Solution 2 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
1
a23*r4106 + ---*a33*r446
2
r426=--------------------------
a23
1
a23*r4108 + ---*a33*r448
2
r428=--------------------------
a23
1
a23*r4109 + ---*a33*r449
2
r429=--------------------------
a23
r430=0
r431=0
r432=0
r433=0
r434=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r451=0
r452=0
r453=0
r454=0
1
a23*r425 - ---*a33*r445
2
r455=-------------------------
a23
r456=r4106
1
a23*r427 - ---*a33*r447
2
r457=-------------------------
a23
r458=r4108
r459=r4109
r460=0
r461=0
r462=0
r463=0
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
r478=0
r479=0
r480=0
r481=0
r482=0
r483=0
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=0
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
1
a23*r425 - ---*a33*r445
2
r4105=-------------------------
a23
1
a23*r427 - ---*a33*r447
2
r4107=-------------------------
a23
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
c22=0
c11=0
b33=0
b32=0
b31=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:
r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r410,
r411, r412, r413, r425, r427, r445, r446, r447, r448,
r449, r4106, r4108, r4109, a33, a23
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.
{a33,a23}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2
HAM=2*u2*u3*a23 + u3 *a33
has apart from the Hamiltonian and Casimirs the following 24 first integrals:
2 2 2
FI=u1 *v1*v2 + u2 *v1*v2 + u3 *v1*v2
2 2 2
= a product of the elements of: {v2,v1,u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2
FI=u1 *v1*v3 + u2 *v1*v3 + u3 *v1*v3
2 2 2
= a product of the elements of: {v3,v1,u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2
FI=u1 *v2*v3 + u2 *v2*v3 + u3 *v2*v3
2 2 2
= a product of the elements of: {v3,v2,u1 + u2 + u3 }
{HAM,FI} = 0
1 2
FI=u2*u3*v1*v2*a23 + ---*u3 *v1*v2*a33
2
1
= a product of the elements of: {---,
2
v2,
v1,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
1 2
FI=u2*u3*v1*v3*a23 + ---*u3 *v1*v3*a33
2
1
= a product of the elements of: {---,
2
v3,
v1,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
1 2 2 1 2 2 2
FI= - ---*u1 *v2 *a33 - ---*u2 *v2 *a33 + u2*u3*v2 *a23
2 2
1
= a product of the elements of: {---,
2
v2,
v2,
2 2
- u1 *a33 - u2 *a33 + 2*u2*u3*a23}
{HAM,FI} = 0
1 2
FI=u2*u3*v2*v3*a23 + ---*u3 *v2*v3*a33
2
1
= a product of the elements of: {---,
2
v3,
v2,
u3,
2*u2*a23 + u3*a33}
{HAM,FI} = 0
1 2 2 1 2 2 2
FI= - ---*u1 *v3 *a33 - ---*u2 *v3 *a33 + u2*u3*v3 *a23
2 2
1
= a product of the elements of: {---,
2
v3,
v3,
2 2
- u1 *a33 - u2 *a33 + 2*u2*u3*a23}
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1 *v2 + u2 *v2 + u3 *v2
2 2 2
= a product of the elements of: {v2,v2,u1 + u2 + u3 }
{HAM,FI} = 0
2 2 2 2 2 2
FI=u1 *v3 + u2 *v3 + u3 *v3
2 2 2
= a product of the elements of: {v3,v3,u1 + u2 + u3 }
{HAM,FI} = 0
3
FI=v1 *v2
= a product of the elements of: {v1,v1,v1,v2}
{HAM,FI} = 0
3
FI=v1 *v3
= a product of the elements of: {v1,v1,v1,v3}
{HAM,FI} = 0
2 2
FI=v1 *v2
= a product of the elements of: {v2,v2,v1,v1}
{HAM,FI} = 0
2
FI=v1 *v2*v3
= a product of the elements of: {v1,v1,v3,v2}
{HAM,FI} = 0
2 2
FI=v1 *v3
= a product of the elements of: {v3,v3,v1,v1}
{HAM,FI} = 0
3
FI=v1*v2
= a product of the elements of: {v2,v2,v2,v1}
{HAM,FI} = 0
2
FI=v1*v2 *v3
= a product of the elements of: {v2,v2,v3,v1}
{HAM,FI} = 0
2
FI=v1*v2*v3
= a product of the elements of: {v3,v3,v2,v1}
{HAM,FI} = 0
3
FI=v1*v3
= a product of the elements of: {v3,v3,v3,v1}
{HAM,FI} = 0
4
FI=v2
= a product of the elements of: {v2,v2,v2,v2}
{HAM,FI} = 0
3
FI=v2 *v3
= a product of the elements of: {v2,v2,v2,v3}
{HAM,FI} = 0
2 2
FI=v2 *v3
= a product of the elements of: {v3,v3,v2,v2}
{HAM,FI} = 0
3
FI=v2*v3
= a product of the elements of: {v3,v3,v3,v2}
{HAM,FI} = 0
4
FI=v3
= a product of the elements of: {v3,v3,v3,v3}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33$
FI=u1**2*v1*v2 + u2**2*v1*v2 + u3**2*v1*v2$
FI=u1**2*v1*v3 + u2**2*v1*v3 + u3**2*v1*v3$
FI=u1**2*v2*v3 + u2**2*v2*v3 + u3**2*v2*v3$
FI=u2*u3*v1*v2*a23 + 1/2*u3**2*v1*v2*a33$
FI=u2*u3*v1*v3*a23 + 1/2*u3**2*v1*v3*a33$
FI= - 1/2*u1**2*v2**2*a33 - 1/2*u2**2*v2**2*a33 + u2*u3*v2**2*a23$
FI=u2*u3*v2*v3*a23 + 1/2*u3**2*v2*v3*a33$
FI= - 1/2*u1**2*v3**2*a33 - 1/2*u2**2*v3**2*a33 + u2*u3*v3**2*a23$
FI=u1**2*v2**2 + u2**2*v2**2 + u3**2*v2**2$
FI=u1**2*v3**2 + u2**2*v3**2 + u3**2*v3**2$
FI=v1**3*v2$
FI=v1**3*v3$
FI=v1**2*v2**2$
FI=v1**2*v2*v3$
FI=v1**2*v3**2$
FI=v1*v2**3$
FI=v1*v2**2*v3$
FI=v1*v2*v3**2$
FI=v1*v3**3$
FI=v2**4$
FI=v2**3*v3$
FI=v2**2*v3**2$
FI=v2*v3**3$
FI=v3**4$