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
r48=0
1 3 1 2 2 1 2 2
---*a12 *b11*c22*r4112 - ---*a12 *a13 *b11*r477 - ---*a12 *a13*b11 *r4107
8 4 8
r49=---------------------------------------------------------------------------
5
a13
r410=0
1 3 1 2 2 1 2 2
---*a12 *b11*c22*r4112 - ---*a12 *a13 *b11*r477 - ---*a12 *a13*b11 *r4107
8 4 8
r411=---------------------------------------------------------------------------
5
a13
r412=0
r413=0
1 3 1 2 2
r414=(----*a12 *b11*c22*r4112 - ---*a12 *a13 *b11*r477
16 8
1 2 2 1 2
- ----*a12 *a13*b11 *r4107 + ----*a12*a13 *b11*c22*r4112
16 16
1 4 5
+ ---*a13 *b11*r477)/a13
8
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
r425=0
r426=0
r427=0
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
r434=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 2 1
- ---*a12*c22*r4112 + a13 *r477 + ---*a13*b11*r4107
2 2
r470=------------------------------------------------------
2
a13
4 3 2 3
r471=( - a12 *c22*r4112 + 2*a12 *a13 *r477 + a12 *a13*b11*r4107
3 2 2 4 3 3
- ---*a12 *a13 *c22*r4112 + 3*a12*a13 *r477 + ---*a12*a13 *b11*r4107)/
2 2
5
a13
1 3 2 2 1 2
---*a12 *c22*r4112 - a12 *a13 *r477 - ---*a12 *a13*b11*r4107
2 2
r472=--------------------------------------------------------------
4
a13
1 2 2 1
- ---*a12 *c22*r4112 + a12*a13 *r477 + ---*a12*a13*b11*r4107
2 2
r473=---------------------------------------------------------------
3
a13
r474=0
r475=0
r476=0
a12*r477
r478=----------
a13
1 2
- ---*a12*b11 *r4112
8
r479=-----------------------
3
a13
r480=0
r481=0
r483=0
r484=0
1
- ---*a12*b11*r4112
4
r485=----------------------
2
a13
r486=0
r487=0
r488=0
r489=0
r490=0
r493=0
1 2
- ---*a12 *b11*r4112
4
r495=-----------------------
3
a13
r496=0
r498=0
r499=0
r4100=0
r4102=0
r4103=0
r4104=0
2
a12 *r4107
r4105=------------
2
a13
- 2*a12*r4107
r4106=----------------
a13
1
- ---*a12*b11*r4112
2
r4108=----------------------
2
a13
1
---*b11*r4112
2
r4109=---------------
a13
r4110=0
- 2*a12*r4112
r4111=----------------
a13
r4113=0
2
- a12 *r4112
r4115=---------------
2
a13
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
- a12*r4112
r4122=--------------
a13
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:
r4107, r477, 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.
