Solution 4 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=0
r41=0
r42=0
r43=0
r45=0
r46=0
r47=0
1 2 3 1 2 3
----*a12 *b11 *r4112 + ----*a13 *b11 *r4112
96 96
r48=---------------------------------------------
2 3
a12 *a13
1
---*b11*r473
4
r49=--------------
a12
r410=0
1 2 1 2
---*a12 *b11*r473 + ---*a13 *b11*r473
4 2
r411=---------------------------------------
2
a12*a13
r412=0
r413=0
7 2 1 2 3 3
r414=( - ----*a12 *a13*b11*r473 + ---*a12 *b11*c22*r4112 - ----*a13 *b11*r473
16 8 16
1 2 2 3
+ ----*a13 *b11 *r4106)/(a12*a13 )
32
r415=0
1 2 2 1 2 2
----*a12 *b11 *r4112 + ----*a13 *b11 *r4112
48 48
r416=---------------------------------------------
2 2
a12 *a13
r417=0
1 2 2 1 2 2
----*a12 *b11 *r4112 + ----*a13 *b11 *r4112
48 48
r418=---------------------------------------------
2 2
a12 *a13
r419=0
r420=0
2 2
a12 *r473 + a13 *r473
r421=-----------------------
a12*a13
1 4 2 1 2 3
r422=( - ----*a12 *b11 *r4112 + ---*a12 *a13 *b11*r492
96 4
1 2 2 2 1 4 2 3 3
- ----*a12 *a13 *b11 *r4112 - ----*a13 *b11 *r4112)/(a12 *a13 )
24 32
1 4 2 1 4 2
- ----*a12 *b11 *r4112 + ----*a13 *b11 *r4112
48 48
r423=------------------------------------------------
2 4
a12 *a13
r424=0
r425=0
r426=0
r427=0
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
r434=0
1 2 2 1 2 2
- ----*a12 *b11 *r4112 - ----*a13 *b11 *r4112
48 48
r435=------------------------------------------------
2 2
a12 *a13
r436=0
1 2 2 1 2 2
- ----*a12 *b11 *r4112 - ----*a13 *b11 *r4112
48 48
r437=------------------------------------------------
2 2
a12 *a13
r439=0
r440=0
r441=0
1 4 2 1 4 2
----*a12 *b11 *r4112 - ----*a13 *b11 *r4112
48 48
r442=---------------------------------------------
2 4
a12 *a13
1 4 2 1 2 3
r443=( - ----*a12 *b11 *r4112 + ---*a12 *a13 *b11*r492
96 4
1 2 2 2 1 4 2 3 3
- ----*a12 *a13 *b11 *r4112 - ----*a13 *b11 *r4112)/(a12 *a13 )
24 32
r444=0
r445=0
r446=0
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
r453=0
r454=0
r455=0
r456=0
r458=0
2 2
- a12 *r492 - a13 *r492
r459=--------------------------
2
a13
r460=0
r461=0
r462=0
r463=0
r464=0
r465=0
r466=0
r467=0
r468=0
r469=0
- a12*r473
r470=-------------
a13
r471=r473
a13*r473
r472=----------
a12
1 2 2 1 2 2
----*a12 *b11 *r4112 + ----*a13 *b11 *r4112
48 48
r474=---------------------------------------------
3
a12 *a13
r475
1 4 2 1 2 2 2 1 4 2
