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:
1 4 3 1 4 3
- -----*a12 *b11 *r487 - -----*a13 *b11 *r487
432 432
r40=------------------------------------------------
6
a12*a13
1 3
-----*b11 *r487
216
r41=-----------------
3
a13
1 3 3 1 2 3
- -----*a12 *b11 *r487 + -----*a12*a13 *b11 *r487
432 432
r42=----------------------------------------------------
6
a13
1 4 3
-----*a12 *b11 *r487
216
r43=----------------------
7
a13
r45=0
r46=0
r47=0
r48=0
31 6 3 19 4 2 3
r49=( - ------*a12 *b11 *r487 - ------*a12 *a13 *b11 *r487
1296 1296
11 2 4 3 35 6 3 8
- ------*a12 *a13 *b11 *r487 - ------*a13 *b11 *r487)/(a12*a13 )
1728 1728
1 2 3 1 2 3
----*a12 *b11 *r487 + ----*a13 *b11 *r487
54 54
r410=-------------------------------------------
5
a13
r411
17 4 3 17 2 2 3 5 4 3
- -----*a12 *b11 *r487 + -----*a12 *a13 *b11 *r487 + -----*a13 *b11 *r487
864 864 144
=----------------------------------------------------------------------------
6
a12*a13
r412=0
r413=0
31 6 3 7 4 2 3
r414=( - ------*a12 *b11 *r487 - ------*a12 *a13 *b11 *r487
2592 2592
1 2 4 3 31 6 3 8
+ -----*a12 *a13 *b11 *r487 - ------*a13 *b11 *r487)/(a12*a13 )
384 3456
r415=0
r416=0
r417=0
r418=0
1 2 2 1 2 2
----*a12 *b11 *r487 + -----*a13 *b11 *r487
27 108
r419=--------------------------------------------
3
a12*a13
r420=0
5 4 2 1 2 2 2 5 4 2
- ----*a12 *b11 *r487 + ----*a12 *a13 *b11 *r487 + ----*a13 *b11 *r487
72 36 72
r421=-------------------------------------------------------------------------
5
a12*a13
r422=0
r423=0
r424=0
r425=0
1
- ---*b11*r487
6
r426=-----------------
a13
1 3
- ----*a12 *b11*r487
12
r427=-----------------------
4
a13
r428=0
r429=0
1 4 1 2 2 1 4
- ----*a12 *b11*r487 - ---*a12 *a13 *b11*r487 - ---*a13 *b11*r487
12 3 4
r430=--------------------------------------------------------------------
4
a12*a13
r431=0
r432=0
r433=0
r434=0
r435=0
r436=0
r437=0
1 2 2
- ----*a12 *b11 *r487
36
r439=------------------------
4
a13
1 2 2 1 2 2
-----*a12 *b11 *r487 + -----*a13 *b11 *r487
108 108
r440=---------------------------------------------
3
a12*a13
1 2
----*b11 *r487
36
r441=----------------
2
a13
r442=0
r443=0
r444=0
1
---*b11*r487
6
r445=--------------
a13
1 3
---*a12 *b11*r487
6
r446=-------------------
4
a13
1
- ---*b11*r487
6
r447=-----------------
a13
r448=0
r449=0
r450=0
r451=0
r452=0
r453=0
r454=0
1 3
- ----*a12 *b11*r487
12
