Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
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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
r421=0
r422=0
r423=0
r424=0
r425=0
r426=0
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
r435=0
r439=0
r442=0
r444=0
r445=0
r448=0
r450=0
r451=0
r453=0
r454=0
r455=0
r458=0
r460=0
r461=0
r463=0
r464=0
r465=0
r467=0
r468=0
r469=0
1 4 2 11 4 3
r470=(---*a12 *a33 *b12*r482 - ------*a12 *b12 *r4103
6 2160
1 2 2 2 1 2 2 3
+ ---*a12 *a13 *a33 *b12*r482 - ---*a12 *a13 *b12 *r4103
6 5
421 4 3 3 4
- ------*a13 *b12 *r4103)/(a12 *a33 )
2160
1 4 2 11 4 3
r471=( - ---*a12 *a13*a33 *b12*r482 + ------*a12 *a13*b12 *r4103
6 2160
1 2 3 2 1 2 3 3
- ---*a12 *a13 *a33 *b12*r482 + ---*a12 *a13 *b12 *r4103
6 5
421 5 3 4 4
+ ------*a13 *b12 *r4103)/(a12 *a33 )
2160
1 4 2 11 4 3
r472=(---*a12 *a33 *b12*r482 - ------*a12 *b12 *r4103
6 2160
1 2 2 2 1 2 2 3
+ ---*a12 *a13 *a33 *b12*r482 - ---*a12 *a13 *b12 *r4103
6 5
421 4 3 3 4
- ------*a13 *b12 *r4103)/(a12 *a33 )
2160
1 4 2 11 4 3
r473=( - ---*a12 *a13*a33 *b12*r482 + ------*a12 *a13*b12 *r4103
6 2160
1 2 3 2 1 2 3 3
- ---*a12 *a13 *a33 *b12*r482 + ---*a12 *a13 *b12 *r4103
6 5
421 5 3 4 4
+ ------*a13 *b12 *r4103)/(a12 *a33 )
2160
1 2 1 2
- ---*a12 *b12*r482 - ---*a13 *b12*r482
2 2
r474=------------------------------------------
a12*a13*a33
r475=0
1 2 1 2
- ---*a12 *b12*r482 - ---*a13 *b12*r482
2 2
r476=------------------------------------------
a12*a13*a33
r477=
11 4 3 3 2 2 3 421 4 3
- ------*a12 *b12 *r4103 - ----*a12 *a13 *b12 *r4103 - ------*a13 *b12 *r4103
1440 10 1440
--------------------------------------------------------------------------------
3 4
a12 *a33
11 4 3 3 2 3 3
r478=(------*a12 *a13*b12 *r4103 + ----*a12 *a13 *b12 *r4103
1440 10
421 5 3 4 4
+ ------*a13 *b12 *r4103)/(a12 *a33 )
1440
r479
11 4 3 3 2 2 3 421 4 3
------*a12 *b12 *r4103 + ----*a12 *a13 *b12 *r4103 + ------*a13 *b12 *r4103
1440 10 1440
=-----------------------------------------------------------------------------
3 3
a12 *a13*a33
r480=r482
r483=0
r484=0
4 2 125 2 2 2 125 4 2
a12 *a33 *r482 + ------*a12 *a13 *b12 *r4103 + ------*a13 *b12 *r4103
2392 2392
r485=-----------------------------------------------------------------------
4 2
a12 *a33
r486=0
r487=0
r488=0
