Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
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Equations
The following unsolved equations remain:
2 2
0=a12 + a13
Expressions
The solution is given through the following expressions:
4 2 2 2 2 4 2
a12 *c23 *r464 + a12 *a13 *c23 *r464 - a13 *c23 *r464
r40=-------------------------------------------------------
2 4
a12 *a13
2
2*a12*c23 *r464
r41=-----------------
3
a13
2
c23 *r464
r42=-----------
2
a12
r43=0
4 4 2 2
r45=(8*a12 *c23*r427 - 5*a12 *c23*r483 + 16*a12 *a13 *c23*r427
17 2 2 4 10 4
- ----*a12 *a13 *c23*r483 + 24*a13 *c23*r427 - ----*a13 *c23*r483)/(
3 3
3 2 13 4
a12 *a13 - ----*a12*a13 )
3
2 64 2 19 2
- a12 *c23*r483 + ----*a13 *c23*r427 - ----*a13 *c23*r483
3 3
r46=------------------------------------------------------------
2 13 3
a12 *a13 - ----*a13
3
4 4 16 2 2
r47=(2*a12 *c23*r427 - a12 *c23*r483 + ----*a12 *a13 *c23*r427
3
11 2 2 26 4 3 2 13 4
- ----*a12 *a13 *c23*r483 + ----*a13 *c23*r427)/(a12 *a13 - ----*a12*a13
3 3 3
)
2 32 2 2
- a12 *c23*r483 + ----*a13 *c23*r427 - a13 *c23*r483
3
r48=-------------------------------------------------------
2 13 3
a12 *a13 - ----*a13
3
2
c23 *r464
r49=-----------
2
a12
2
- 3*a13*c23 *r464
r410=--------------------
3
a12
2
2*c23 *r464
r411=-------------
2
a13
4 4 22 2 2
r412=( - 2*a12 *c23*r427 + a12 *c23*r483 - ----*a12 *a13 *c23*r427
3
8 2 2 4 1 4
+ ---*a12 *a13 *c23*r483 + 8*a13 *c23*r427 + ---*a13 *c23*r483)/(
3 3
3 2 13 4
a12 *a13 - ----*a12*a13 )
3
4 4 22 2 2
r413=(2*a12 *c23*r427 - a12 *c23*r483 + ----*a12 *a13 *c23*r427
3
8 2 2 4 1 4 4
- ---*a12 *a13 *c23*r483 - 8*a13 *c23*r427 - ---*a13 *c23*r483)/(a12 *a13
3 3
13 2 3
- ----*a12 *a13 )
3
4 2 2 2 2 1 4 2
a12 *c23 *r464 + a12 *a13 *c23 *r464 + ---*a13 *c23 *r464
4
r414=-----------------------------------------------------------
4 2
a12 *a13
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
2 34 2 4 2
2*a12 *r427 - ----*a13 *r427 + ---*a13 *r483
3 3
r425=----------------------------------------------
2 13 2
a12 - ----*a13
3
8 4
- ---*a12*a13*r427 + ---*a12*a13*r483
3 3
r426=----------------------------------------
2 13 2
a12 - ----*a13
3
- 2*c23*r464
r428=---------------
a12
2*a13*c23*r464
r429=----------------
2
a12
4 1 4 11 2 2 4 2 2
r430=(a12 *r427 - ---*a12 *r483 + ----*a12 *a13 *r427 - ---*a12 *a13 *r483
2 3 3
4 1 4 2 2 13 4
- 4*a13 *r427 - ---*a13 *r483)/(a12 *a13 - ----*a13 )
6 3
r431=0
r432=0
r433=0
1
r434=---*r464
2
r435=0
r439=0
r442=0
r444=0
8 4
---*a12*a13*r427 - ---*a12*a13*r483
3 3
r445=-------------------------------------
2 13 2
a12 - ----*a13
3
- 2*a13*c23*r464
r448=-------------------
2
a12
r450=0
r451=0
r453=0
r454=0
2 2 4 2
a12 *r427 - 7*a13 *r427 + ---*a13 *r483
3
r455=-----------------------------------------
2 13 2
a12 - ----*a13
3
- 2*c23*r464
r458=---------------
a12
4 1 4 11 2 2 4 2 2
