Solution 2 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:
r40=0
r41=0
r42=0
r43=0
r45=0
r46=0
r47=0
r48=0
r49=0
r410=0
r411=0
r412=0
r413=0
r414=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
7 2 7 2
---*a12 *r473 + ---*a13 *r473
9 9
r421=-------------------------------
a12*a13
r422=0
r423=0
r424=0
r425=0
1 2 1 2
----*a12 *r4106 + ----*a13 *r4106
10 10
r426=-----------------------------------
2 2 2
a12 + ---*a13
5
r428=0
r429=0
3 1 2 2 2 1 3
a12 *r427 + ----*a12 *a13*r4106 + ---*a12*a13 *r427 + ----*a13 *r4106
10 5 10
r430=-----------------------------------------------------------------------
3 2 2
a12 + ---*a12*a13
5
r431=r452
r432=0
r433=0
1
r434=---*r464
2
r435=0
r436=0
r437=0
r439=0
r440=0
2 2 2 2
- ---*a12 *r473 - ---*a13 *r473
9 9
r441=----------------------------------
2
a13
r442=0
r443=0
r444=0
1 2 1 2
- ----*a12 *r4106 - ----*a13 *r4106
10 10
r445=--------------------------------------
2 2 2
a12 + ---*a13
5
r446= - 2*r427
1 2 1 2
----*a12 *r4106 + ----*a13 *r4106
10 10
r447=-----------------------------------
2 2 2
a12 + ---*a13
5
r448=0
r449=0
r450=0
r451=0
r453=0
r454=0
r455=r427
1 2 1 2
- ----*a12 *r4106 - ----*a13 *r4106
10 10
r456=--------------------------------------
2 2 2
a12 + ---*a13
5
r458=0
r459=0
3 1 2 2 2 1 3
a12 *r427 + ----*a12 *a13*r4106 + ---*a12*a13 *r427 + ----*a13 *r4106
10 5 10
r460=-----------------------------------------------------------------------
3 2 2
a12 + ---*a12*a13
5
r461=r452
r462=0
r463=0
r465=0
r466=r452
r467=0
r468=0
1
r469=---*r464
2
2 2 7 2
- ---*a12 *r473 + ---*a13 *r473
9 9
r470=----------------------------------
a12*a13
2 2 7 2
---*a12 *r473 - ---*a13 *r473
9 9
r471=-------------------------------
2
a12
2 2 7 2
- ---*a12 *r473 + ---*a13 *r473
9 9
r472=----------------------------------
a12*a13
r474=0
r475=0
r476=0
r478=r473
r479=0
1 2 1 2
- ---*a12 *r492 + ---*a13 *r492
2 2
r480=----------------------------------
a12*a13
r483
3 3 2 4 2 3 3
- 2*a12 *r427 - ----*a12 *a13*r4106 - ---*a12*a13 *r427 - ----*a13 *r4106
10 5 10
=----------------------------------------------------------------------------
3 2 2
a12 + ---*a12*a13
5
1 2 1 2
- ----*a12 *r4106 - ----*a13 *r4106
10 10
r484=--------------------------------------
2 2 2
a12 + ---*a13
5
r485=0
- a12*r487
r486=-------------
a13
r488=r452
r489=r4103
1 2 1 2
- ---*a12 *r492 - ---*a13 *r492
2 2
r490=----------------------------------
2
a13
- a12*r492
r491=-------------
a13
1 2 1 2
----*a12 *r4106 + ----*a13 *r4106
10 10
r493=-----------------------------------
2 2 2
a12 + ---*a13
5
3 3 2 3 2 4 3
----*a12 *r4106 - 2*a12 *a13*r427 + ----*a12*a13 *r4106 - ---*a13 *r427
10 10 5
r494=-------------------------------------------------------------------------
2 2 3
a12 *a13 + ---*a13
5
r495=0
2 2
- a12 *r487 - a13 *r487
r496=--------------------------
2
a13
r498=0
a12*r4103
r499=-----------
a13
- a12*r487
r4100=-------------
a13
2
- a12 *r487
r4101=--------------
2
a13
r4102=r452
a12*r4103
r4104=-----------
a13
1 5 1 3 2 2 3 2 5
----*a12 *r4106 - ---*a12 *a13 *r4106 + a12 *a13 *r427 + ---*a13 *r427
10 5 5
r4105=------------------------------------------------------------------------
2 3 2 5
a12 *a13 + ---*a13
5
a13*r492
r4108=----------
a12
r4109=r492
r4110=0
- a12*r4112 + a13*r452
r4111=-------------------------
a13
r4113=0
2
- a12 *r4112
r4115=---------------
2
a13
a12*r4112 + a13*r452
r4116=----------------------
a13
r4117=0
- 2*a12*r464
r4118=---------------
a13
2 2
- a12 *r464 + a13 *r464
r4119=--------------------------
2
a13
3 2
6*a12 *r487 + 5*a12*a13 *r487
r4120=-------------------------------
3
a13
2 2
- 2*a12 *r487 - a13 *r487
r4121=----------------------------
2
a13
r4122=r452
r4123=r4103
a12*r4103
r4124=-----------
a13
1
r4125=---*r464
2
c33=0
c23=0
c22=0
c13=0
c12=0
b33=0
b31=0
b21=0
b13=0
b11=0
r481=r492
3 3 2 2 3
----*a12 *r4106 + a12 *a13*r427 + ---*a13 *r427
10 5
r4107=-------------------------------------------------
2 2 3
a12 *a13 + ---*a13
5
a13*r473
r477=----------
a12
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:
r473, r4106, r492, r427, r4112, r487, r452, r4103, r464,
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.