{b11,a12,a13}
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 3 first integrals:
3 4 2 2 3 2 5
FI= - u1 *v1*a12*a13 - u1 *u2*v3*a12 *a13 + u1 *u3*v2*a13
2 4 1 2 4 1 2 3
- 2*u1 *u3*v3*a12*a13 + ---*u1 *v1*v2*a13 *b11 - ---*u1 *v1*v3*a12*a13 *b11
2 2
1 2 2 2 1 2 3
- ---*u1*u2*v1 *a12 *a13 *b11 - ---*u1*u3*v1 *a12*a13 *b11
4 4
1 3 2 2 1 3 2 2
- ---*u1*v1 *a12*a13 *b11 - ---*u1*v2 *a12 *a13 *c22
8 2
1 2 3 2 4 3 2 2
+ ---*u1*v2 *v3*a12 *a13*c22 + u1*v2*v3 *( - a12 *c22 - ---*a12 *a13 *c22)
2 2
1 3 3 4 1 3 1 2
- ---*u1*v3 *a12*a13 *c22 + v1 *(----*a12 *b11*c22 + ----*a12*a13 *b11*c22)
2 16 16
1 2 2 3 1 2 2 3
+ ---*v1 *v2 *a12 *b11*c22 + ---*v1 *v3 *a12 *b11*c22
8 8
{HAM,FI} = too large to simplify
2 2 2 3 3 2 2 2
FI=u1*v1 *v2*a12*a13 + u1*v1 *v3*a13 + u1*v2 *a12*a13 - u1*v2 *v3*a12 *a13
2 3 2 3 3
+ u1*v2*v3 *(2*a12 + 3*a12*a13 ) + u1*v3 *a13
4 1 2 1 2 1 2 2 2
+ v1 *( - ---*a12 *b11 + ---*a13 *b11) - ---*v1 *v2 *a12 *b11
8 8 4
1 2 2 2
- ---*v1 *v3 *a12 *b11
4
{HAM,FI} = too large to simplify
2 2 4 2 3 2 2 2 2
FI=u1 *v2 *a13 - 2*u1 *v2*v3*a12*a13 + u1 *v3 *a12 *a13
1 3 2 1 2 2
+ ---*u1*v2 *a12*a13 *b11 - ---*u1*v2 *v3*a12 *a13*b11
2 2
2 3 3 2 1 3 3
+ u1*v2*v3 *(a12 *b11 + ---*a12*a13 *b11) + ---*u1*v3 *a13 *b11
2 2
1 4 2 2 1 2 2 2 2 1 2 2 2 2
- ----*v1 *a12 *b11 - ---*v1 *v2 *a12 *b11 - ---*v1 *v3 *a12 *b11
16 8 8
{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**3*v1*a12*a13**4 - u1**2*u2*v3*a12**2*a13**3 + u1**2*u3*v2*a13**5 - 2*
u1**2*u3*v3*a12*a13**4 + 1/2*u1**2*v1*v2*a13**4*b11 - 1/2*u1**2*v1*v3*a12*a13**3
*b11 - 1/4*u1*u2*v1**2*a12**2*a13**2*b11 - 1/4*u1*u3*v1**2*a12*a13**3*b11 - 1/8*
u1*v1**3*a12*a13**2*b11**2 - 1/2*u1*v2**3*a12**2*a13**2*c22 + 1/2*u1*v2**2*v3*
a12**3*a13*c22 + u1*v2*v3**2*( - a12**4*c22 - 3/2*a12**2*a13**2*c22) - 1/2*u1*v3
**3*a12*a13**3*c22 + v1**4*(1/16*a12**3*b11*c22 + 1/16*a12*a13**2*b11*c22) + 1/8
*v1**2*v2**2*a12**3*b11*c22 + 1/8*v1**2*v3**2*a12**3*b11*c22$
FI=u1*v1**2*v2*a12*a13**2 + u1*v1**2*v3*a13**3 + u1*v2**3*a12*a13**2 - u1*v2**2*
v3*a12**2*a13 + u1*v2*v3**2*(2*a12**3 + 3*a12*a13**2) + u1*v3**3*a13**3 + v1**4*
( - 1/8*a12**2*b11 + 1/8*a13**2*b11) - 1/4*v1**2*v2**2*a12**2*b11 - 1/4*v1**2*v3
**2*a12**2*b11$
FI=u1**2*v2**2*a13**4 - 2*u1**2*v2*v3*a12*a13**3 + u1**2*v3**2*a12**2*a13**2 + 1
/2*u1*v2**3*a12*a13**2*b11 - 1/2*u1*v2**2*v3*a12**2*a13*b11 + u1*v2*v3**2*(a12**
3*b11 + 3/2*a12*a13**2*b11) + 1/2*u1*v3**3*a13**3*b11 - 1/16*v1**4*a12**2*b11**2
- 1/8*v1**2*v2**2*a12**2*b11**2 - 1/8*v1**2*v3**2*a12**2*b11**2$