r460=(a12 *r427 - ---*a12 *r483 + ----*a12 *a13 *r427 - ---*a12 *a13 *r483
2 3 3
4 1 4 2 2 13 4
- 4*a13 *r427 - ---*a13 *r483)/(a12 *a13 - ----*a13 )
6 3
r461=0
r463=0
r465=0
r467=0
r468=0
1
r469=---*r464
2
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r478=0
r479=0
2 2
2*a12 *c23*r464 - 2*a13 *c23*r464
r480=-----------------------------------
2
a12*a13
2*a13*c23*r464
r481=----------------
2
a12
3 3 22 2 11 2
2*a12 *r427 - a12 *r483 + ----*a12*a13 *r427 - ----*a12*a13 *r483
3 3
r484=-------------------------------------------------------------------
2 13 3
a12 *a13 - ----*a13
3
2
a13 *c23*r464
r485=---------------
3
a12
r486=0
r487=0
r488=0
8 2 4 2
- ---*a12*a13 *r427 + ---*a12*a13 *r483
3 3
r489=------------------------------------------
2 13 2
a12 *c23 - ----*a13 *c23
3
2
2*a12 *c23*r464
r490=-----------------
3
a13
8*a12*a13*r427 - 4*a12*a13*r483
r493=---------------------------------
2 13 2
a12 - ----*a13
3
- a13*c23*r464
r495=-----------------
2
a12
r496=0
r498=0
8 2 4 2
- ---*a12 *a13*r427 + ---*a12 *a13*r483
3 3
r499=------------------------------------------
2 13 2
a12 *c23 - ----*a13 *c23
3
r4100=0
r4102=0
8 2 4 2
- ---*a12 *a13*r427 + ---*a12 *a13*r483
3 3
r4104=------------------------------------------
2 13 2
a12 *c23 - ----*a13 *c23
3
2 1 2 2 5 2
2*a12 *r427 - ---*a12 *r483 - 6*a13 *r427 + ---*a13 *r483
2 6
r4105=-----------------------------------------------------------
2 13 2
a12 - ----*a13
3
3 3 4 2 2 2
4*a12 *r427 - 2*a12 *r483 + ---*a12*a13 *r427 - ---*a12*a13 *r483
3 3
r4106=-------------------------------------------------------------------
2 13 3
a12 *a13 - ----*a13
3
1 2 16 2 1 2
---*a12 *r483 - ----*a13 *r427 + ---*a13 *r483
2 3 2
r4107=------------------------------------------------
2 13 2
a12 - ----*a13
3
- 4*c23*r464
r4108=---------------
a12
- 2*c23*r464
r4109=---------------
a13
4 3 4 2 2 3 2 2
r4110=(3*a12 *r427 - ---*a12 *r483 + 5*a12 *a13 *r427 - ---*a12 *a13 *r483
2 2
14 4 4 2 2 13 4
- ----*a13 *r427 - 2*a13 *r483)/(a12 *a13 - ----*a13 )
3 3
r4111=0
r4112=0
r4113=0
r4115=0
r4117=0
- 2*a12*r464
r4118=---------------
a13
2 2
- a12 *r464 + a13 *r464
r4119=--------------------------
2
a13
r4120=0
r4121=0
r4122=0
8 2 4 2
- ---*a12*a13 *r427 + ---*a12*a13 *r483
3 3
r4123=------------------------------------------
2 13 2
a12 *c23 - ----*a13 *c23
3
8 2 4 2
- ---*a12 *a13*r427 + ---*a12 *a13*r483
3 3
r4124=------------------------------------------
2 13 2
a12 *c23 - ----*a13 *c23
3
1
r4125=---*r464
2
a13*c23
c33=---------
a12
- a13*c23
c22=------------
a12
c13=0
c12=0
b33=0
b31=0
b21=0
b13=0
b11=0
8 2 4 2
- ---*a12*a13 *r427 + ---*a12*a13 *r483
3 3
r4103=------------------------------------------
2 13 2
a12 *c23 - ----*a13 *c23
3
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:
r427, r483, r464, c23, a12, a13
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.
