Solution 2 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
r49=0
r410=0
r411=0
r412=0
r413=0
r414=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
r425=2*r427
r426=0
r428=0
r429=0
r430=r427
r431=0
r432=0
r433=0
1
r434=---*r464
2
r435=0
r439=0
r442=0
r444=0
r445=0
r448=0
r450=0
r451=0
r453=0
r454=0
r455=r427
r458=0
r460=r427
r461=0
r463=0
r465=0
r467=0
r468=0
1
r469=---*r464
2
r470=r477
- a13*r477
r471=-------------
a12
r472=r477
- a13*r477
r473=-------------
a12
r474=0
r475=0
r476=0
- a13*r477
r478=-------------
a12
r479=0
r480=0
r481=0
r484=0
r485=0
a12*r496
r486=----------
a13
r487= - r496
r488=0
r489=r4103
r490=0
r493=0
r495=0
r498=0
a12*r4103
r499=-----------
a13
a12*r496
r4100=----------
a13
r4102=0
a12*r4103
r4104=-----------
a13
2 2 2
a12 *r4107 - a12 *r427 + a13 *r427
r4105=------------------------------------
2
a13
- 2*a12*r4107 + 2*a12*r427
r4106=-----------------------------
a13
r4108=0
r4109=0
r4110=2*r427
- 2*a12*r4112
r4111=----------------
a13
2 3 2 2
- ---*a12 *r496 - ---*a12*a13 *r496
3 3
r4113=--------------------------------------
3
a13
2
- a12 *r4112
r4115=---------------
2
a13
2
- a12 *r496
r4117=--------------
2
a13
- 2*a12*r464
r4118=---------------
a13
2 2
- a12 *r464 + a13 *r464
r4119=--------------------------
2
a13
1 3 2 2
- ---*a12 *r496 + ---*a12*a13 *r496
3 3
r4120=--------------------------------------
3
a13
r4121= - r496
- a12*r4112
r4122=--------------
a13
r4123=r4103
a12*r4103
r4124=-----------
a13
1
r4125=---*r464
2
c33=0
c23=0
c22=0
c13=0
c12=0
b33=0
b31=0
b21=0
b13=0
b11=0
r483=2*r427
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:
r427, r4107, r477, r4112, r496, r4103, r464, 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}
Relevance for the application:
Modulo the following equation:
2 2
0=a12 + a13
the system of equations related to the Hamiltonian HAM:
HAM=2*u1*u2*a12 + 2*u1*u3*a13
has apart from the Hamiltonian and Casimirs the following 7 first integrals:
1 4 2 2 2 2 2 2 1 4 2
FI=---*u1 *a13 + u1 *u2 *( - a12 + a13 ) - 2*u1 *u2*u3*a12*a13 + ---*u2 *a13
2 2
2 2 2 1 4 2
+ u2 *u3 *a13 + ---*u3 *a13
2
{HAM,FI} = too large to simplify
3 3 3 2 2
FI=u1 *u2*a12 + u1 *u3*a13 + u1*u2 *a12 + u1*u2 *u3*a13 + u1*u2*u3 *a12
3
+ u1*u3 *a13
{HAM,FI} = too large to simplify
3 3 3 1 3 2 2 2 2
FI= - u1 *v2*a13 + u1 *v3*( - ---*a12 + ---*a12*a13 ) - u1 *u2*v1*a12 *a13
3 3
2 2 3 2 2 2 2
+ u1 *u3*v1*( - ---*a12 - ---*a12*a13 ) + u1*u2 *v3*a12*a13
3 3
3 2 3 2 2
+ u1*u2*u3*v3*a13 - u1*u3 *v2*a13 + u1*u3 *v3*a12*a13
{HAM,FI} = too large to simplify
3 2 2 2 2 2
FI= - u1 *v1*a12*a13 - u1 *u2*v3*a12 + u1 *u3*v2*a13 - 2*u1 *u3*v3*a12*a13
{HAM,FI} = too large to simplify
2 2 3 2 2
FI= - u1*v1 *v2*a13 + u1*v1 *v3*a12 - u1*v2 *a13 + u1*v2 *v3*a12 - u1*v2*v3 *a13
3
+ u1*v3 *a12
{HAM,FI} = too large to simplify
2 2 2 2 2 2 2
FI=u1 *v2 *a13 - 2*u1 *v2*v3*a12*a13 + u1 *v3 *a12
{HAM,FI} = too large to simplify
2 2 2 2 2 2 2 2
FI=2*u1 *v1 *a13 + 2*u1 *v2*v3*a12*a13 + u1 *v3 *( - a12 + a13 )
2 2 2 2 2 2 2 2 2 2
+ 2*u1*u3*v1*v3*a13 + u2 *v1 *a13 + u2 *v3 *a13 + u3 *v1 *a13
2 2 2 2 2 2
+ u3 *v2 *a13 + 2*u3 *v3 *a13
{HAM,FI} = too large to simplify
And again in machine readable form:
HAM=2*u1*u2*a12 + 2*u1*u3*a13$
FI=1/2*u1**4*a13**2 + u1**2*u2**2*( - a12**2 + a13**2) - 2*u1**2*u2*u3*a12*a13 +
1/2*u2**4*a13**2 + u2**2*u3**2*a13**2 + 1/2*u3**4*a13**2$
FI=u1**3*u2*a12 + u1**3*u3*a13 + u1*u2**3*a12 + u1*u2**2*u3*a13 + u1*u2*u3**2*
a12 + u1*u3**3*a13$
FI= - u1**3*v2*a13**3 + u1**3*v3*( - 1/3*a12**3 + 2/3*a12*a13**2) - u1**2*u2*v1*
a12**2*a13 + u1**2*u3*v1*( - 2/3*a12**3 - 2/3*a12*a13**2) + u1*u2**2*v3*a12*a13
**2 + u1*u2*u3*v3*a13**3 - u1*u3**2*v2*a13**3 + u1*u3**2*v3*a12*a13**2$
FI= - u1**3*v1*a12*a13 - u1**2*u2*v3*a12**2 + u1**2*u3*v2*a13**2 - 2*u1**2*u3*v3
*a12*a13$
FI= - u1*v1**2*v2*a13 + u1*v1**2*v3*a12 - u1*v2**3*a13 + u1*v2**2*v3*a12 - u1*v2
*v3**2*a13 + u1*v3**3*a12$
FI=u1**2*v2**2*a13**2 - 2*u1**2*v2*v3*a12*a13 + u1**2*v3**2*a12**2$
FI=2*u1**2*v1**2*a13**2 + 2*u1**2*v2*v3*a12*a13 + u1**2*v3**2*( - a12**2 + a13**
2) + 2*u1*u3*v1*v3*a13**2 + u2**2*v1**2*a13**2 + u2**2*v3**2*a13**2 + u3**2*v1**
2*a13**2 + u3**2*v2**2*a13**2 + 2*u3**2*v3**2*a13**2$