- ----*a12 *b11 *r4112 - ----*a12 *a13 *b11 *r4112 - ----*a13 *b11 *r4112
48 24 48
=----------------------------------------------------------------------------
2 4
a12 *a13
1 2 2 1 2 2
- ----*a12 *b11 *r4112 - ----*a13 *b11 *r4112
48 48
r476=------------------------------------------------
3
a12*a13
r477=
5 2 1 2 3 3 1 2
- ---*a12 *a13*r473 + ---*a12 *c22*r4112 - ---*a13 *r473 + ---*a13 *b11*r4106
2 2 2 4
--------------------------------------------------------------------------------
2
a12*a13
r478=
5 2 1 2 3 3 1 2
- ---*a12 *a13*r473 + ---*a12 *c22*r4112 - ---*a13 *r473 + ---*a13 *b11*r4106
2 2 2 4
--------------------------------------------------------------------------------
3
a13
1 6 2 13 4 2 2
r479=( - ----*a12 *b11 *r4112 - ----*a12 *a13 *b11 *r4112
48 96
1 2 5 1 2 4 2
+ ---*a12 *a13 *b11*r492 - ----*a12 *a13 *b11 *r4112
4 48
1 6 2 3 5
- ----*a13 *b11 *r4112)/(a12 *a13 )
32
- a12*r492
r480=-------------
a13
r481=r492
r483=0
r484=0
1
- ---*a12*b11*r4112
4
r485=----------------------
2
a13
r486=0
r487=0
r488=0
r489=0
r490=0
- a12*r492
r491=-------------
a13
r493=0
2 2
a12 *r4106 + a13 *r4106
r494=-------------------------
a12*a13
1 2
- ---*a12 *b11*r4112
4
r495=-----------------------
3
a13
r496=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
1
- ---*a12*r4106
2
r4105=------------------
a13
1
- ---*a13*r4106
2
r4107=------------------
a12
1
- a12*a13*r492 - ---*a12*b11*r4112
2
r4108=-------------------------------------
2
a13
1
a13*r492 + ---*b11*r4112
2
r4109=--------------------------
a13
1 2 1 2
---*a12 *r4106 + ---*a13 *r4106
2 2
r4110=---------------------------------
a12*a13
- a12*r4112
r4111=--------------
a13
r4113=0
2
- a12 *r4112
r4115=---------------
2
a13
a12*r4112
r4116=-----------
a13
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
c33=c22
c23=0
c13=0
c12=0
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:
r473, r4106, r492, r4112, c22, 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.
{a13,a12,b11,c22}
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 + 2*u1*u3*a13 + u1*v1*b11 + v2 *c22 + v3 *c22
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 4 4 2 5 3 2 3 5
FI=u1 *u2*v2*a12 *a13 - u1 *u2*v3*a12 *a13 + u1 *u3*v2*a12 *a13
2 4 4 1 2 3 4
- u1 *u3*v3*a12 *a13 + ---*u1 *v1*v2*a12 *a13 *b11
2
1 2 4 3 1 2 5 2
- ---*u1 *v1*v3*a12 *a13 *b11 - ---*u1*u2*v1 *a12 *a13 *b11
