Solution 5 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r40=0
r41=0
1
---*b32*r443
2
r42=--------------
a23
r43=0
1
---*b32*r438
2
r44=--------------
a23
r45=0
r46=0
r47=0
r48=0
r49=0
r410=0
1
---*b32*r443
2
r411=--------------
a23
r412=0
r413=0
r414=0
r415=0
1
---*a33*r443
2
r416=--------------
a23
r417=0
1 2 1 2 1 2
---*a23 *a33*r438 + ---*a23*b32 *r462 - ---*a33*b32 *r4116
2 4 8
r418=------------------------------------------------------------
3
a23
r419=0
r420=0
r421=0
r422=0
1
---*a33*r443
2
r423=--------------
a23
r424=0
r425=0
r426=0
1 2 1 1 2
---*a23 *b32*r4116 + ---*a23*a33*b32*r462 - ---*a33 *b32*r4116
2 2 4
r427=----------------------------------------------------------------
3
a23
r428=0
r429=0
r430=0
r431=0
1 2 1 2 1 3
---*a23 *a33*r4116 + ---*a23*a33 *r462 - ---*a33 *r4116
2 4 8
r432=---------------------------------------------------------
3
a23
r433=0
r434=0
r435=0
r436=r443
r437=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
r446=0
1
a23*b32*r462 - ---*a33*b32*r4116
2
r447=----------------------------------
2
a23
r448=0
r449=0
r450=0
r451=0
2 1 2
a23 *r4116 + a23*a33*r462 - ---*a33 *r4116
2
r452=--------------------------------------------
2
a23
r453=0
r454=0
r455=0
r456=0
1
---*b32*r4116
2
r457=---------------
a23
r458=0
r459=0
r460=0
r461=0
r463=0
r464=0
r465=0
r466=r4116
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
r4105=0
r4106=0
1
---*b32*r4116
2
r4107=---------------
a23
r4108=0
r4109=0
r4110=0
r4111=0
1
---*a33*r4116
2
r4112=---------------
a23
r4113=0
r4114=0
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
c11=0
b33=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:
r438, r443, r462, r4116, b32, a33, c22, 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.
{c22,a23}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v2*b32 + v2 *c22
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 3 1 2 2 2 1 2 2 2
FI=u1 *u2*u3*v2*a23 + ---*u1 *u3 *v2*a23 *a33 + ---*u1 *u3*v2 *a23 *b32
2 2
3 3 1 2 2 2 3 3 1 2
+ u2 *u3*v2*a23 + ---*u2 *u3*v2 *a23 *b32 + u2*u3 *v2*(a23 - ---*a23*a33 )
2 2
1 2 2 4 1 2 1 3
- ---*u2*u3 *v2 *a23*a33*b32 + u3 *v2*(---*a23 *a33 - ---*a33 )
2 2 8
3 2 1 2 1 2 1 2 3 2
+ u3 *v2 *(---*a23 *b32 - ---*a33 *b32) - ---*u3 *v2 *a33*b32
2 4 8
1
= a product of the elements of: {---,
8
v2,
u3,
2*u2*a23 + u3*a33 + v2*b32,
2 2 2 2 2 2 2
4*u1 *a23 + 4*u2 *a23 - 2*u2*u3*a23*a33 + u3 *(4*a23 - a33 )
- u3*v2*a33*b32}
{HAM,FI} = 0
2 2 2 3 2 2 1 4 2
FI=u2 *u3 *v2*a23 + u2*u3 *v2*a23*a33 + u2*u3 *v2 *a23*b32 + ---*u3 *v2*a33
4
1 3 2 1 2 3 2
+ ---*u3 *v2 *a33*b32 + ---*u3 *v2 *b32
2 4
1
= a product of the elements of: {---,
4
u3,
u3,
2*u2*a23 + u3*a33 + v2*b32,
2*u2*a23 + u3*a33 + v2*b32,
v2}
{HAM,FI} = 0
2 2 1 2 2 1 2 2
FI=u2*u3*v1 *v2*a23 + u2*u3*v2*v3 *a23 + ---*u3 *v1 *v2*a33 + ---*u3 *v2*v3 *a33
2 2
1 2 2 1 2 2
+ ---*u3*v1 *v2 *b32 + ---*u3*v2 *v3 *b32
2 2
1
= a product of the elements of: {---,
2
v2,
2 2
v1 + v3 ,
u3,
2*u2*a23 + u3*a33 + v2*b32}
{HAM,FI} = 0
3 1 2 3 1 4
FI=u2*u3*v2 *a23 + ---*u3 *v2 *a33 + ---*u3*v2 *b32
2 2
1
= a product of the elements of: {---,
2
v2,
v2,
v2,
u3,
2*u2*a23 + u3*a33 + v2*b32}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v2*b32 + v2**2*c22$
FI=u1**2*u2*u3*v2*a23**3 + 1/2*u1**2*u3**2*v2*a23**2*a33 + 1/2*u1**2*u3*v2**2*
a23**2*b32 + u2**3*u3*v2*a23**3 + 1/2*u2**2*u3*v2**2*a23**2*b32 + u2*u3**3*v2*(
a23**3 - 1/2*a23*a33**2) - 1/2*u2*u3**2*v2**2*a23*a33*b32 + u3**4*v2*(1/2*a23**2
*a33 - 1/8*a33**3) + u3**3*v2**2*(1/2*a23**2*b32 - 1/4*a33**2*b32) - 1/8*u3**2*
v2**3*a33*b32**2$
FI=u2**2*u3**2*v2*a23**2 + u2*u3**3*v2*a23*a33 + u2*u3**2*v2**2*a23*b32 + 1/4*u3
**4*v2*a33**2 + 1/2*u3**3*v2**2*a33*b32 + 1/4*u3**2*v2**3*b32**2$
FI=u2*u3*v1**2*v2*a23 + u2*u3*v2*v3**2*a23 + 1/2*u3**2*v1**2*v2*a33 + 1/2*u3**2*
v2*v3**2*a33 + 1/2*u3*v1**2*v2**2*b32 + 1/2*u3*v2**2*v3**2*b32$
FI=u2*u3*v2**3*a23 + 1/2*u3**2*v2**3*a33 + 1/2*u3*v2**4*b32$