Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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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
1
---*b31*r439
2
r49=--------------
a23
r410=0
1
---*b31*r439
2
r411=--------------
a23
r412=0
r413=0
1
---*b31*r444
2
r414=--------------
a23
r415=0
r416=0
r417=0
r418=0
1
---*a33*r439
2
r419=--------------
a23
r420=0
1
---*a33*r439
2
r421=--------------
a23
r422=0
r423=0
1 2 1 2 1 2
---*a23 *a33*r444 + ---*a23*b31 *r463 - ---*a33*b31 *r4117
2 4 8
r424=------------------------------------------------------------
3
a23
r425=0
r426=0
r427=0
r428=0
r429=0
1 2 1 1 2
---*a23 *b31*r4117 + ---*a23*a33*b31*r463 - ---*a33 *b31*r4117
2 2 4
r430=----------------------------------------------------------------
3
a23
r431=0
r432=0
1 2 1 2 1 3
---*a23 *a33*r4117 + ---*a23*a33 *r463 - ---*a33 *r4117
2 4 8
r433=---------------------------------------------------------
3
a23
r434=0
r435=0
r436=0
r437=0
r438=0
r440=0
r442=0
r443=0
r445=0
r446=0
r447=0
r448=0
r449=0
1
a23*b31*r463 - ---*a33*b31*r4117
2
r450=----------------------------------
2
a23
r451=0
r452=0
2 1 2
a23 *r4117 + a23*a33*r463 - ---*a33 *r4117
2
r453=--------------------------------------------
2
a23
r454=0
r455=0
r456=0
r457=0
r458=0
r459=0
1
---*b31*r4117
2
r460=---------------
a23
r461=0
r462=0
r464=0
r465=0
r466=0
r467=r4117
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
r4107=0
r4108=0
r4109=0
1
---*b31*r4117
2
r4110=---------------
a23
r4111=0
r4112=0
1
---*a33*r4117
2
r4113=---------------
a23
r4114=0
r4115=0
r4116=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
c22=0
b33=0
b32=0
r441=r439
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:
r439, r444, r463, r4117, b31, a33, c11, 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.
{a23,c11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2
HAM=2*u2*u3*a23 + u3 *a33 + u3*v1*b31 + v1 *c11
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*v1*a23 + ---*u1 *u3 *v1*a23 *a33 + ---*u1 *u3*v1 *a23 *b31
2 2
3 3 1 2 2 2 3 3 1 2
+ u2 *u3*v1*a23 + ---*u2 *u3*v1 *a23 *b31 + u2*u3 *v1*(a23 - ---*a23*a33 )
2 2
1 2 2 4 1 2 1 3
- ---*u2*u3 *v1 *a23*a33*b31 + u3 *v1*(---*a23 *a33 - ---*a33 )
2 2 8
3 2 1 2 1 2 1 2 3 2
+ u3 *v1 *(---*a23 *b31 - ---*a33 *b31) - ---*u3 *v1 *a33*b31
2 4 8
1
= a product of the elements of: {---,
8
v1,
u3,
2*u2*a23 + u3*a33 + v1*b31,
2 2 2 2 2 2 2
4*u1 *a23 + 4*u2 *a23 - 2*u2*u3*a23*a33 + u3 *(4*a23 - a33 )
- u3*v1*a33*b31}
{HAM,FI} = 0
2 2 2 3 2 2 1 4 2
FI=u2 *u3 *v1*a23 + u2*u3 *v1*a23*a33 + u2*u3 *v1 *a23*b31 + ---*u3 *v1*a33
4
1 3 2 1 2 3 2
+ ---*u3 *v1 *a33*b31 + ---*u3 *v1 *b31
2 4
1
= a product of the elements of: {---,
4
u3,
u3,
2*u2*a23 + u3*a33 + v1*b31,
2*u2*a23 + u3*a33 + v1*b31,
v1}
{HAM,FI} = 0
3 1 2 3 1 4
FI=u2*u3*v1 *a23 + ---*u3 *v1 *a33 + ---*u3*v1 *b31
2 2
1
= a product of the elements of: {---,
2
v1,
v1,
v1,
u3,
2*u2*a23 + u3*a33 + v1*b31}
{HAM,FI} = 0
2 2 1 2 2 1 2 2
FI=u2*u3*v1*v2 *a23 + u2*u3*v1*v3 *a23 + ---*u3 *v1*v2 *a33 + ---*u3 *v1*v3 *a33
2 2
1 2 2 1 2 2
+ ---*u3*v1 *v2 *b31 + ---*u3*v1 *v3 *b31
2 2
1
= a product of the elements of: {---,
2
2 2
v2 + v3 ,
v1,
u3,
2*u2*a23 + u3*a33 + v1*b31}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u2*u3*a23 + u3**2*a33 + u3*v1*b31 + v1**2*c11$
FI=u1**2*u2*u3*v1*a23**3 + 1/2*u1**2*u3**2*v1*a23**2*a33 + 1/2*u1**2*u3*v1**2*
a23**2*b31 + u2**3*u3*v1*a23**3 + 1/2*u2**2*u3*v1**2*a23**2*b31 + u2*u3**3*v1*(
a23**3 - 1/2*a23*a33**2) - 1/2*u2*u3**2*v1**2*a23*a33*b31 + u3**4*v1*(1/2*a23**2
*a33 - 1/8*a33**3) + u3**3*v1**2*(1/2*a23**2*b31 - 1/4*a33**2*b31) - 1/8*u3**2*
v1**3*a33*b31**2$
FI=u2**2*u3**2*v1*a23**2 + u2*u3**3*v1*a23*a33 + u2*u3**2*v1**2*a23*b31 + 1/4*u3
**4*v1*a33**2 + 1/2*u3**3*v1**2*a33*b31 + 1/4*u3**2*v1**3*b31**2$
FI=u2*u3*v1**3*a23 + 1/2*u3**2*v1**3*a33 + 1/2*u3*v1**4*b31$
FI=u2*u3*v1*v2**2*a23 + u2*u3*v1*v3**2*a23 + 1/2*u3**2*v1*v2**2*a33 + 1/2*u3**2*
v1*v3**2*a33 + 1/2*u3*v1**2*v2**2*b31 + 1/2*u3*v1**2*v3**2*b31$