{a13,a12}
Relevance for the application:
Modulo the following equation:
2 2
0=a12 + a13
the system of equations related to the Hamiltonian HAM:
HAM=2*u1*u2*a12 + 2*u1*u3*a13
has apart from the Hamiltonian and Casimirs the following 9 first integrals:
1 4 2 2 2 2 2 2 1 4 2
FI=---*u1 *a13 + u1 *u2 *( - a12 + a13 ) - 2*u1 *u2*u3*a12*a13 + ---*u2 *a13
2 2
2 2 2 1 4 2
+ u2 *u3 *a13 + ---*u3 *a13
2
{HAM,FI} = too large to simplify
3 3 3 2 2
FI=u1 *u2*a12 + u1 *u3*a13 + u1*u2 *a12 + u1*u2 *u3*a13 + u1*u2*u3 *a12
3
+ u1*u3 *a13
{HAM,FI} = too large to simplify
3 2 2 2 2 3 2
FI=u1 *v1 + u1 *u2*v2 + u1 *u3*v3 + u1*u2 *v1 + u1*u3 *v1 + u2 *v2 + u2 *u3*v3
2 3
+ u2*u3 *v2 + u3 *v3
{HAM,FI} = too large to simplify
3 2 3 3 3 2
FI=u1 *v2*( - 2*a12 *a13 - a13 ) + u1 *v3*(6*a12 + 5*a12*a13 )
2 2 2 2 2 3
- u1*u2 *v2*a12 *a13 - u1*u2 *v3*a12*a13 + u1*u2*u3*v3*( - a12 *a13 - a13 )
2 3 2 2
+ u1*u3 *v2*a13 - u1*u3 *v3*a12*a13
{HAM,FI} = too large to simplify
2 2 2 2 2 2
FI=u1 *u2*v2*a12*a13 - u1 *u2*v3*a12 + u1 *u3*v2*a13 - u1 *u3*v3*a12*a13
{HAM,FI} = too large to simplify
2 2 2 2 2 2 2 2
FI=u1 *v2 + u1 *v3 - 2*u1*u2*v1*v2 - 2*u1*u3*v1*v3 + u2 *v1 + u2 *v3
2 2 2 2
- 2*u2*u3*v2*v3 + u3 *v1 + u3 *v2
{HAM,FI} = too large to simplify
2 2 2 3 2 2
FI=u1 *v1*v2*a12*a13 + u1 *v1*v3*a13 + u1*u2*v2 *a12*a13
2 2 1 3 1 2
- u1*u2*v2*v3*a12 *a13 + u1*u2*v3 *( - ---*a12 - ---*a12*a13 )
2 2
2 2 1 2 1 3
+ u1*u3*v2*v3*a12*a13 + u1*u3*v3 *( - ---*a12 *a13 + ---*a13 )
2 2
{HAM,FI} = too large to simplify
3 2 2 4 2 2 3 3 2 5
FI=----*u1 *v2 *a12 *a13 + u1 *v2*v3*(a12 *a13 + ---*a12*a13 )
10 5
2 2 1 6 1 4 2
+ u1 *v3 *(----*a12 - ---*a12 *a13 )
10 5
3 4 2 3 2 4
+ u1*u2*v1*v2*(----*a12 *a13 + ----*a12 *a13 )
10 10
1 3 3 1 5
+ u1*u2*v1*v3*(----*a12 *a13 + ----*a12*a13 )
10 10
1 3 3 1 5
+ u1*u3*v1*v2*( - ----*a12 *a13 - ----*a12*a13 )
10 10
3 2 4 3 6
+ u1*u3*v1*v3*( - ----*a12 *a13 - ----*a13 )
10 10
2 2 1 2 4 1 6
+ u2 *v1 *(----*a12 *a13 + ----*a13 )
10 10
2 1 3 3 1 5
+ u2 *v2*v3*( - ----*a12 *a13 - ----*a12*a13 )
10 10
2 1 3 3 1 5
+ u2*u3*v2 *(----*a12 *a13 + ----*a12*a13 )
10 10
2 1 3 3 1 5
+ u2*u3*v3 *( - ----*a12 *a13 - ----*a12*a13 )
10 10
2 2 1 2 4 1 6
+ u3 *v1 *(----*a12 *a13 + ----*a13 )
10 10
2 1 3 3 1 5
+ u3 *v2*v3*(----*a12 *a13 + ----*a12*a13 )
10 10
{HAM,FI} = too large to simplify
2 2 2 2 3 3 2 2
FI=u1*v1 *v2*a12 *a13 + u1*v1 *v3*a12*a13 + u1*v2 *a12 *a13
2 2 3 7 3