2 4
1 2 4 3 3 1 6 2
- ---*u1*u3*v1 *a12 *a13 *b11 + u1*v1 *( - ----*a12 *b11
4 48
13 4 2 2 1 2 4 2 1 6 2
- ----*a12 *a13 *b11 - ----*a12 *a13 *b11 - ----*a13 *b11 )
96 48 32
1 2 5 2 1 2 4 3
+ ---*u1*v1 *v2*a12 *a13 *c22 + ---*u1*v1 *v3*a12 *a13 *c22
2 2
2 1 4 2 2 1 2 4 2
+ u1*v1*v2 *( - ----*a12 *a13 *b11 - ----*a12 *a13 *b11 ) + u1*v1*v2*v3
48 48
1 5 2 1 3 3 2 1 5 2
*( - ----*a12 *a13*b11 - ----*a12 *a13 *b11 - ----*a12*a13 *b11 )
48 24 48
2 1 2 4 2 1 6 2
+ u1*v1*v3 *(----*a12 *a13 *b11 + ----*a13 *b11 )
48 48
2 1 4 2 2 1 2 4 2 1 6 2
+ u2*v1 *v2*( - ----*a12 *a13 *b11 - ----*a12 *a13 *b11 - ----*a13 *b11 )
96 24 32
2 1 5 2 1 5 2
+ u2*v1 *v3*(----*a12 *a13*b11 - ----*a12*a13 *b11 )
48 48
2 1 3 3 2 1 5 2
+ u2*v2 *v3*( - ----*a12 *a13 *b11 - ----*a12*a13 *b11 )
48 48
3 1 3 3 2 1 5 2
+ u2*v3 *( - ----*a12 *a13 *b11 - ----*a12*a13 *b11 )
48 48
2 1 5 2 1 5 2
+ u3*v1 *v2*( - ----*a12 *a13*b11 + ----*a12*a13 *b11 )
48 48
2 1 4 2 2 1 2 4 2 1 6 2
+ u3*v1 *v3*( - ----*a12 *a13 *b11 - ----*a12 *a13 *b11 - ----*a13 *b11 )
96 24 32
3 1 3 3 2 1 5 2
+ u3*v2 *(----*a12 *a13 *b11 + ----*a12*a13 *b11 )
48 48
2 1 3 3 2 1 5 2
+ u3*v2*v3 *(----*a12 *a13 *b11 + ----*a12*a13 *b11 )
48 48
1 4 4 2
+ ---*v1 *a12 *a13 *b11*c22
8
3 1 3 2 3 1 4 3
+ v1*v2 *(----*a12 *a13 *b11 + ----*a12*a13 *b11 )
96 96
{HAM,FI} = too large to simplify
2 2 2 2 2 2
FI=u1 *v1*v2*a12*a13 - u1 *v1*v3*a12 *a13 + u1*u2*v2 *a12*a13
2 2 2 2
- u1*u2*v2*v3*a12 *a13 + u1*u3*v2*v3*a12*a13 - u1*u3*v3 *a12 *a13
1 3 2 2 3 2
+ ---*u1*v1 *a13 *b11 + u2 *v1*v2*( - a12 - a12*a13 )
4
1 2 2 1 2 2
+ ---*u2*v1 *v2*a13 *b11 + ---*u3*v1 *v3*a13 *b11
4 4
{HAM,FI} = too large to simplify
2 2 1 2 1 2 1 2 2 2 2
FI=u1 *v1 *(---*a12 + ---*a13 ) - ---*u1 *v2 *a13 + u1 *v2*v3*a12*a13
2 2 2
1 2 2 2 2 2 1 2
- ---*u1 *v3 *a12 + u1*u2*v1*v2*(a12 + a13 ) + ---*u1*v1 *v2*a12*b11
2 4
1 2 1 4 2
+ ---*u1*v1 *v3*a13*b11 + ----*v1 *b11
4 32
{HAM,FI} = too large to simplify
2 5 3 3 2
FI=u1*v1 *v2*( - ---*a12 - ---*a12*a13 )
2 2
2 5 2 3 3 3 2 2 3
+ u1*v1 *v3*( - ---*a12 *a13 - ---*a13 ) + u1*v2 *a12*a13 + u1*v2 *v3*a13
2 2
2 2 3 2 2 2 3
+ u1*v2*v3 *a12*a13 - u1*v3 *a12 *a13 + u3*v1*v2 *(a12 *a13 + a13 )
4 7 2 3 2
+ v1 *( - ----*a12 *b11 - ----*a13 *b11)
16 16
2 2 1 2 1 2 1 2 2 2
+ v1 *v2 *(---*a12 *b11 + ---*a13 *b11) + ---*v1 *v3 *a13 *b11
4 2 4
{HAM,FI} = too large to simplify