r455=-----------------------
4
a13
1
---*b11*r487
6
r456=--------------
a13
r458=0
r459=0
1 4 1 2 2 1 4
- ----*a12 *b11*r487 - ---*a12 *a13 *b11*r487 - ---*a13 *b11*r487
12 3 4
r460=--------------------------------------------------------------------
4
a12*a13
r461=0
r462=0
r463=0
r464=0
r465=0
r466=0
r467=0
r468=0
r469=0
35 3 2 11 2 2
-----*a12 *b11 *r487 + -----*a12*a13 *b11 *r487
432 432
r470=-------------------------------------------------
5
a13
11 2 2 35 2 2
- -----*a12 *b11 *r487 - -----*a13 *b11 *r487
432 432
r471=------------------------------------------------
4
a13
1 2 2 5 2 2
----*a12 *b11 *r487 + ----*a13 *b11 *r487
72 72
r472=-------------------------------------------
3
a12*a13
1 2 2 1 2 2
----*a12 *b11 *r487 - ----*a13 *b11 *r487
72 24
r473=-------------------------------------------
4
a13
r474=0
r475=0
r476=0
r477=0
r478=0
r479=0
r480=0
r481=0
1 4 1 4
---*a12 *b11*r487 + ---*a13 *b11*r487
6 2
r483=---------------------------------------
4
a12*a13
1
---*b11*r487
6
r484=--------------
a13
r485=0
- a12*r487
r486=-------------
a13
r488=0
r489=0
r490=0
r491=0
r492=0
1
- ---*b11*r487
6
r493=-----------------
a13
1 3 1 2
---*a12 *b11*r487 - ---*a12*a13 *b11*r487
6 2
r494=-------------------------------------------
4
a13
r495=0
2 2
- a12 *r487 - a13 *r487
r496=--------------------------
2
a13
r498=0
r499=0
- a12*r487
r4100=-------------
a13
2
- a12 *r487
r4101=--------------
2
a13
r4102=0
r4103=0
r4104=0
1 3 1 2
- ----*a12 *b11*r487 + ---*a12*a13 *b11*r487
12 3
r4105=-----------------------------------------------
4
a13
r4106=0
1 3 1 2
- ----*a12 *b11*r487 + ---*a12*a13 *b11*r487
12 3
r4107=-----------------------------------------------
4
a13
r4108=0
r4109=0
1 2 1 2
- ---*a12 *b11*r487 - ---*a13 *b11*r487
2 4
r4110=------------------------------------------
2
a12*a13
r4111=0
r4112=0
r4113=0
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
3 2
6*a12 *r487 + 5*a12*a13 *r487
r4120=-------------------------------
3
a13
2 2
- 2*a12 *r487 - a13 *r487
r4121=----------------------------
2
a13
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:
r487, 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.
{b11,a12,a13,r487}
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 6 8 3 4 5 2 7
FI=u1 *v2*( - 2*a12 *a13 - a12*a13 ) + u1 *v3*(6*a12 *a13 + 5*a12 *a13 )
2 2 1 2 6 1 8
