*v2**2*v3*b33 + u3*v2**2*n3$
FI=2*u2*u3*v3**2*a23 + u3**2*v3**2*a33 + u3*v3**3*b33 + u3*v3**2*n3$
FI=2*u2*u3*v3*a23 + u3**2*v3*a33 + u3*v3**2*b33 + u3*v3*n3$
FI=2*u2*u3*a23 + u3**2*a33 + u3*v3*b33 + u3*n3$