Solution 3 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=a12 + a13
Expressions
The solution is given through the following expressions:
r40
1 5 3 1 3 2 3 1 4 3
-----*a12 *b11 *r496 + -----*a12 *a13 *b11 *r496 - -----*a12*a13 *b11 *r496
864 432 288
=-----------------------------------------------------------------------------
8
a13
1 4 3 1 2 2 3
-----*a12 *b11 *r496 + -----*a12 *a13 *b11 *r496
648 162
r41=--------------------------------------------------
7
a13
1 3 3 1 2 3
-----*a12 *b11 *r496 - -----*a12*a13 *b11 *r496
432 432
r42=-------------------------------------------------
6
a13
1 2 3
-----*a12 *b11 *r496
216
r43=----------------------
5
a13
r45=0
r46=0
r47=0
r48=0
1 4 3 11 2 2 3 5 4 3
-----*a12 *b11 *r496 - -----*a12 *a13 *b11 *r496 - ----*a13 *b11 *r496
108 288 96
r49=------------------------------------------------------------------------
3 4
a12 *a13
1 2 3 1 2 3
- -----*a12 *b11 *r496 - -----*a13 *b11 *r496
216 216
r410=------------------------------------------------
5
a13
r411
1 4 3 47 2 2 3 55 4 3
- -----*a12 *b11 *r496 + -----*a12 *a13 *b11 *r496 + -----*a13 *b11 *r496
216 864 864
=----------------------------------------------------------------------------
6
a12*a13
r412=0
r413=0
r414
1 4 3 49 2 2 3 55 4 3
- -----*a12 *b11 *r496 + ------*a12 *a13 *b11 *r496 + ------*a13 *b11 *r496
864 1728 1728
=------------------------------------------------------------------------------
6
a12*a13
r415=0
r416=0
r417=0
r418=0
1 2
- ----*a12*b11 *r496
36
r419=-----------------------
3
a13
1 2
----*b11 *r496
36
r420=----------------
2
a13
1 2
- ----*a12*b11 *r496
36
r421=-----------------------
3
a13
r422=0
r423=0
r424=0
1
- ---*a12*b11*r496
6
r425=---------------------
2
a13
r426=0
1
- ----*a12*b11*r496
12
r427=----------------------
2
a13
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
r434=0
r435=0
1 2
- ----*b11 *r496
36
r439=-------------------
2
a13
r442=0
r444=0
r445=0
r448=0
r450=0
r451=0
r453=0
r454=0
1
- ----*a12*b11*r496
12
r455=----------------------
2
a13
r458=0
r460=0
r461=0
r463=0
r464=0
r465=0
r467=0
r468=0
r469=0
19 2 2 5 2 2
----*a12 *b11 *r496 + ----*a13 *b11 *r496
72 24
r470=-------------------------------------------
3
a12*a13
11 2 2 5 2 2
- ----*a12 *b11 *r496 - ----*a13 *b11 *r496
72 24
r471=----------------------------------------------
2 2
a12 *a13
19 2 2 5 2 2
----*a12 *b11 *r496 + ----*a13 *b11 *r496
72 24
r472=-------------------------------------------
3
a12*a13
11 2 2 5 2 2
- ----*a12 *b11 *r496 - ----*a13 *b11 *r496
72 24
r473=----------------------------------------------
2 2
a12 *a13
r474=0
r475=0
r476=0
r477=0
1 2
----*b11 *r496
36
r478=----------------
2
a13
r479=0
r480=0
r481=0
1
- ---*a12*b11*r496
6
r483=---------------------
2
a13
1
- ---*b11*r496
3
r484=-----------------
a13
r485=0
a12*r496
r486=----------
a13
r487= - r496
r488=0
r489=0
r490=0
1 2 5 2
----*a12 *b11*r496 + ----*a13 *b11*r496
12 12
r493=-----------------------------------------
3
a13
r495=0
r498=0
r499=0
a12*r496
r4100=----------
a13
r4102=0
r4103=0
r4104=0
1 3 1 2
----*a12 *b11*r496 - ---*a12*a13 *b11*r496
12 3
r4105=--------------------------------------------
4
a13
1 2 1 2
- ---*a12 *b11*r496 - ---*a13 *b11*r496
6 6
r4106=------------------------------------------
3
a13
5
----*b11*r496
12
r4107=---------------
a12
r4108=0
r4109=0
1 3 1 2
- ----*a12 *b11*r496 - ---*a12*a13 *b11*r496
12 2
r4110=-----------------------------------------------
4
a13
r4111=0
r4112=0
3 2
- a12 *r496 - a12*a13 *r496
r4113=------------------------------
3
a13
r4115=0
3 2 1 2
- ---*a12 *r496 - ---*a13 *r496
2 2
r4117=----------------------------------
2
a13
r4118=0
r4119=0
1 3 1 2
- ---*a12 *r496 + ---*a12*a13 *r496
2 2
r4120=--------------------------------------
3
a13
r4121= - r496
r4122=0
r4123=0
r4124=0
r4125=0
c33=0
c23=0
c22=0
1 2
- ----*b11
36
c13=--------------
a13
1 2
----*a12*b11
36
c12=---------------
2
a13
b33=0
b31=0
b21=0
b13=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:
r496, b11, 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.