{c23,a13,a12}
Relevance for the application:
Modulo the following equation:
2 2
0=a12 + a13
the system of equations related to the Hamiltonian HAM:
2 2
HAM=(2*u1*u2*a12 + 2*u1*u3*a12*a13 - v2 *a13*c23 + 2*v2*v3*a12*c23
2
+ v3 *a13*c23)/a12
has apart from the Hamiltonian and Casimirs the following 3 first integrals:
1 4 4 4 2 2 6 2 4 4
FI=---*u1 *a12 *a13 + u1 *u2 *( - a12 *a13 + a12 *a13 )
2
2 5 3 2 4 3
- 2*u1 *u2*u3*a12 *a13 - 2*u1 *v1*v2*a12 *a13 *c23
2 3 4 2 2 5
- 4*u1 *v1*v3*a12 *a13 *c23 - u1*u2*v1 *a12 *a13 *c23
2 6 2 6
+ 2*u1*u2*v3 *a12 *a13*c23 + u1*u3*v1 *a12*a13 *c23
2 5
+ 2*u1*u3*v2*v3*a12 *a13 *c23
2 5 2 3 4 1 4 4 4
+ u1*u3*v3 *(2*a12 *a13 *c23 - 2*a12 *a13 *c23) + ---*u2 *a12 *a13
2
2 2 4 4 2 3 4
+ u2 *u3 *a12 *a13 - 2*u2 *v1*v3*a12 *a13 *c23
2 5 1 4 4 4
- 2*u2*u3*v1*v3*a12 *a13 *c23 + ---*u3 *a12 *a13
2
2 2 5 2 3 4
+ 2*u3 *v1*v2*a12 *a13 *c23 - 2*u3 *v1*v3*a12 *a13 *c23
4 4 2 2 2 4 2 1 6 2
+ v1 *(a12 *a13 *c23 + a12 *a13 *c23 + ---*a13 *c23 )
4
2 2 4 2 2 2 5 2
+ 2*v1 *v2 *a12 *a13 *c23 - 3*v1 *v2*v3*a12*a13 *c23
2 2 2 4 2 2 2 2 4 2 3 5 2
+ v1 *v3 *a12 *a13 *c23 + v2 *v3 *a12 *a13 *c23 + 2*v2*v3 *a12 *a13*c23
4 6 2 4 2 2 2 4 2
+ v3 *(a12 *c23 + a12 *a13 *c23 - a12 *a13 *c23 )
{HAM,FI} = too large to simplify
4 3 4 3 4 3 3 4
FI=---*u1 *u2*a12 *a13 + ---*u1 *u3*a12 *a13
3 3
2 2 3 6 3 4 2 2 4
+ u1 *v1 *( - ---*a12 *c23 - ---*a12 *a13 *c23 - 2*a12 *a13 *c23)
2 2
2 2 1 4 2 1 2 4
+ u1 *v2 *(---*a12 *a13 *c23 + ---*a12 *a13 *c23)
2 2
2 5 2 3 3
+ u1 *v2*v3*( - 2*a12 *a13*c23 - ---*a12 *a13 *c23)
3
2 2 1 4 2 5 2 4 4 3 4 3
+ u1 *v3 *( - ---*a12 *a13 *c23 + ---*a12 *a13 *c23) + ---*u1*u2 *a12 *a13
2 6 3
4 2 3 4 4 2 4 3
+ ---*u1*u2 *u3*a12 *a13 + ---*u1*u2*u3 *a12 *a13
3 3
3 3 4 3 3 4
- 4*u1*u2*v1*v3*a12 *a13 *c23 + ---*u1*u3 *a12 *a13
3
5 11 3 3
+ u1*u3*v1*v2*( - a12 *a13*c23 - ----*a12 *a13 *c23)
3
4 2 13 2 4
+ u1*u3*v1*v3*(a12 *a13 *c23 - ----*a12 *a13 *c23)
3
2 2 1 6 4 4 2 1 2 4
+ u2 *v1 *( - ---*a12 *c23 - ---*a12 *a13 *c23 - ---*a12 *a13 *c23)