r489=r4103
a12*r482
r490=----------
a13
r493=0
4 2 125 2 2 2 125 4 2
a12 *a33 *r482 + ------*a12 *a13 *b12 *r4103 + ------*a13 *b12 *r4103
2392 2392
r495=-----------------------------------------------------------------------
3 2
a12 *a13*a33
r496=0
r498=0
a12*r4103
r499=-----------
a13
r4100=0
r4102=0
a12*r4103
r4104=-----------
a13
1 6 2 1 4 2 2
r4105=(---*a12 *a33 *r482 + ---*a12 *a13 *a33 *r482
2 2
427 4 2 2 1 4 2 2
+ ------*a12 *a13 *b12 *r4103 - ----*a12 *a33 *b12 *r4103
1688 16
1 2 4 2 1 2 2 2 2
- ---*a12 *a13 *b12 *r4103 - ----*a12 *a13 *a33 *b12 *r4103
2 16
1271 6 2 4 3
- ------*a13 *b12 *r4103)/(a12 *a13*a33 )
1688
2 2
- a12 *r482 - a13 *r482
r4106=--------------------------
a12*a33
1 4 1 2 2 1 2 2
r4107=( - ---*a12 *r482 - ---*a12 *a13 *r482 - ----*a12 *b12 *r4103
2 2 16
1 2 2 2
- ----*a13 *b12 *r4103)/(a12 *a13*a33)
16
4 2 2 2 2 427 2 2 2
r4108=(2*a12 *a33 *r482 + 2*a12 *a13 *a33 *r482 + -----*a12 *a13 *b12 *r4103
844
427 4 2 2 2 2
+ -----*a13 *b12 *r4103)/(a12 *a13 *a33 )
844
4 2 1271 2 2 2 1271 4 2
- a12 *a33 *r482 - ------*a12 *a13 *b12 *r4103 - ------*a13 *b12 *r4103
844 844
r4109=--------------------------------------------------------------------------
3 2
a12 *a13*a33
1 6 2 3 4 2 2
r4110=(---*a12 *a33 *r482 + ---*a12 *a13 *a33 *r482
2 2
427 4 2 2 1 4 2 2
+ ------*a12 *a13 *b12 *r4103 - ----*a12 *a33 *b12 *r4103
1688 16
2 4 2 50649 2 4 2
+ a12 *a13 *a33 *r482 + --------*a12 *a13 *b12 *r4103
252356
1 2 2 2 2 125 6 2 4
- ----*a12 *a13 *a33 *b12 *r4103 - ------*a13 *b12 *r4103)/(a12 *a13
16 2392
3
*a33 )
2*a13*b12*r4103
r4111=-----------------
a12*a33
b12*r4103
r4112=-----------
a33
r4113=0
b12*r4103
r4115=-----------
a33
2 2
- a12 *b12*r4114 - a13 *b12*r4114
r4117=------------------------------------
2
a13 *a33
2*a12*r4114
r4118=-------------
a13
2
a12 *r4114
r4119=------------
2
a13
r4120=0
2 2
- a12 *b12*r4103 - a13 *b12*r4103
r4121=------------------------------------
2
a13*a33
- a12*b12*r4103
r4122=------------------
a13*a33
r4123=r4103
a12*r4103
r4124=-----------
a13
r4125=0
c33=0
c23=0
c22=0
c13=0
c12=0
b33=0
b23=0
a13*b12
b13=---------
a12
b11=0
- a12*r482
r481=-------------
a13
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:
r482, r4114, r4103, b12, a12, a13, a33
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.