{a12,r496,a13,b11}
Relevance for the application:
Modulo the following equation:
2 2
0=a12 + a13
the system of equations related to the Hamiltonian HAM:
2 3 2 1 2
HAM=(2*u1*u2*a12*a13 + 2*u1*u3*a13 + u1*v1*a13 *b11 + ----*v1*v2*a12*b11
18
1 2 2
- ----*v1*v3*a13*b11 )/a13
18
has apart from the Hamiltonian and Casimirs only the following first integral:
3 3 8 3 1 6 5 1 4 7
FI= - u1 *v2*a12 *a13 + u1 *v3*( - ---*a12 *a13 + ---*a12 *a13 )
2 2
2 3 5 6 1 3 8
+ u1 *u2*v1*( - ---*a12 *a13 - ---*a12 *a13 )
2 2
2 6 5 4 7
+ u1 *u3*v1*( - a12 *a13 - a12 *a13 )
2 2 1 6 4 1 4 6
+ u1 *v1 *( - ----*a12 *a13 *b11 - ---*a12 *a13 *b11)
12 2
5 2 2 2 8
+ ----*u1 *v2 *a12 *a13 *b11
12
2 1 5 5 1 3 7
+ u1 *v2*v3*( - ---*a12 *a13 *b11 - ---*a12 *a13 *b11)
6 6
2 2 1 6 4 1 4 6 2 4 7
+ u1 *v3 *(----*a12 *a13 *b11 - ---*a12 *a13 *b11) + u1*u2 *v3*a12 *a13
12 3
3 8
+ u1*u2*u3*v3*a12 *a13
1 5 5 5 3 7
+ u1*u2*v1*v3*(----*a12 *a13 *b11 + ----*a12 *a13 *b11)
12 12
2 3 8 2 4 7 1 3 7
- u1*u3 *v2*a12 *a13 + u1*u3 *v3*a12 *a13 - ---*u1*u3*v1*v2*a12 *a13 *b11
3
1 4 6 1 2 3 6 2
- ---*u1*u3*v1*v3*a12 *a13 *b11 + ----*u1*v1 *v2*a12 *a13 *b11
6 36
3 11 3 6 2 5 8 2
+ u1*v2 *( - ----*a12 *a13 *b11 - ----*a12*a13 *b11 )
72 24
2 19 4 5 2 5 2 7 2
+ u1*v2 *v3*(----*a12 *a13 *b11 + ----*a12 *a13 *b11 )
72 24
2 11 3 6 2 5 8 2
+ u1*v2*v3 *( - ----*a12 *a13 *b11 - ----*a12*a13 *b11 )
72 24
3 19 4 5 2 5 2 7 2
+ u1*v3 *(----*a12 *a13 *b11 + ----*a12 *a13 *b11 )
72 24
1 2 2 4 6 1 2 3 6 2
- ----*u2 *v3 *a12 *a13 *b11 - ----*u2*v1*v3 *a12 *a13 *b11
12 36
1 2 2 4 6 1 2 2 4 6
- ----*u3 *v2 *a12 *a13 *b11 - ---*u3 *v3 *a12 *a13 *b11
12 6
1 2 4 5 2 1 3 6 2
- ----*u3*v1*v2 *a12 *a13 *b11 + ----*u3*v1*v2*v3*a12 *a13 *b11
36 36
1 2 4 5 2 4
- ----*u3*v1*v3 *a12 *a13 *b11 + v1
36
1 6 2 3 49 4 4 3 55 2 6 3
*( - -----*a12 *a13 *b11 + ------*a12 *a13 *b11 + ------*a12 *a13 *b11 ) +
864 1728 1728
2 2
v1 *v2
1 6 2 3 47 4 4 3 55 2 6 3
*( - -----*a12 *a13 *b11 + -----*a12 *a13 *b11 + -----*a12 *a13 *b11 )
216 864 864
2 1 5 3 3 1 3 5 3
+ v1 *v2*v3*( - -----*a12 *a13 *b11 - -----*a12 *a13 *b11 )
216 216
2 2 1 4 4 3 11 2 6 3 5 8 3
+ v1 *v3 *(-----*a12 *a13 *b11 - -----*a12 *a13 *b11 - ----*a13 *b11 )
108 288 96
1 3 5 3 3
+ -----*v2 *v3*a12 *a13 *b11
216
2 2 1 6 2 3 1 4 4 3
+ v2 *v3 *(-----*a12 *a13 *b11 - -----*a12 *a13 *b11 )
