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:
5 4 2 5 2 2 2 4 2
- ---*a12 *c23 *r464 - ---*a12 *a13 *c23 *r464 - a13 *c23 *r464
4 4
r40=------------------------------------------------------------------
2 4
a12 *a13
2 2 2 2
a12 *c23 *r464 - a13 *c23 *r464
r41=---------------------------------
3
a12*a13
2
- c23 *r464
r42=--------------
2
a13
r43=0
2
2*a12*c23*r427 + 2*c23 *r4103
r45=-------------------------------
2
a13
2 2
4*a12*c23 *r4103 - 2*a13 *c23*r427
r46=------------------------------------
3
a13
2
2*a12*c23*r427 - 2*c23 *r4103
r47=-------------------------------
2
a13
- 2*c23*r427
r48=---------------
a13
2 2 2 2
- 2*a12 *c23 *r464 - 3*a13 *c23 *r464
r49=----------------------------------------
4
a13
2
- 3*a12*c23 *r464
r410=--------------------
3
a13
2
2*c23 *r464
r411=-------------
2
a13
2 2
a12*c23 *r4103 - 2*a13 *c23*r427
r412=----------------------------------
2
a12*a13
2 2
a12*c23 *r4103 - 2*a13 *c23*r427
r413=----------------------------------
3
a13
3 2 2 2 2
---*a12 *c23 *r464 + a13 *c23 *r464
4
r414=-------------------------------------
4
a13
r415=0
r416=0
r417=0
r418=0
- 2*c23*r452
r419=---------------
a12
- 2*c23*r452
r420=---------------
a13
r421=0
r422=0
r423=0
r424=0
- a12*c23*r4103
r425=------------------
2
a13
c23*r4103
r426=-----------
a13
- 2*c23*r464
r428=---------------
a12
- 2*c23*r464
r429=---------------
a13
3 2 2 2
- ---*a12 *c23*r4103 + a12*a13 *r427 - a13 *c23*r4103
2
r430=--------------------------------------------------------
2
a12*a13
r431=r452
r432=0
r433=0
1
r434=---*r464
2
r435=0
r436=0
r437=0
r439=0
- 2*c23*r452
r440=---------------
a12
- 2*c23*r452
r441=---------------
a13
r442=0
r443=0
r444=0
r445=0
r446= - 2*r427
r447=0
r448=0
r449=0
r450=0
r451=0
r453=0
r454=0
2
- a12*c23*r4103 + a13 *r427
r455=------------------------------
2
a13
c23*r4103
r456=-----------
a13
2*a12*c23*r464
r458=----------------
2
a13
- 2*c23*r464
r459=---------------
a13
3 2 2 2
- ---*a12 *c23*r4103 + a12*a13 *r427 - a13 *c23*r4103
2
r460=--------------------------------------------------------
2
a12*a13
r461=r452
r462=0
r463=0
r465=0
r466=r452
r467=0
r468=0
1
r469=---*r464
2
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
- 2*c23*r452
r477=---------------
a12