2 3 6
4 2 2 2 4 4 2 3 3
+ ---*u2 *v3 *a12 *a13 *c23 - ---*u2*u3*v3 *a12 *a13 *c23
3 3
2 2 1 6 4 4 2 1 2 4
+ u3 *v1 *( - ---*a12 *c23 - ---*a12 *a13 *c23 - ---*a12 *a13 *c23)
2 3 6
4 2 3 3 4 2 2 2 4
+ ---*u3 *v2*v3*a12 *a13 *c23 + ---*u3 *v3 *a12 *a13 *c23
3 3
3 4 2 8 2 3 2 1 5 2
+ v1 *v2*( - a12 *a13*c23 - ---*a12 *a13 *c23 - ---*a13 *c23 )
3 3
3 5 2 8 3 2 2 1 4 2
+ v1 *v3*(a12 *c23 + ---*a12 *a13 *c23 + ---*a12*a13 *c23 )
3 3
3 4 2 2 3 2
+ v1*v2 *( - a12 *a13*c23 - a12 *a13 *c23 )
2 5 2 11 3 2 2
+ v1*v2 *v3*( - a12 *c23 - ----*a12 *a13 *c23 )
3
2 4 2 19 2 3 2
+ v1*v2*v3 *( - a12 *a13*c23 - ----*a12 *a13 *c23 )
3
3 5 2 17 3 2 2 10 4 2
+ v1*v3 *( - 5*a12 *c23 - ----*a12 *a13 *c23 - ----*a12*a13 *c23 )
3 3
{HAM,FI} = too large to simplify
8 3 4 3 8 3 3 4
FI= - ---*u1 *u2*a12 *a13 - ---*u1 *u3*a12 *a13
3 3
2 2 6 4 2 14 2 4
+ u1 *v1 *(3*a12 *c23 + 5*a12 *a13 *c23 - ----*a12 *a13 *c23)
3
16 2 2 2 4
- ----*u1 *v2 *a12 *a13 *c23
3
2 5 4 3 3
+ u1 *v2*v3*(4*a12 *a13*c23 + ---*a12 *a13 *c23)
3
2 2 4 2 2 4 8 3 4 3
+ u1 *v3 *(2*a12 *a13 *c23 - 6*a12 *a13 *c23) - ---*u1*u2 *a12 *a13
3
8 2 3 4 8 2 4 3
- ---*u1*u2 *u3*a12 *a13 - ---*u1*u2*u3 *a12 *a13
3 3
3 3 8 3 3 4
+ 8*u1*u2*v1*v3*a12 *a13 *c23 - ---*u1*u3 *a12 *a13
3
5 22 3 3
+ u1*u3*v1*v2*(2*a12 *a13*c23 + ----*a12 *a13 *c23)
3
2 2 6 11 4 2 2 4
+ u2 *v1 *(a12 *c23 + ----*a12 *a13 *c23 - 4*a12 *a13 *c23)
3
2 2 4 2 2 4 8 2 3 3
+ u2 *v3 *(a12 *a13 *c23 - 7*a12 *a13 *c23) + ---*u2*u3*v3 *a12 *a13 *c23
3
2 2 6 11 4 2 2 4
+ u3 *v1 *(a12 *c23 + ----*a12 *a13 *c23 - 4*a12 *a13 *c23)
3
2 2 4 2 13 2 4 8 2 3 3
+ u3 *v2 *(a12 *a13 *c23 - ----*a12 *a13 *c23) - ---*u3 *v2*v3*a12 *a13 *c23
3 3
2 2 4 2 34 2 4
+ u3 *v3 *(2*a12 *a13 *c23 - ----*a12 *a13 *c23)
3
3 4 2 22 2 3 2 5 2
+ v1 *v2*(2*a12 *a13*c23 + ----*a12 *a13 *c23 - 8*a13 *c23 )
3
3 5 2 22 3 2 2 4 2
+ v1 *v3*( - 2*a12 *c23 - ----*a12 *a13 *c23 + 8*a12*a13 *c23 )
3
32 3 2 3 2
+ ----*v1*v2 *a12 *a13 *c23
3
2 5 2 16 3 2 2 26 4 2