{a12,a13,a33,b12}
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=(u1 *a12*a33 + 2*u1*u2*a12 + 2*u1*u3*a12*a13 + u1*v2*a12*b12
2 2
+ u1*v3*a13*b12 + u2 *a12*a33 + u3 *a12*a33)/a12
has apart from the Hamiltonian and Casimirs the following 3 first integrals:
3 5 4 3 4 4 3 5 3
FI=u1 *u2*a12 *a33 + u1 *u3*a12 *a13*a33 - u1 *v1*a12 *a33 *b12
3 6 2 4 2 2
+ u1 *v2*( - a12 *a33 *b12 - a12 *a13 *a33 *b12)
2 4 3 2 4 3
+ u1 *u2*v3*a12 *a13*a33 *b12 + u1 *u3*v2*a12 *a13*a33 *b12
2 3 2 3 2 2 427 4 2 2
+ 2*u1 *u3*v3*a12 *a13 *a33 *b12 + u1 *v1 *(------*a12 *a13 *a33*b12
1688
1 4 3 2 50649 2 4 2
- ----*a12 *a33 *b12 + --------*a12 *a13 *a33*b12
16 252356
1 2 2 3 2 125 6 2
- ----*a12 *a13 *a33 *b12 - ------*a13 *a33*b12 )
16 2392
2 1271 3 2 2 2 1271 4 2 2
+ u1 *v1*v2*( - ------*a12 *a13 *a33 *b12 - ------*a12*a13 *a33 *b12 )
844 844
2 427 4 2 2 427 2 3 2 2
+ u1 *v1*v3*(-----*a12 *a13*a33 *b12 + -----*a12 *a13 *a33 *b12 )
844 844
2 2 1 4 3 2 1 2 2 3 2 2 2
+ u1 *v2 *( - ----*a12 *a33 *b12 - ----*a12 *a13 *a33 *b12 ) + u1 *v3 *(
16 16
427 4 2 2 1 4 3 2 1 2 4 2
------*a12 *a13 *a33*b12 - ----*a12 *a33 *b12 - ---*a12 *a13 *a33*b12
1688 16 2
1 2 2 3 2 1271 6 2 3 5 4
- ----*a12 *a13 *a33 *b12 - ------*a13 *a33*b12 ) + u1*u2 *a12 *a33
16 1688
2 4 4 2 5 4
+ u1*u2 *u3*a12 *a13*a33 + u1*u2*u3 *a12 *a33
2 125 3 2 2 2 125 4 2 2
+ u1*u2*v1 *(------*a12 *a13 *a33 *b12 + ------*a12*a13 *a33 *b12 )
2392 2392
3 4 4
+ u1*u3 *a12 *a13*a33
2 125 2 3 2 2 125 5 2 2 3
+ u1*u3*v1 *(------*a12 *a13 *a33 *b12 + ------*a13 *a33 *b12 ) + u1*v1
2392 2392
11 5 3 3 3 2 3 421 4 3
*(------*a12 *a33*b12 + ----*a12 *a13 *a33*b12 + ------*a12*a13 *a33*b12 )
1440 10 1440
2 11 4 2 3 3 2 4 3 421 6 3
+ u1*v1 *v2*(------*a12 *a13 *b12 + ----*a12 *a13 *b12 + ------*a13 *b12 )
1440 10 1440
2
+ u1*v1 *v3
11 5 3 3 3 3 3 421 5 3
*( - ------*a12 *a13*b12 - ----*a12 *a13 *b12 - ------*a12*a13 *b12 )
1440 10 1440
3 11 4 2 3 1 2 4 3 421 6 3
+ u1*v2 *(------*a12 *a13 *b12 + ---*a12 *a13 *b12 + ------*a13 *b12 ) +
2160 5 2160
2
u1*v2 *v3
11 5 3 1 3 3 3 421 5 3
*( - ------*a12 *a13*b12 - ---*a12 *a13 *b12 - ------*a12*a13 *b12 )
2160 5 2160
2 11 4 2 3 1 2 4 3 421 6 3
+ u1*v2*v3 *(------*a12 *a13 *b12 + ---*a12 *a13 *b12 + ------*a13 *b12 )