- 2*c23*r452
r478=---------------
a13
r479=0
2 2
a12 *c23*r464 - a13 *c23*r464
r480=-------------------------------
2
a12*a13
- 2*c23*r464
r481=---------------
a13
2
2*a12*c23*r4103 - 2*a13 *r427
r483=-------------------------------
2
a13
- 2*c23*r4103
r484=----------------
a13
a12*c23*r464
r485=--------------
2
a13
r486=0
r487=0
r488=r452
r489=r4103
2 2
a12 *c23*r464 - a13 *c23*r464
r490=-------------------------------
3
a13
- 2*a12*c23*r464
r491=-------------------
2
a13
r492=0
- 2*c23*r4103
r493=----------------
a13
2
- 2*a12*c23*r4103 - 2*a13 *r427
r494=----------------------------------
2
a13
2
a12 *c23*r464
r495=---------------
3
a13
r496=0
r498=0
a12*r4103
r499=-----------
a13
r4100=0
r4101=0
r4102=r452
a12*r4103
r4104=-----------
a13
2
- a12*c23*r4103 + a13 *r427
r4105=------------------------------
2
a13
c23*r4103
r4106=-----------
a13
r4107=r427
- 2*c23*r464
r4108=---------------
a12
- 2*c23*r464
r4109=---------------
a13
1
- ---*a12*c23*r4103
2
r4110=----------------------
2
a13
r4111=r452
r4112=0
r4113=0
r4115=0
r4116=r452
r4117=0
- 2*a12*r464
r4118=---------------
a13
2 2
- a12 *r464 + a13 *r464
r4119=--------------------------
2
a13
r4120=0
r4121=0
r4122=r452
r4123=r4103
a12*r4103
r4124=-----------
a13
1
r4125=---*r464
2
- a12*c23
c33=------------
a13
a12*c23
c22=---------
a13
c13=0
c12=0
b33=0
b31=0
b21=0
b13=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:
r427, r452, r4103, 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*a13 + 2*u1*u3*a13 + v2 *a12*c23 + 2*v2*v3*a13*c23
2
- v3 *a12*c23)/a13
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
1 4 2 4 2 2 4 2 2 4
FI=---*u1 *a12 *a13 + u1 *u2 *( - a12 *a13 + a12 *a13 )
2
2 3 3 2 2 3
- 2*u1 *u2*u3*a12 *a13 - 2*u1 *v1*v2*a12 *a13 *c23
2 4 2 4
- 2*u1 *v1*v3*a12*a13 *c23 + u1*u2*v1 *a12 *a13*c23
3 2 2 4 2 3
- 2*u1*u2*v2*v3*a12 *a13 *c23 + u1*u2*v3 *(a12 *a13*c23 - a12 *a13 *c23)
2 3 2 2 3
+ u1*u3*v1 *a12 *a13 *c23 - 2*u1*u3*v2*v3*a12 *a13 *c23
2 3 2 4 1 4 2 4
+ u1*u3*v3 *(a12 *a13 *c23 - a12*a13 *c23) + ---*u2 *a12 *a13
2
2 2 2 4 2 2 3 2 3 2
+ u2 *u3 *a12 *a13 - 2*u2 *v1*v2*a12 *a13 *c23 + 2*u2 *v1*v3*a12 *a13 *c23
1 4 2 4 2 2 3 2 4
+ ---*u3 *a12 *a13 - 2*u3 *v1*v2*a12 *a13 *c23 - 2*u3 *v1*v3*a12*a13 *c23
2