+ u1*v2 *v3*( - ---*a12 *a13 + ---*a12*a13 )
9 9
2 2 2 2 7 4
+ u1*v2*v3 *(---*a12 *a13 - ---*a13 )
9 9
3 2 3 7 3
+ u1*v3 *( - ---*a12 *a13 + ---*a12*a13 )
9 9
2 2 4 2 2 2
+ u2*v1*v2 *( - ---*a12 - ---*a12 *a13 )
9 9
2 7 3 7 3
+ u3*v1*v2 *(---*a12 *a13 + ---*a12*a13 )
9 9
{HAM,FI} = too large to simplify
And again in machine readable form:
HAM=2*u1*u2*a12 + 2*u1*u3*a13$
FI=1/2*u1**4*a13**2 + u1**2*u2**2*( - a12**2 + a13**2) - 2*u1**2*u2*u3*a12*a13 +
1/2*u2**4*a13**2 + u2**2*u3**2*a13**2 + 1/2*u3**4*a13**2$
FI=u1**3*u2*a12 + u1**3*u3*a13 + u1*u2**3*a12 + u1*u2**2*u3*a13 + u1*u2*u3**2*
a12 + u1*u3**3*a13$
FI=u1**3*v1 + u1**2*u2*v2 + u1**2*u3*v3 + u1*u2**2*v1 + u1*u3**2*v1 + u2**3*v2 +
u2**2*u3*v3 + u2*u3**2*v2 + u3**3*v3$
FI=u1**3*v2*( - 2*a12**2*a13 - a13**3) + u1**3*v3*(6*a12**3 + 5*a12*a13**2) - u1
*u2**2*v2*a12**2*a13 - u1*u2**2*v3*a12*a13**2 + u1*u2*u3*v3*( - a12**2*a13 - a13
**3) + u1*u3**2*v2*a13**3 - u1*u3**2*v3*a12*a13**2$
FI=u1**2*u2*v2*a12*a13 - u1**2*u2*v3*a12**2 + u1**2*u3*v2*a13**2 - u1**2*u3*v3*
a12*a13$
FI=u1**2*v2**2 + u1**2*v3**2 - 2*u1*u2*v1*v2 - 2*u1*u3*v1*v3 + u2**2*v1**2 + u2
**2*v3**2 - 2*u2*u3*v2*v3 + u3**2*v1**2 + u3**2*v2**2$
FI=u1**2*v1*v2*a12*a13**2 + u1**2*v1*v3*a13**3 + u1*u2*v2**2*a12*a13**2 - u1*u2*
v2*v3*a12**2*a13 + u1*u2*v3**2*( - 1/2*a12**3 - 1/2*a12*a13**2) + u1*u3*v2*v3*
a12*a13**2 + u1*u3*v3**2*( - 1/2*a12**2*a13 + 1/2*a13**3)$
FI=3/10*u1**2*v2**2*a12**4*a13**2 + u1**2*v2*v3*(a12**3*a13**3 + 2/5*a12*a13**5)
+ u1**2*v3**2*(1/10*a12**6 - 1/5*a12**4*a13**2) + u1*u2*v1*v2*(3/10*a12**4*a13
**2 + 3/10*a12**2*a13**4) + u1*u2*v1*v3*(1/10*a12**3*a13**3 + 1/10*a12*a13**5) +
u1*u3*v1*v2*( - 1/10*a12**3*a13**3 - 1/10*a12*a13**5) + u1*u3*v1*v3*( - 3/10*
a12**2*a13**4 - 3/10*a13**6) + u2**2*v1**2*(1/10*a12**2*a13**4 + 1/10*a13**6) +
u2**2*v2*v3*( - 1/10*a12**3*a13**3 - 1/10*a12*a13**5) + u2*u3*v2**2*(1/10*a12**3
*a13**3 + 1/10*a12*a13**5) + u2*u3*v3**2*( - 1/10*a12**3*a13**3 - 1/10*a12*a13**
5) + u3**2*v1**2*(1/10*a12**2*a13**4 + 1/10*a13**6) + u3**2*v2*v3*(1/10*a12**3*
a13**3 + 1/10*a12*a13**5)$
FI=u1*v1**2*v2*a12**2*a13**2 + u1*v1**2*v3*a12*a13**3 + u1*v2**3*a12**2*a13**2 +
u1*v2**2*v3*( - 2/9*a12**3*a13 + 7/9*a12*a13**3) + u1*v2*v3**2*(2/9*a12**2*a13
**2 - 7/9*a13**4) + u1*v3**3*( - 2/9*a12**3*a13 + 7/9*a12*a13**3) + u2*v1*v2**2*
( - 2/9*a12**4 - 2/9*a12**2*a13**2) + u3*v1*v2**2*(7/9*a12**3*a13 + 7/9*a12*a13
**3)$