2160 5 2160
3
+ u1*v3
11 5 3 1 3 3 3 421 5 3
*( - ------*a12 *a13*b12 - ---*a12 *a13 *b12 - ------*a12*a13 *b12 )
2160 5 2160
{HAM,FI} = too large to simplify
2 2 2 2
FI=u1 *u2 *a12 *a33 + 2*u1 *u2*u3*a12*a13*a33
2 2 2 2 2 2
+ u1 *u2*v1*( - a12 *b12 - a13 *b12) + u1 *u3 *a13 *a33
{HAM,FI} = too large to simplify
2 2 1 4 3 2 3 5
FI=u1 *v1 *(---*a12 *a13*a33 + ---*a12 *a13 *a33 + a13 *a33)
2 2
2 3 2 2 4 2 2 2 2
- u1 *v1*v2*a12 *a13*a33 + u1 *v1*v3*(2*a12 *a33 + 2*a12 *a13 *a33 )
2 2 1 4 1 2 3
+ u1 *v2 *( - ---*a12 *a13*a33 - ---*a12 *a13 *a33)
2 2
2 3 2 4
+ u1 *v2*v3*( - a12 *a13 *a33 - a12*a13 *a33)
2 2 1 4 1 2 3 2 3 2
+ u1 *v3 *(---*a12 *a13*a33 + ---*a12 *a13 *a33) + u1*u2*v1 *a12 *a13*a33
2 2
2 3 2 2 2 2 2
+ u1*u2*v3 *a12 *a13*a33 + u1*u3*v1 *a12 *a13 *a33
2 2 2 2 3 2
+ u1*u3*v2 *a12 *a13 *a33 - u1*u3*v2*v3*a12 *a13*a33
2 2 2 2
+ u1*u3*v3 *a12 *a13 *a33
2 1 3 1 3
+ u1*v1*v2 *( - ---*a12 *a13*a33*b12 - ---*a12*a13 *a33*b12)
2 2
2 1 3 1 3
+ u1*v1*v3 *( - ---*a12 *a13*a33*b12 - ---*a12*a13 *a33*b12)
2 2
3 1 2 3 1 5
+ u1*v2 *( - ---*a12 *a13 *b12 - ---*a13 *b12)
6 6
2 1 3 2 1 4
+ u1*v2 *v3*(---*a12 *a13 *b12 + ---*a12*a13 *b12)
6 6
2 1 2 3 1 5
+ u1*v2*v3 *( - ---*a12 *a13 *b12 - ---*a13 *b12)
6 6
3 1 3 2 1 4
+ u1*v3 *(---*a12 *a13 *b12 + ---*a12*a13 *b12)
6 6
{HAM,FI} = too large to simplify
And again in machine readable form:
HAM=(u1**2*a12*a33 + 2*u1*u2*a12**2 + 2*u1*u3*a12*a13 + u1*v2*a12*b12 + u1*v3*
a13*b12 + u2**2*a12*a33 + u3**2*a12*a33)/a12$
FI=u1**3*u2*a12**5*a33**4 + u1**3*u3*a12**4*a13*a33**4 - u1**3*v1*a12**5*a33**3*
b12 + u1**3*v2*( - a12**6*a33**2*b12 - a12**4*a13**2*a33**2*b12) + u1**2*u2*v3*
a12**4*a13*a33**3*b12 + u1**2*u3*v2*a12**4*a13*a33**3*b12 + 2*u1**2*u3*v3*a12**3
*a13**2*a33**3*b12 + u1**2*v1**2*(427/1688*a12**4*a13**2*a33*b12**2 - 1/16*a12**
4*a33**3*b12**2 + 50649/252356*a12**2*a13**4*a33*b12**2 - 1/16*a12**2*a13**2*a33
**3*b12**2 - 125/2392*a13**6*a33*b12**2) + u1**2*v1*v2*( - 1271/844*a12**3*a13**
2*a33**2*b12**2 - 1271/844*a12*a13**4*a33**2*b12**2) + u1**2*v1*v3*(427/844*a12
**4*a13*a33**2*b12**2 + 427/844*a12**2*a13**3*a33**2*b12**2) + u1**2*v2**2*( - 1
/16*a12**4*a33**3*b12**2 - 1/16*a12**2*a13**2*a33**3*b12**2) + u1**2*v3**2*(427/
