Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2 2
0=a11 + 4*a13 + 4*a23 + 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
r418=0
r421=0
r423=0
r424=0
1
r427=---*r494
2
r429=0
1
r430=---*r494
2
2*a13*r494
r432=------------
b12
- 2*a23*r494
r433=---------------
b12
2 2
2*a13 *r494 + 2*a23 *r494
r434=---------------------------
2
b12
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r447=0
r449=0
r450=0
r452=0
- a11*r494
r453=-------------
b12
2*a11*a23*r494
r454=----------------
2
b12
r455=0
r456=0
1
r457=---*r494
2
r458=0
r459=0
r460=0
2*a13*r494
r462=------------
b12
r463=0
2 2 2 1 2
a11 *r494 + 4*a13 *r494 + 2*a23 *r494 + ---*b12 *kap*r494
2
r464=-----------------------------------------------------------
2
b12
r465=0
r466=0
r467=0
r468=0
r469=0
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r479=0
r482=0
r484=0
r485=0
a11*r494
r487=----------
b12
r488=0
2*a11*a13*r494
r489=----------------
2
b12
r490=0
r491=0
r492=0
r493=0
r495=0
- 2*a23*r494
r497=---------------
b12
2*a13*r494
r498=------------
b12
- 4*a13*a23*r494
r499=-------------------
2
b12
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
1
r4110=---*r494
2
r4112=0
- 2*a23*r494
r4113=---------------
b12
2 2 2 1 2
a11 *r494 + 2*a13 *r494 + 4*a23 *r494 + ---*b12 *kap*r494
2
r4114=-----------------------------------------------------------
2
b12
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
a33=0
a22=a11
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:
r494, a11, a13, a23, 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.
{a13,r494,b12}
Relevance for the application:
Modulo the following equation:
2 2 2 2
0=a11 + 4*a13 + 4*a23 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2
HAM=u1 *a11 + 2*u1*u3*a13 + u1*v2*b12 + u2 *a11 + 2*u2*u3*a23 - u2*v1*b12
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 2 1 2 2
FI=u1 *u3 *(a11 + 2*a13 + 4*a23 + ---*b12 *kap) - 2*u1 *u3*v1*a23*b12
2
1 2 2 2 2
+ ---*u1 *v1 *b12 - 4*u1*u2*u3 *a13*a23 + 2*u1*u2*u3*v1*a13*b12
2
2 3
- 2*u1*u2*u3*v2*a23*b12 + u1*u2*v1*v2*b12 + 2*u1*u3 *a11*a13
2 2 2 2 2 2 1 2
+ u1*u3 *v2*a11*b12 + u2 *u3 *(a11 + 4*a13 + 2*a23 + ---*b12 *kap)
2
2 1 2 2 2 3
+ 2*u2 *u3*v2*a13*b12 + ---*u2 *v2 *b12 + 2*u2*u3 *a11*a23
2
2 4 2 2 3
- u2*u3 *v1*a11*b12 + u3 *(2*a13 + 2*a23 ) - 2*u3 *v1*a23*b12
3 1 2 2 2 1 2 2 2
+ 2*u3 *v2*a13*b12 + ---*u3 *v1 *b12 + ---*u3 *v2 *b12
2 2
3 2 2 3
{HAM,FI} = u1 *u3*v1*( - 4*a13 *b12 - 4*a23 *b12 - b12 *kap)
2 2 2 3
+ u1 *u2*u3*v2*( - 4*a13 *b12 - 4*a23 *b12 - b12 *kap)
2 2
+ 4*u1 *u3 *v1*a11*a13*b12
2 2 2 2 3
+ u1 *u3 *v3*( - 4*a13 *b12 - 4*a23 *b12 - b12 *kap)
2 2
+ 2*u1 *u3*v1*v2*a11*b12
2 2 2 3
+ u1*u2 *u3*v1*( - 4*a13 *b12 - 4*a23 *b12 - b12 *kap)
2 2
+ 4*u1*u2*u3 *v1*a11*a23*b12 + 4*u1*u2*u3 *v2*a11*a13*b12
2 2 2 2
- 2*u1*u2*u3*v1 *a11*b12 + 2*u1*u2*u3*v2 *a11*b12
3 2 2 3
+ u1*u3 *v1*(4*a13 *b12 + 4*a23 *b12) + 4*u1*u3 *v3*a11*a13*b12
2 2 2 2 2
- 4*u1*u3 *v1 *a23*b12 + 4*u1*u3 *v1*v2*a13*b12
2 2 3 3 2 3
+ 2*u1*u3 *v2*v3*a11*b12 + u1*u3*v1 *b12 + u1*u3*v1*v2 *b12
3 2 2 3
+ u2 *u3*v2*( - 4*a13 *b12 - 4*a23 *b12 - b12 *kap)
2 2
+ 4*u2 *u3 *v2*a11*a23*b12
2 2 2 2 3
+ u2 *u3 *v3*( - 4*a13 *b12 - 4*a23 *b12 - b12 *kap)
2 2 3 2 2
- 2*u2 *u3*v1*v2*a11*b12 + u2*u3 *v2*(4*a13 *b12 + 4*a23 *b12)
3 2 2
+ 4*u2*u3 *v3*a11*a23*b12 - 4*u2*u3 *v1*v2*a23*b12
2 2 2 2 2
- 2*u2*u3 *v1*v3*a11*b12 + 4*u2*u3 *v2 *a13*b12
2 3 3 3
+ u2*u3*v1 *v2*b12 + u2*u3*v2 *b12
4 2 2 3 2
+ u3 *v3*(4*a13 *b12 + 4*a23 *b12) - 4*u3 *v1*v3*a23*b12
3 2 2 2 3 2 2 3
+ 4*u3 *v2*v3*a13*b12 + u3 *v1 *v3*b12 + u3 *v2 *v3*b12
And again in machine readable form:
HAM=u1**2*a11 + 2*u1*u3*a13 + u1*v2*b12 + u2**2*a11 + 2*u2*u3*a23 - u2*v1*b12$
FI=u1**2*u3**2*(a11**2 + 2*a13**2 + 4*a23**2 + 1/2*b12**2*kap) - 2*u1**2*u3*v1*
a23*b12 + 1/2*u1**2*v1**2*b12**2 - 4*u1*u2*u3**2*a13*a23 + 2*u1*u2*u3*v1*a13*b12
- 2*u1*u2*u3*v2*a23*b12 + u1*u2*v1*v2*b12**2 + 2*u1*u3**3*a11*a13 + u1*u3**2*v2
*a11*b12 + u2**2*u3**2*(a11**2 + 4*a13**2 + 2*a23**2 + 1/2*b12**2*kap) + 2*u2**2
*u3*v2*a13*b12 + 1/2*u2**2*v2**2*b12**2 + 2*u2*u3**3*a11*a23 - u2*u3**2*v1*a11*
b12 + u3**4*(2*a13**2 + 2*a23**2) - 2*u3**3*v1*a23*b12 + 2*u3**3*v2*a13*b12 + 1/
2*u3**2*v1**2*b12**2 + 1/2*u3**2*v2**2*b12**2$