And again in machine readable form:
HAM=2*u1*u2*a12 + 2*u1*u3*a13 + u1*v1*b11 + v2**2*c22 + v3**2*c22$
FI=u1**2*u2*v2*a12**4*a13**4 - u1**2*u2*v3*a12**5*a13**3 + u1**2*u3*v2*a12**3*
a13**5 - u1**2*u3*v3*a12**4*a13**4 + 1/2*u1**2*v1*v2*a12**3*a13**4*b11 - 1/2*u1
**2*v1*v3*a12**4*a13**3*b11 - 1/4*u1*u2*v1**2*a12**5*a13**2*b11 - 1/4*u1*u3*v1**
2*a12**4*a13**3*b11 + u1*v1**3*( - 1/48*a12**6*b11**2 - 13/96*a12**4*a13**2*b11
**2 - 1/48*a12**2*a13**4*b11**2 - 1/32*a13**6*b11**2) + 1/2*u1*v1**2*v2*a12**5*
a13**2*c22 + 1/2*u1*v1**2*v3*a12**4*a13**3*c22 + u1*v1*v2**2*( - 1/48*a12**4*a13
**2*b11**2 - 1/48*a12**2*a13**4*b11**2) + u1*v1*v2*v3*( - 1/48*a12**5*a13*b11**2
- 1/24*a12**3*a13**3*b11**2 - 1/48*a12*a13**5*b11**2) + u1*v1*v3**2*(1/48*a12**
2*a13**4*b11**2 + 1/48*a13**6*b11**2) + u2*v1**2*v2*( - 1/96*a12**4*a13**2*b11**
2 - 1/24*a12**2*a13**4*b11**2 - 1/32*a13**6*b11**2) + u2*v1**2*v3*(1/48*a12**5*
a13*b11**2 - 1/48*a12*a13**5*b11**2) + u2*v2**2*v3*( - 1/48*a12**3*a13**3*b11**2
- 1/48*a12*a13**5*b11**2) + u2*v3**3*( - 1/48*a12**3*a13**3*b11**2 - 1/48*a12*
a13**5*b11**2) + u3*v1**2*v2*( - 1/48*a12**5*a13*b11**2 + 1/48*a12*a13**5*b11**2
) + u3*v1**2*v3*( - 1/96*a12**4*a13**2*b11**2 - 1/24*a12**2*a13**4*b11**2 - 1/32
*a13**6*b11**2) + u3*v2**3*(1/48*a12**3*a13**3*b11**2 + 1/48*a12*a13**5*b11**2)
+ u3*v2*v3**2*(1/48*a12**3*a13**3*b11**2 + 1/48*a12*a13**5*b11**2) + 1/8*v1**4*
a12**4*a13**2*b11*c22 + v1*v2**3*(1/96*a12**3*a13**2*b11**3 + 1/96*a12*a13**4*
b11**3)$
FI=u1**2*v1*v2*a12*a13**2 - u1**2*v1*v3*a12**2*a13 + u1*u2*v2**2*a12*a13**2 - u1
*u2*v2*v3*a12**2*a13 + u1*u3*v2*v3*a12*a13**2 - u1*u3*v3**2*a12**2*a13 + 1/4*u1*
v1**3*a13**2*b11 + u2**2*v1*v2*( - a12**3 - a12*a13**2) + 1/4*u2*v1**2*v2*a13**2
*b11 + 1/4*u3*v1**2*v3*a13**2*b11$
FI=u1**2*v1**2*(1/2*a12**2 + 1/2*a13**2) - 1/2*u1**2*v2**2*a13**2 + u1**2*v2*v3*
a12*a13 - 1/2*u1**2*v3**2*a12**2 + u1*u2*v1*v2*(a12**2 + a13**2) + 1/4*u1*v1**2*
v2*a12*b11 + 1/4*u1*v1**2*v3*a13*b11 + 1/32*v1**4*b11**2$
FI=u1*v1**2*v2*( - 5/2*a12**3 - 3/2*a12*a13**2) + u1*v1**2*v3*( - 5/2*a12**2*a13
- 3/2*a13**3) + u1*v2**3*a12*a13**2 + u1*v2**2*v3*a13**3 + u1*v2*v3**2*a12*a13
**2 - u1*v3**3*a12**2*a13 + u3*v1*v2**2*(a12**2*a13 + a13**3) + v1**4*( - 7/16*
a12**2*b11 - 3/16*a13**2*b11) + v1**2*v2**2*(1/4*a12**2*b11 + 1/2*a13**2*b11) +
1/4*v1**2*v3**2*a13**2*b11$