432 432
3 1 7 3 1 5 3 3
+ v2*v3 *(-----*a12 *a13*b11 + -----*a12 *a13 *b11 )
648 162
4 1 8 3 1 6 2 3 1 4 4 3
+ v3 *(-----*a12 *b11 + -----*a12 *a13 *b11 - -----*a12 *a13 *b11 )
864 432 288
{HAM,FI} = too large to simplify
And again in machine readable form:
HAM=(2*u1*u2*a12*a13**2 + 2*u1*u3*a13**3 + u1*v1*a13**2*b11 + 1/18*v1*v2*a12*b11
**2 - 1/18*v1*v3*a13*b11**2)/a13**2$
FI= - u1**3*v2*a12**3*a13**8 + u1**3*v3*( - 1/2*a12**6*a13**5 + 1/2*a12**4*a13**
7) + u1**2*u2*v1*( - 3/2*a12**5*a13**6 - 1/2*a12**3*a13**8) + u1**2*u3*v1*( -
a12**6*a13**5 - a12**4*a13**7) + u1**2*v1**2*( - 1/12*a12**6*a13**4*b11 - 1/2*
a12**4*a13**6*b11) + 5/12*u1**2*v2**2*a12**2*a13**8*b11 + u1**2*v2*v3*( - 1/6*
a12**5*a13**5*b11 - 1/6*a12**3*a13**7*b11) + u1**2*v3**2*(1/12*a12**6*a13**4*b11
- 1/3*a12**4*a13**6*b11) + u1*u2**2*v3*a12**4*a13**7 + u1*u2*u3*v3*a12**3*a13**
8 + u1*u2*v1*v3*(1/12*a12**5*a13**5*b11 + 5/12*a12**3*a13**7*b11) - u1*u3**2*v2*
a12**3*a13**8 + u1*u3**2*v3*a12**4*a13**7 - 1/3*u1*u3*v1*v2*a12**3*a13**7*b11 -
1/6*u1*u3*v1*v3*a12**4*a13**6*b11 + 1/36*u1*v1**2*v2*a12**3*a13**6*b11**2 + u1*
v2**3*( - 11/72*a12**3*a13**6*b11**2 - 5/24*a12*a13**8*b11**2) + u1*v2**2*v3*(19
/72*a12**4*a13**5*b11**2 + 5/24*a12**2*a13**7*b11**2) + u1*v2*v3**2*( - 11/72*
a12**3*a13**6*b11**2 - 5/24*a12*a13**8*b11**2) + u1*v3**3*(19/72*a12**4*a13**5*
b11**2 + 5/24*a12**2*a13**7*b11**2) - 1/12*u2**2*v3**2*a12**4*a13**6*b11 - 1/36*
u2*v1*v3**2*a12**3*a13**6*b11**2 - 1/12*u3**2*v2**2*a12**4*a13**6*b11 - 1/6*u3**
2*v3**2*a12**4*a13**6*b11 - 1/36*u3*v1*v2**2*a12**4*a13**5*b11**2 + 1/36*u3*v1*
v2*v3*a12**3*a13**6*b11**2 - 1/36*u3*v1*v3**2*a12**4*a13**5*b11**2 + v1**4*( - 1
/864*a12**6*a13**2*b11**3 + 49/1728*a12**4*a13**4*b11**3 + 55/1728*a12**2*a13**6
*b11**3) + v1**2*v2**2*( - 1/216*a12**6*a13**2*b11**3 + 47/864*a12**4*a13**4*b11
**3 + 55/864*a12**2*a13**6*b11**3) + v1**2*v2*v3*( - 1/216*a12**5*a13**3*b11**3
- 1/216*a12**3*a13**5*b11**3) + v1**2*v3**2*(1/108*a12**4*a13**4*b11**3 - 11/288
*a12**2*a13**6*b11**3 - 5/96*a13**8*b11**3) + 1/216*v2**3*v3*a12**5*a13**3*b11**
3 + v2**2*v3**2*(1/432*a12**6*a13**2*b11**3 - 1/432*a12**4*a13**4*b11**3) + v2*
v3**3*(1/648*a12**7*a13*b11**3 + 1/162*a12**5*a13**3*b11**3) + v3**4*(1/864*a12
**8*b11**3 + 1/432*a12**6*a13**2*b11**3 - 1/288*a12**4*a13**4*b11**3)$