+ v1*v2 *v3*(2*a12 *c23 + ----*a12 *a13 *c23 + ----*a12*a13 *c23 )
3 3
64 2 2 3 2
+ ----*v1*v2*v3 *a12 *a13 *c23
3
3 5 2 3 2 2 4 2
+ v1*v3 *(8*a12 *c23 + 16*a12 *a13 *c23 + 24*a12*a13 *c23 )
{HAM,FI} = too large to simplify
And again in machine readable form:
HAM=(2*u1*u2*a12**2 + 2*u1*u3*a12*a13 - v2**2*a13*c23 + 2*v2*v3*a12*c23 + v3**2*
a13*c23)/a12$
FI=1/2*u1**4*a12**4*a13**4 + u1**2*u2**2*( - a12**6*a13**2 + a12**4*a13**4) - 2*
u1**2*u2*u3*a12**5*a13**3 - 2*u1**2*v1*v2*a12**4*a13**3*c23 - 4*u1**2*v1*v3*a12
**3*a13**4*c23 - u1*u2*v1**2*a12**2*a13**5*c23 + 2*u1*u2*v3**2*a12**6*a13*c23 +
u1*u3*v1**2*a12*a13**6*c23 + 2*u1*u3*v2*v3*a12**2*a13**5*c23 + u1*u3*v3**2*(2*
a12**5*a13**2*c23 - 2*a12**3*a13**4*c23) + 1/2*u2**4*a12**4*a13**4 + u2**2*u3**2
*a12**4*a13**4 - 2*u2**2*v1*v3*a12**3*a13**4*c23 - 2*u2*u3*v1*v3*a12**2*a13**5*
c23 + 1/2*u3**4*a12**4*a13**4 + 2*u3**2*v1*v2*a12**2*a13**5*c23 - 2*u3**2*v1*v3*
a12**3*a13**4*c23 + v1**4*(a12**4*a13**2*c23**2 + a12**2*a13**4*c23**2 + 1/4*a13
**6*c23**2) + 2*v1**2*v2**2*a12**4*a13**2*c23**2 - 3*v1**2*v2*v3*a12*a13**5*c23
**2 + v1**2*v3**2*a12**2*a13**4*c23**2 + v2**2*v3**2*a12**2*a13**4*c23**2 + 2*v2
*v3**3*a12**5*a13*c23**2 + v3**4*(a12**6*c23**2 + a12**4*a13**2*c23**2 - a12**2*
a13**4*c23**2)$
FI=4/3*u1**3*u2*a12**4*a13**3 + 4/3*u1**3*u3*a12**3*a13**4 + u1**2*v1**2*( - 3/2
*a12**6*c23 - 3/2*a12**4*a13**2*c23 - 2*a12**2*a13**4*c23) + u1**2*v2**2*(1/2*
a12**4*a13**2*c23 + 1/2*a12**2*a13**4*c23) + u1**2*v2*v3*( - 2*a12**5*a13*c23 -
2/3*a12**3*a13**3*c23) + u1**2*v3**2*( - 1/2*a12**4*a13**2*c23 + 5/6*a12**2*a13
**4*c23) + 4/3*u1*u2**3*a12**4*a13**3 + 4/3*u1*u2**2*u3*a12**3*a13**4 + 4/3*u1*
u2*u3**2*a12**4*a13**3 - 4*u1*u2*v1*v3*a12**3*a13**3*c23 + 4/3*u1*u3**3*a12**3*
a13**4 + u1*u3*v1*v2*( - a12**5*a13*c23 - 11/3*a12**3*a13**3*c23) + u1*u3*v1*v3*
(a12**4*a13**2*c23 - 13/3*a12**2*a13**4*c23) + u2**2*v1**2*( - 1/2*a12**6*c23 -
4/3*a12**4*a13**2*c23 - 1/6*a12**2*a13**4*c23) + 4/3*u2**2*v3**2*a12**2*a13**4*