+ u1 *v1 *( - ---*a12 *a13 *b11 - ---*a13 *b11)
2 4
2 2 1 4 4 1 2 6
+ u1 *v2 *( - ----*a12 *a13 *b11 + ---*a12 *a13 *b11)
12 3
2 2 1 4 4 1 2 6 2 3 6
+ u1 *v3 *( - ----*a12 *a13 *b11 + ---*a12 *a13 *b11) - u1*u2 *v2*a12 *a13
12 3
2 2 7 3 6 8
- u1*u2 *v3*a12 *a13 + u1*u2*u3*v3*( - a12 *a13 - a12*a13 )
1 4 4 1 2 6
+ u1*u2*v1*v2*(---*a12 *a13 *b11 - ---*a12 *a13 *b11)
6 2
1 7 2 8 2 2 7
- ---*u1*u2*v1*v3*a12*a13 *b11 + u1*u3 *v2*a12*a13 - u1*u3 *v3*a12 *a13
6
1 7
+ ---*u1*u3*v1*v2*a12*a13 *b11
6
1 4 4 1 8
+ u1*u3*v1*v3*(---*a12 *a13 *b11 + ---*a13 *b11)
6 2
3 1 3 4 2 1 6 2
+ u1*v2 *(----*a12 *a13 *b11 - ----*a12*a13 *b11 )
72 24
2 1 2 5 2 5 7 2
+ u1*v2 *v3*(----*a12 *a13 *b11 + ----*a13 *b11 )
72 72
2 11 3 4 2 35 6 2
+ u1*v2*v3 *( - -----*a12 *a13 *b11 - -----*a12*a13 *b11 )
432 432
3 35 4 3 2 11 2 5 2
+ u1*v3 *(-----*a12 *a13 *b11 + -----*a12 *a13 *b11 )
432 432
2 2 1 4 4 1 2 6 1 8
+ u2 *v1 *( - ----*a12 *a13 *b11 - ---*a12 *a13 *b11 - ---*a13 *b11)
12 3 4
1 2 7 1 2 2 4 4
+ ---*u2 *v2*v3*a12*a13 *b11 - ----*u2 *v3 *a12 *a13 *b11
6 12
1 2 7 1 4 4
- ---*u2*u3*v2 *a12*a13 *b11 + ---*u2*u3*v2*v3*a12 *a13 *b11
6 6
1 2 7 1 2 6 2
+ ---*u2*u3*v3 *a12*a13 *b11 + ----*u2*v1*v2 *a12*a13 *b11
6 36
1 2 5 2 1 7 2
+ u2*v1*v2*v3*(-----*a12 *a13 *b11 + -----*a13 *b11 )
108 108
1 2 3 4 2
- ----*u2*v1*v3 *a12 *a13 *b11
36
2 2 1 4 4 1 2 6 1 8
+ u3 *v1 *( - ----*a12 *a13 *b11 - ---*a12 *a13 *b11 - ---*a13 *b11)
12 3 4
1 2 2 4 4 1 2 7
- ----*u3 *v2 *a12 *a13 *b11 - ---*u3 *v2*v3*a12*a13 *b11
12 6
2 5 4 3 2 1 2 5 2 5 7 2
+ u3*v1*v2 *( - ----*a12 *a13 *b11 + ----*a12 *a13 *b11 + ----*a13 *b11 )
72 36 72
2 1 2 5 2 1 7 2 4
+ u3*v1*v3 *(----*a12 *a13 *b11 + -----*a13 *b11 ) + v1 *(
27 108
31 6 3 7 4 2 3 1 2 4 3
- ------*a12 *b11 - ------*a12 *a13 *b11 + -----*a12 *a13 *b11
2592 2592 384
31 6 3
- ------*a13 *b11 )
3456
2 2 17 4 2 3 17 2 4 3 5 6 3
+ v1 *v2 *( - -----*a12 *a13 *b11 + -----*a12 *a13 *b11 + -----*a13 *b11 )
864 864 144
2 1 3 3 3 1 5 3 2 2
+ v1 *v2*v3*(----*a12 *a13 *b11 + ----*a12*a13 *b11 ) + v1 *v3 *(
54 54
31 6 3 19 4 2 3 11 2 4 3
- ------*a12 *b11 - ------*a12 *a13 *b11 - ------*a12 *a13 *b11
1296 1296 1728
35 6 3 1 3 5 3
- ------*a13 *b11 ) + -----*v2 *v3*a12 *a13*b11
1728 216
2 2 1 4 2 3 1 2 4 3
+ v2 *v3 *( - -----*a12 *a13 *b11 + -----*a12 *a13 *b11 )
432 432
1 3 5 3
+ -----*v2*v3 *a12*a13 *b11
216
4 1 4 2 3 1 6 3