4 3 4 2 2 2 2 2 2 2 2 2
+ v1 *(---*a12 *c23 + a12 *a13 *c23 ) + 2*v1 *v2 *a12 *a13 *c23
4
2 3 2 2 2 4 2 2 2 2
- 3*v1 *v2*v3*a12 *a13*c23 + v1 *v3 *( - 2*a12 *c23 - 3*a12 *a13 *c23 )
2 2 2 2 2 3 3 2 3 2
- v2 *v3 *a12 *a13 *c23 + v2*v3 *(a12 *a13*c23 - a12*a13 *c23 )
4 5 4 2 5 2 2 2 4 2
+ v3 *( - ---*a12 *c23 - ---*a12 *a13 *c23 - a13 *c23 )
4 4
{HAM,FI} = too large to simplify
3 2 2 3 3 1 2 2 2
FI=u1 *u2*a12 *a13 + u1 *u3*a12*a13 - ---*u1 *v1 *a12 *a13*c23
2
2 2 2 2 2 3 2 2
+ u1 *v2*v3*a12*a13 *c23 - u1 *v3 *a12 *a13*c23 + u1*u2 *a12 *a13
2 3 2 2 2 2
+ u1*u2 *u3*a12*a13 + u1*u2*u3 *a12 *a13 - 2*u1*u2*v1*v2*a12 *a13*c23
2 3 3 2
- 2*u1*u2*v1*v3*a12*a13 *c23 + u1*u3 *a12*a13 - 2*u1*u3*v1*v2*a12*a13 *c23
2 2 2 3 2 3
+ 2*u1*u3*v1*v3*a12 *a13*c23 + u2 *v1 *( - ---*a12 *a13*c23 - a13 *c23)
2
2 2 2 2 2
+ u2 *v2*v3*a12*a13 *c23 - u2 *v3 *a12 *a13*c23
2 2 3 2 3 2 2
+ u3 *v1 *( - ---*a12 *a13*c23 - a13 *c23) + u3 *v2*v3*a12*a13 *c23
2
2 2 2 3 2 2 3 2
- u3 *v3 *a12 *a13*c23 + v1 *v2*a12 *c23 + v1 *v3*a12*a13*c23
2 2 2 2 2 3 2
- 2*v1*v2 *v3*a12*a13*c23 + 4*v1*v2*v3 *a12 *c23 + 2*v1*v3 *a12*a13*c23
{HAM,FI} = too large to simplify
3 2 2 2
FI=u1 *v1*a12*a13 + u1 *u2*v2*a12*a13 + u1 *u3*v3*a12*a13 + u1*u2 *v1*a12*a13
2 2 2
+ u1*u3 *v1*a12*a13 - 2*u1*v1 *v2*a12*c23 - 2*u1*v1 *v3*a13*c23
3 2 2
+ u2 *v2*a12*a13 + u2 *u3*v3*a12*a13 + u2*u3 *v2*a12*a13
2 3
- 2*u2*v1*v2 *a12*c23 - 2*u2*v1*v2*v3*a13*c23 + u3 *v3*a12*a13
2
- 2*u3*v1*v2*v3*a12*c23 - 2*u3*v1*v3 *a13*c23
{HAM,FI} = too large to simplify
2 2 2 2 2 2 2
FI=u1 *v2 *a12*a13 + u1 *v3 *a12*a13 - 2*u1*u2*v1*v2*a12*a13
2 2 2 2 2 2 2
- 2*u1*u3*v1*v3*a12*a13 + u2 *v1 *a12*a13 + u2 *v3 *a12*a13
2 2 2 2 2 2 2
- 2*u2*u3*v2*v3*a12*a13 + u3 *v1 *a12*a13 + u3 *v2 *a12*a13
3 3 2 3
- 2*v1 *v2*a12*a13*c23 - 2*v1 *v3*a13 *c23 - 2*v1*v2 *a12*a13*c23
2 2 2 3 2
+ 2*v1*v2 *v3*a12 *c23 - 2*v1*v2*v3 *a12*a13*c23 + 2*v1*v3 *a12 *c23
{HAM,FI} = too large to simplify
And again in machine readable form:
HAM=(2*u1*u2*a12*a13 + 2*u1*u3*a13**2 + v2**2*a12*c23 + 2*v2*v3*a13*c23 - v3**2*
a12*c23)/a13$
FI=1/2*u1**4*a12**2*a13**4 + u1**2*u2**2*( - a12**4*a13**2 + a12**2*a13**4) - 2*
u1**2*u2*u3*a12**3*a13**3 - 2*u1**2*v1*v2*a12**2*a13**3*c23 - 2*u1**2*v1*v3*a12*
