Solution 1 to problem h
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem h
Equations
The following unsolved equations remain:
2 2
0=a11 + b12 *kap
Expressions
The solution is given through the following expressions:
r40=0
r41=0
r42=0
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
r49=0
r410=0
r411=0
r412=0
r413=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r423=0
r424=0
r426=0
r427= - r4110 - r460
r428=0
r429=r4109
r430= - r4110
- a11*r4109 + b12*r488
r431=-------------------------
b12
r432=0
r433=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r446=2*r4110 + 2*r460
r447=0
r448= - 2*r4109
r449=0
r450=0
r451=0
- 2*a11*r4109 + b12*r488
r452=---------------------------
b12
- 2*a11*r460 - b12*r487
r453=--------------------------
b12
r454=0
r455=0
r456=0
r457=r4110 + r460
r458=0
r459= - r4109
2*a11*r4109
r461=-------------
b12
r462=0
r463=0
2 2 2
a11 *r460 + a11*b12*r487 - b12 *kap*r4110 - b12 *kap*r460
r464=-----------------------------------------------------------
2
b12
r465=0
a11*r4109
r466=-----------
b12
- 2*a11*r460
r467=---------------
b12
r468=0
2
a11 *r460
r469=-----------
2
b12
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r479=0
r480=0
r482=0
r483=2*r4110
r484=0
r485=0
r486=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=2*r4110
r495= - 2*r4109
2*a11*r460
r496=------------
b12
r497=0
r498=0
2 2
2*a11 *r4109 + 3*b12 *kap*r4109
r499=---------------------------------
2
b12
r4100=0
2*a11*r460
r4101=------------
b12
3*a11*r4109
r4102=-------------
b12
r4103=0
2
- a11 *r4109
r4104=---------------
2
b12
r4105=0
r4106=0
r4108=0
r4111=0
r4112=0
r4113=0
2 2
a11 *r460 + a11*b12*r487 - b12 *kap*r4110
r4114=-------------------------------------------
2
b12
r4115=0
- a11*r4109
r4116=--------------
b12
r4117=0
r4118=0
2
a11 *r460
r4119=-----------
2
b12
r4120=0
r4121=0
a11*r4109
r4122=-----------
b12
r4123=0
2
- a11 *r4109
r4124=---------------
2
b12
r4125=0
1
a33=---*a11
2
a23=0
a22=a11
a13=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:
r434, r460, r487, r488, r4110, r4109, a11, b12
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.
{b12,r4109,a11}
Relevance for the application:
Modulo the following equation:
2 2
0=a11 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2 1 2
HAM=u1 *a11 + u1*v2*b12 + u2 *a11 - u2*v1*b12 + ---*u3 *a11
2
has apart from the Hamiltonian and Casimirs the following 6 first integrals:
3 2 3 2 2 2
FI= - u1 *u2*a11 + u1 *v1*a11*b12 - u1 *u2*v2*a11*b12 + u1 *v1*v2*b12
3 2 2 2 2 2
- u1*u2 *a11 + 3*u1*u2 *v1*a11*b12 + u1*u2*u3 *(2*a11 + 3*b12 *kap)
2 2 3 2 2 2
- 2*u1*u2*v1 *b12 + u2 *v2*a11*b12 + 2*u2 *u3*v3*a11*b12 - u2 *v1*v2*b12
2 2 3 2 2
- 2*u2*u3 *v2*a11*b12 - 2*u2*u3*v1*v3*b12 - u3 *v3*a11*b12 + u3 *v1*v2*b12
which is not further factorizable.
{HAM,FI} = 0
2 2 2 2 2 2
FI= - u1 *u3 *kap + u1 *v1 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2 *u3 *kap
2 2 2 2 2 2
+ u2 *v2 + 2*u2*u3*v2*v3 - u3 *v1 - u3 *v2
which is not further factorizable.
{HAM,FI} = 0
2 2 3
FI=u1*u3 *v1 + u2*u3 *v2 + u3 *v3
2
=u3 *(u2*v2 + u3*v3 + u1*v1)
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1 *u3 *a11 + u1*u3 *v2*b12 + u2 *u3 *a11 - u2*u3 *v1*b12
2 2 2
=u3 *((u1 + u2 )*a11 + (u1*v2 - u2*v1)*b12)
{HAM,FI} = 0
2 2 2 2 2 2 2
FI=u1 *u2 *a11 + u1 *u3 *a11 + 2*u1*u2 *v2*a11*b12 + 2*u1*u2*u3*v3*a11*b12
4 2 3 2 2 2 2 2 2 2
+ u2 *a11 - 2*u2 *v1*a11*b12 + u2 *u3 *(a11 - b12 *kap) + u2 *v1 *b12
2 2 2 2 2 2 2 2
+ u2 *v2 *b12 - 2*u2*u3 *v1*a11*b12 + 2*u2*u3*v2*v3*b12 - u3 *v2 *b12
which is not further factorizable.
{HAM,FI} = 0
4
FI=u3
4
=u3
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*v2*b12 + u2**2*a11 - u2*v1*b12 + 1/2*u3**2*a11$
FI= - u1**3*u2*a11**2 + u1**3*v1*a11*b12 - u1**2*u2*v2*a11*b12 + u1**2*v1*v2*b12
**2 - u1*u2**3*a11**2 + 3*u1*u2**2*v1*a11*b12 + u1*u2*u3**2*(2*a11**2 + 3*b12**2
*kap) - 2*u1*u2*v1**2*b12**2 + u2**3*v2*a11*b12 + 2*u2**2*u3*v3*a11*b12 - u2**2*
v1*v2*b12**2 - 2*u2*u3**2*v2*a11*b12 - 2*u2*u3*v1*v3*b12**2 - u3**3*v3*a11*b12 +
u3**2*v1*v2*b12**2$
FI= - u1**2*u3**2*kap + u1**2*v1**2 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2**2*u3**
2*kap + u2**2*v2**2 + 2*u2*u3*v2*v3 - u3**2*v1**2 - u3**2*v2**2$
FI=u1*u3**2*v1 + u2*u3**2*v2 + u3**3*v3$
FI=u1**2*u3**2*a11 + u1*u3**2*v2*b12 + u2**2*u3**2*a11 - u2*u3**2*v1*b12$
FI=u1**2*u2**2*a11**2 + u1**2*u3**2*a11**2 + 2*u1*u2**2*v2*a11*b12 + 2*u1*u2*u3*
v3*a11*b12 + u2**4*a11**2 - 2*u2**3*v1*a11*b12 + u2**2*u3**2*(a11**2 - b12**2*
kap) + u2**2*v1**2*b12**2 + u2**2*v2**2*b12**2 - 2*u2*u3**2*v1*a11*b12 + 2*u2*u3
*v2*v3*b12**2 - u3**2*v2**2*b12**2$
FI=u3**4$