Solution 21 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
m1*n1*n3*r42
r10=--------------
2
2*a33*c23
2
- m2*n1 *r42
r11=---------------
2
4*a33*c23
2
m1*n1 *r42
r12=------------
2
4*a33*c23
2
- n1 *n3*r42
r13=---------------
2
4*a33*c23
r14=0
3
- n1 *r42
r15=------------
2
4*a33*c23
2 2
- m1 *r42 - m2 *r42
r20=----------------------
2
4*c23
2
- n1 *r42
r21=------------
2*a33*c23
r22=0
r23=0
r24=0
m1*n1*r42
r26=-----------
2
2*c23
r27=0
r28=0
r210=0
- m2*n1*r42
r212=--------------
2
2*c23
r213=0
2
n1 *r42
r214=---------
2
4*c23
r215=0
m2*n1*r42
r216=-----------
2
2*c23
r217=0
r218=0
r219=0
2
n1 *r42
r220=---------
2
4*c23
r30=0
r31=0
m2*r42
r32=--------
c23
r33=0
r34=0
r35=0
r36=0
m2*r42
r37=--------
c23
r38=0
r39=0
r310=0
r311=0
r312=0
r313=0
r314=0
r315=0
r316=0
r317=0
r318=0
r319=0
r320=0
- n1*r42
r323=-----------
c23
r325=0
r326=0
r328=0
r329=0
r330=0
r332=0
r333=0
r334=0
r335=0
n1*r42
r336=--------
c23
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
r40=0
r41=0
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
r49=r42
r410=0
r411=0
r412=0
r413=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
r425=0
r426=0
r427=0
r428=0
r429=0
r431=0
r432=0
r433=0
r435=0
r439=0
r442=0
r444=0
r445=0
r448=0
r450=0
r451=0
r453=0
r454=0
r455=0
r458=0
r460=0
r461=0
r463=0
r464=0
r465=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
r493=0
r495=0
r496=0
r498=0
r499=0
r4100=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4107=0
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m3=0
n2=0
c33=0
c22=0
c13=0
c12=0
c11=0
b33=0
b32=0
b31=0
b23=0
b22=0
b21=0
b13=0
b12=0
b11=0
a23=0
a22=0
a13=0
a12=0
a11=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:
r42, m1, m2, n3, n1, c23, a33
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.
{r42,n1,a33,c23}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11,
a12,
a13,
a22,
a23,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
b33,
c11,
c12,
c13,
c22,
c33,
n2,
m3}$
The system of equations related to the Hamiltonian HAM:
2
HAM=u1*n1 + u3 *a33 + u3*n3 + v1*m1 + 2*v2*v3*c23 + v2*m2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 3
FI=u1 *a33*n1 + 4*u1*v2*v3*a33*c23*n1 + 2*u1*v2*a33*m2*n1 - u1*n1
2 2
+ u2 *a33*n1 - 4*u2*v1*v3*a33*c23*n1 - 2*u2*v1*a33*m2*n1
2 2 2 2 2
+ 2*u3*v3*a33*m1*n1 - u3*n1 *n3 + 4*v1 *v3 *a33*c23 + 4*v1 *v3*a33*c23*m2
2 2 2 2 2 2
+ v1*m1*n1 + 4*v2 *v3 *a33*c23 + 4*v2 *v3*a33*c23*m2 - 2*v2*v3*c23*n1
2 2 2 2
- v2*m2*n1 + v3 *( - a33*m1 - a33*m2 ) + 2*v3*m1*n1*n3
= a product of the elements of: {4,
2 2 3
u1 *a33*n1 u1*v2*a33*m2*n1 - u1*n1
------------- + u1*v2*v3*a33*c23*n1 + ----------------- + -----------
4 2 4
2 2
u2 *a33*n1 - u2*v1*a33*m2*n1
+ ------------- - u2*v1*v3*a33*c23*n1 + --------------------
4 2
2
u3*v3*a33*m1*n1 - u3*n1 *n3 2 2 2 2
+ ----------------- + -------------- + v1 *v3 *a33*c23 + v1 *v3*a33*c23*m2
2 4
2 2
v1*m1*n1 2 2 2 2 - v2*v3*c23*n1
+ ----------- + v2 *v3 *a33*c23 + v2 *v3*a33*c23*m2 + ------------------
4 2
2 2 2
- v2*m2*n1 2 - a33*m1 - a33*m2 v3*m1*n1*n3
+ -------------- + v3 *---------------------- + -------------}
4 4 2
{HAM,FI} = 0
And again in machine readable form:
HAM=u1*n1 + u3**2*a33 + u3*n3 + v1*m1 + 2*v2*v3*c23 + v2*m2$
FI=u1**2*a33*n1**2 + 4*u1*v2*v3*a33*c23*n1 + 2*u1*v2*a33*m2*n1 - u1*n1**3 + u2**
2*a33*n1**2 - 4*u2*v1*v3*a33*c23*n1 - 2*u2*v1*a33*m2*n1 + 2*u3*v3*a33*m1*n1 - u3
*n1**2*n3 + 4*v1**2*v3**2*a33*c23**2 + 4*v1**2*v3*a33*c23*m2 + v1*m1*n1**2 + 4*
v2**2*v3**2*a33*c23**2 + 4*v2**2*v3*a33*c23*m2 - 2*v2*v3*c23*n1**2 - v2*m2*n1**2
+ v3**2*( - a33*m1**2 - a33*m2**2) + 2*v3*m1*n1*n3$