a13**4*c23 + u1*u2*v1**2*a12**4*a13*c23 - 2*u1*u2*v2*v3*a12**3*a13**2*c23 + u1*
u2*v3**2*(a12**4*a13*c23 - a12**2*a13**3*c23) + u1*u3*v1**2*a12**3*a13**2*c23 -
2*u1*u3*v2*v3*a12**2*a13**3*c23 + u1*u3*v3**2*(a12**3*a13**2*c23 - a12*a13**4*
c23) + 1/2*u2**4*a12**2*a13**4 + u2**2*u3**2*a12**2*a13**4 - 2*u2**2*v1*v2*a12**
2*a13**3*c23 + 2*u2**2*v1*v3*a12**3*a13**2*c23 + 1/2*u3**4*a12**2*a13**4 - 2*u3
**2*v1*v2*a12**2*a13**3*c23 - 2*u3**2*v1*v3*a12*a13**4*c23 + v1**4*(3/4*a12**4*
c23**2 + a12**2*a13**2*c23**2) + 2*v1**2*v2**2*a12**2*a13**2*c23**2 - 3*v1**2*v2
*v3*a12**3*a13*c23**2 + v1**2*v3**2*( - 2*a12**4*c23**2 - 3*a12**2*a13**2*c23**2
) - v2**2*v3**2*a12**2*a13**2*c23**2 + v2*v3**3*(a12**3*a13*c23**2 - a12*a13**3*
c23**2) + v3**4*( - 5/4*a12**4*c23**2 - 5/4*a12**2*a13**2*c23**2 - a13**4*c23**2
)$
FI=u1**3*u2*a12**2*a13**2 + u1**3*u3*a12*a13**3 - 1/2*u1**2*v1**2*a12**2*a13*c23
+ u1**2*v2*v3*a12*a13**2*c23 - u1**2*v3**2*a12**2*a13*c23 + u1*u2**3*a12**2*a13
**2 + u1*u2**2*u3*a12*a13**3 + u1*u2*u3**2*a12**2*a13**2 - 2*u1*u2*v1*v2*a12**2*
a13*c23 - 2*u1*u2*v1*v3*a12*a13**2*c23 + u1*u3**3*a12*a13**3 - 2*u1*u3*v1*v2*a12
*a13**2*c23 + 2*u1*u3*v1*v3*a12**2*a13*c23 + u2**2*v1**2*( - 3/2*a12**2*a13*c23
- a13**3*c23) + u2**2*v2*v3*a12*a13**2*c23 - u2**2*v3**2*a12**2*a13*c23 + u3**2*
v1**2*( - 3/2*a12**2*a13*c23 - a13**3*c23) + u3**2*v2*v3*a12*a13**2*c23 - u3**2*
v3**2*a12**2*a13*c23 + v1**3*v2*a12**2*c23**2 + v1**3*v3*a12*a13*c23**2 - 2*v1*
v2**2*v3*a12*a13*c23**2 + 4*v1*v2*v3**2*a12**2*c23**2 + 2*v1*v3**3*a12*a13*c23**
2$
FI=u1**3*v1*a12*a13 + u1**2*u2*v2*a12*a13 + u1**2*u3*v3*a12*a13 + u1*u2**2*v1*
a12*a13 + u1*u3**2*v1*a12*a13 - 2*u1*v1**2*v2*a12*c23 - 2*u1*v1**2*v3*a13*c23 +
u2**3*v2*a12*a13 + u2**2*u3*v3*a12*a13 + u2*u3**2*v2*a12*a13 - 2*u2*v1*v2**2*a12
*c23 - 2*u2*v1*v2*v3*a13*c23 + u3**3*v3*a12*a13 - 2*u3*v1*v2*v3*a12*c23 - 2*u3*
v1*v3**2*a13*c23$
FI=u1**2*v2**2*a12*a13**2 + u1**2*v3**2*a12*a13**2 - 2*u1*u2*v1*v2*a12*a13**2 -
2*u1*u3*v1*v3*a12*a13**2 + u2**2*v1**2*a12*a13**2 + u2**2*v3**2*a12*a13**2 - 2*
u2*u3*v2*v3*a12*a13**2 + u3**2*v1**2*a12*a13**2 + u3**2*v2**2*a12*a13**2 - 2*v1
**3*v2*a12*a13*c23 - 2*v1**3*v3*a13**2*c23 - 2*v1*v2**3*a12*a13*c23 + 2*v1*v2**2
*v3*a12**2*c23 - 2*v1*v2*v3**2*a12*a13*c23 + 2*v1*v3**3*a12**2*c23$