1688*a12**4*a13**2*a33*b12**2 - 1/16*a12**4*a33**3*b12**2 - 1/2*a12**2*a13**4*
a33*b12**2 - 1/16*a12**2*a13**2*a33**3*b12**2 - 1271/1688*a13**6*a33*b12**2) +
u1*u2**3*a12**5*a33**4 + u1*u2**2*u3*a12**4*a13*a33**4 + u1*u2*u3**2*a12**5*a33
**4 + u1*u2*v1**2*(125/2392*a12**3*a13**2*a33**2*b12**2 + 125/2392*a12*a13**4*
a33**2*b12**2) + u1*u3**3*a12**4*a13*a33**4 + u1*u3*v1**2*(125/2392*a12**2*a13**
3*a33**2*b12**2 + 125/2392*a13**5*a33**2*b12**2) + u1*v1**3*(11/1440*a12**5*a33*
b12**3 + 3/10*a12**3*a13**2*a33*b12**3 + 421/1440*a12*a13**4*a33*b12**3) + u1*v1
**2*v2*(11/1440*a12**4*a13**2*b12**3 + 3/10*a12**2*a13**4*b12**3 + 421/1440*a13
**6*b12**3) + u1*v1**2*v3*( - 11/1440*a12**5*a13*b12**3 - 3/10*a12**3*a13**3*b12
**3 - 421/1440*a12*a13**5*b12**3) + u1*v2**3*(11/2160*a12**4*a13**2*b12**3 + 1/5
*a12**2*a13**4*b12**3 + 421/2160*a13**6*b12**3) + u1*v2**2*v3*( - 11/2160*a12**5
*a13*b12**3 - 1/5*a12**3*a13**3*b12**3 - 421/2160*a12*a13**5*b12**3) + u1*v2*v3
**2*(11/2160*a12**4*a13**2*b12**3 + 1/5*a12**2*a13**4*b12**3 + 421/2160*a13**6*
b12**3) + u1*v3**3*( - 11/2160*a12**5*a13*b12**3 - 1/5*a12**3*a13**3*b12**3 -
421/2160*a12*a13**5*b12**3)$
FI=u1**2*u2**2*a12**2*a33 + 2*u1**2*u2*u3*a12*a13*a33 + u1**2*u2*v1*( - a12**2*
b12 - a13**2*b12) + u1**2*u3**2*a13**2*a33$
FI=u1**2*v1**2*(1/2*a12**4*a13*a33 + 3/2*a12**2*a13**3*a33 + a13**5*a33) - u1**2
*v1*v2*a12**3*a13*a33**2 + u1**2*v1*v3*(2*a12**4*a33**2 + 2*a12**2*a13**2*a33**2
) + u1**2*v2**2*( - 1/2*a12**4*a13*a33 - 1/2*a12**2*a13**3*a33) + u1**2*v2*v3*(
- a12**3*a13**2*a33 - a12*a13**4*a33) + u1**2*v3**2*(1/2*a12**4*a13*a33 + 1/2*
a12**2*a13**3*a33) + u1*u2*v1**2*a12**3*a13*a33**2 + u1*u2*v3**2*a12**3*a13*a33
**2 + u1*u3*v1**2*a12**2*a13**2*a33**2 + u1*u3*v2**2*a12**2*a13**2*a33**2 - u1*
u3*v2*v3*a12**3*a13*a33**2 + u1*u3*v3**2*a12**2*a13**2*a33**2 + u1*v1*v2**2*( -
1/2*a12**3*a13*a33*b12 - 1/2*a12*a13**3*a33*b12) + u1*v1*v3**2*( - 1/2*a12**3*
a13*a33*b12 - 1/2*a12*a13**3*a33*b12) + u1*v2**3*( - 1/6*a12**2*a13**3*b12 - 1/6
*a13**5*b12) + u1*v2**2*v3*(1/6*a12**3*a13**2*b12 + 1/6*a12*a13**4*b12) + u1*v2*
v3**2*( - 1/6*a12**2*a13**3*b12 - 1/6*a13**5*b12) + u1*v3**3*(1/6*a12**3*a13**2*
b12 + 1/6*a12*a13**4*b12)$