c23 - 4/3*u2*u3*v3**2*a12**3*a13**3*c23 + u3**2*v1**2*( - 1/2*a12**6*c23 - 4/3*
a12**4*a13**2*c23 - 1/6*a12**2*a13**4*c23) + 4/3*u3**2*v2*v3*a12**3*a13**3*c23 +
4/3*u3**2*v3**2*a12**2*a13**4*c23 + v1**3*v2*( - a12**4*a13*c23**2 - 8/3*a12**2
*a13**3*c23**2 - 1/3*a13**5*c23**2) + v1**3*v3*(a12**5*c23**2 + 8/3*a12**3*a13**
2*c23**2 + 1/3*a12*a13**4*c23**2) + v1*v2**3*( - a12**4*a13*c23**2 - a12**2*a13
**3*c23**2) + v1*v2**2*v3*( - a12**5*c23**2 - 11/3*a12**3*a13**2*c23**2) + v1*v2
*v3**2*( - a12**4*a13*c23**2 - 19/3*a12**2*a13**3*c23**2) + v1*v3**3*( - 5*a12**
5*c23**2 - 17/3*a12**3*a13**2*c23**2 - 10/3*a12*a13**4*c23**2)$
FI= - 8/3*u1**3*u2*a12**4*a13**3 - 8/3*u1**3*u3*a12**3*a13**4 + u1**2*v1**2*(3*
a12**6*c23 + 5*a12**4*a13**2*c23 - 14/3*a12**2*a13**4*c23) - 16/3*u1**2*v2**2*
a12**2*a13**4*c23 + u1**2*v2*v3*(4*a12**5*a13*c23 + 4/3*a12**3*a13**3*c23) + u1
**2*v3**2*(2*a12**4*a13**2*c23 - 6*a12**2*a13**4*c23) - 8/3*u1*u2**3*a12**4*a13
**3 - 8/3*u1*u2**2*u3*a12**3*a13**4 - 8/3*u1*u2*u3**2*a12**4*a13**3 + 8*u1*u2*v1
*v3*a12**3*a13**3*c23 - 8/3*u1*u3**3*a12**3*a13**4 + u1*u3*v1*v2*(2*a12**5*a13*
c23 + 22/3*a12**3*a13**3*c23) + u2**2*v1**2*(a12**6*c23 + 11/3*a12**4*a13**2*c23
- 4*a12**2*a13**4*c23) + u2**2*v3**2*(a12**4*a13**2*c23 - 7*a12**2*a13**4*c23)
+ 8/3*u2*u3*v3**2*a12**3*a13**3*c23 + u3**2*v1**2*(a12**6*c23 + 11/3*a12**4*a13
**2*c23 - 4*a12**2*a13**4*c23) + u3**2*v2**2*(a12**4*a13**2*c23 - 13/3*a12**2*
a13**4*c23) - 8/3*u3**2*v2*v3*a12**3*a13**3*c23 + u3**2*v3**2*(2*a12**4*a13**2*
c23 - 34/3*a12**2*a13**4*c23) + v1**3*v2*(2*a12**4*a13*c23**2 + 22/3*a12**2*a13
**3*c23**2 - 8*a13**5*c23**2) + v1**3*v3*( - 2*a12**5*c23**2 - 22/3*a12**3*a13**
2*c23**2 + 8*a12*a13**4*c23**2) + 32/3*v1*v2**3*a12**2*a13**3*c23**2 + v1*v2**2*
v3*(2*a12**5*c23**2 + 16/3*a12**3*a13**2*c23**2 + 26/3*a12*a13**4*c23**2) + 64/3
*v1*v2*v3**2*a12**2*a13**3*c23**2 + v1*v3**3*(8*a12**5*c23**2 + 16*a12**3*a13**2
*c23**2 + 24*a12*a13**4*c23**2)$