+ v3 *( - -----*a12 *a13 *b11 - -----*a13 *b11 )
432 432
{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*( - 2*a12**3*a13**6 - a12*a13**8) + u1**3*v3*(6*a12**4*a13**5 + 5*
a12**2*a13**7) + u1**2*v1**2*( - 1/2*a12**2*a13**6*b11 - 1/4*a13**8*b11) + u1**2
*v2**2*( - 1/12*a12**4*a13**4*b11 + 1/3*a12**2*a13**6*b11) + u1**2*v3**2*( - 1/
12*a12**4*a13**4*b11 + 1/3*a12**2*a13**6*b11) - u1*u2**2*v2*a12**3*a13**6 - u1*
u2**2*v3*a12**2*a13**7 + u1*u2*u3*v3*( - a12**3*a13**6 - a12*a13**8) + u1*u2*v1*
v2*(1/6*a12**4*a13**4*b11 - 1/2*a12**2*a13**6*b11) - 1/6*u1*u2*v1*v3*a12*a13**7*
b11 + u1*u3**2*v2*a12*a13**8 - u1*u3**2*v3*a12**2*a13**7 + 1/6*u1*u3*v1*v2*a12*
a13**7*b11 + u1*u3*v1*v3*(1/6*a12**4*a13**4*b11 + 1/2*a13**8*b11) + u1*v2**3*(1/
72*a12**3*a13**4*b11**2 - 1/24*a12*a13**6*b11**2) + u1*v2**2*v3*(1/72*a12**2*a13
**5*b11**2 + 5/72*a13**7*b11**2) + u1*v2*v3**2*( - 11/432*a12**3*a13**4*b11**2 -
35/432*a12*a13**6*b11**2) + u1*v3**3*(35/432*a12**4*a13**3*b11**2 + 11/432*a12
**2*a13**5*b11**2) + u2**2*v1**2*( - 1/12*a12**4*a13**4*b11 - 1/3*a12**2*a13**6*
b11 - 1/4*a13**8*b11) + 1/6*u2**2*v2*v3*a12*a13**7*b11 - 1/12*u2**2*v3**2*a12**4
*a13**4*b11 - 1/6*u2*u3*v2**2*a12*a13**7*b11 + 1/6*u2*u3*v2*v3*a12**4*a13**4*b11
+ 1/6*u2*u3*v3**2*a12*a13**7*b11 + 1/36*u2*v1*v2**2*a12*a13**6*b11**2 + u2*v1*
v2*v3*(1/108*a12**2*a13**5*b11**2 + 1/108*a13**7*b11**2) - 1/36*u2*v1*v3**2*a12
**3*a13**4*b11**2 + u3**2*v1**2*( - 1/12*a12**4*a13**4*b11 - 1/3*a12**2*a13**6*
b11 - 1/4*a13**8*b11) - 1/12*u3**2*v2**2*a12**4*a13**4*b11 - 1/6*u3**2*v2*v3*a12
*a13**7*b11 + u3*v1*v2**2*( - 5/72*a12**4*a13**3*b11**2 + 1/36*a12**2*a13**5*b11
**2 + 5/72*a13**7*b11**2) + u3*v1*v3**2*(1/27*a12**2*a13**5*b11**2 + 1/108*a13**
7*b11**2) + v1**4*( - 31/2592*a12**6*b11**3 - 7/2592*a12**4*a13**2*b11**3 + 1/
384*a12**2*a13**4*b11**3 - 31/3456*a13**6*b11**3) + v1**2*v2**2*( - 17/864*a12**
4*a13**2*b11**3 + 17/864*a12**2*a13**4*b11**3 + 5/144*a13**6*b11**3) + v1**2*v2*
v3*(1/54*a12**3*a13**3*b11**3 + 1/54*a12*a13**5*b11**3) + v1**2*v3**2*( - 31/
1296*a12**6*b11**3 - 19/1296*a12**4*a13**2*b11**3 - 11/1728*a12**2*a13**4*b11**3
- 35/1728*a13**6*b11**3) + 1/216*v2**3*v3*a12**5*a13*b11**3 + v2**2*v3**2*( - 1
/432*a12**4*a13**2*b11**3 + 1/432*a12**2*a13**4*b11**3) + 1/216*v2*v3**3*a12*a13
**5*b11**3 + v3**4*( - 1/432*a12**4*a13**2*b11**3 - 1/432*a13**6*b11**3)$