Solution 15 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
1
---*m2*n2*n3*r42
2
r10=------------------
2
a33*c13
1 2
---*m2*n2 *r42
4
r11=----------------
2
a33*c13
1 2
- ---*m1*n2 *r42
4
r12=-------------------
2
a33*c13
1 2
- ---*n2 *n3*r42
4
r13=-------------------
2
a33*c13
1 3
- ---*n2 *r42
4
r14=----------------
2
a33*c13
r15=0
1 2 1 2
- ---*m1 *r42 - ---*m2 *r42
4 4
r20=------------------------------
2
c13
r21=0
r22=0
1 2
- ---*n2 *r42
2
r23=----------------
a33*c13
r24=0
1
---*m2*n2*r42
2
r26=---------------
2
c13
r27=0
r28=0
r210=0
1
---*m1*n2*r42
2
r212=---------------
2
c13
r213=0
1 2
---*n2 *r42
4
r214=-------------
2
c13
r215=0
1
- ---*m1*n2*r42
2
r216=------------------
2
c13
r217=0
r218=0
r219=0
1 2
---*n2 *r42
4
r220=-------------
2
c13
r30=0
r31=0
m1*r42
r32=--------
c13
r33=0
r34=0
r35=0
r36=0
m1*r42
r37=--------
c13
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
r321=0
r322=0
n2*r42
r323=--------
c13
r324=0
r325=0
r326=0
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
r335=0
- n2*r42
r336=-----------
c13
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
r346=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
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
r446=0
r447=0
r448=0
r449=0
r450=0
r451=0
r453=0
r454=0
r455=0
r456=0
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
r464=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
r4105=0
r4106=0
r4107=0
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m3=0
n1=0
c33=0
c23=0
c22=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, n2, c13, 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.
{c13,a33,r42,n2}
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,
c22,
c23,
c33,
n1,
m3}$
The system of equations related to the Hamiltonian HAM:
2
HAM=u2*n2 + u3 *a33 + u3*n3 + 2*v1*v3*c13 + v1*m1 + v2*m2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2
FI=u1 *a33*n2 - 4*u1*v2*v3*a33*c13*n2 - 2*u1*v2*a33*m1*n2 + u2 *a33*n2
3
+ 4*u2*v1*v3*a33*c13*n2 + 2*u2*v1*a33*m1*n2 - u2*n2 + 2*u3*v3*a33*m2*n2
2 2 2 2 2 2
- u3*n2 *n3 + 4*v1 *v3 *a33*c13 + 4*v1 *v3*a33*c13*m1 - 2*v1*v3*c13*n2
2 2 2 2 2 2
- v1*m1*n2 + 4*v2 *v3 *a33*c13 + 4*v2 *v3*a33*c13*m1 + v2*m2*n2
2 2 2
+ v3 *( - a33*m1 - a33*m2 ) + 2*v3*m2*n2*n3
= a product of the elements of: {4,
2 2 2 2
u1 *a33*n2 - u1*v2*a33*m1*n2 u2 *a33*n2
------------- - u1*v2*v3*a33*c13*n2 + -------------------- + -------------
4 2 4
3
u2*v1*a33*m1*n2 - u2*n2 u3*v3*a33*m2*n2
+ u2*v1*v3*a33*c13*n2 + ----------------- + ----------- + -----------------
2 4 2
2 2
- u3*n2 *n3 2 2 2 2 - v1*v3*c13*n2
+ -------------- + v1 *v3 *a33*c13 + v1 *v3*a33*c13*m1 + ------------------
4 2
2 2
- v1*m1*n2 2 2 2 2 v2*m2*n2
+ -------------- + v2 *v3 *a33*c13 + v2 *v3*a33*c13*m1 + -----------
4 4
2 2
2 - a33*m1 - a33*m2 v3*m2*n2*n3
+ v3 *---------------------- + -------------}
4 2
{HAM,FI} = 0
And again in machine readable form:
HAM=u2*n2 + u3**2*a33 + u3*n3 + 2*v1*v3*c13 + v1*m1 + v2*m2$
FI=u1**2*a33*n2**2 - 4*u1*v2*v3*a33*c13*n2 - 2*u1*v2*a33*m1*n2 + u2**2*a33*n2**2
+ 4*u2*v1*v3*a33*c13*n2 + 2*u2*v1*a33*m1*n2 - u2*n2**3 + 2*u3*v3*a33*m2*n2 - u3
*n2**2*n3 + 4*v1**2*v3**2*a33*c13**2 + 4*v1**2*v3*a33*c13*m1 - 2*v1*v3*c13*n2**2
- v1*m1*n2**2 + 4*v2**2*v3**2*a33*c13**2 + 4*v2**2*v3*a33*c13*m1 + v2*m2*n2**2
+ v3**2*( - a33*m1**2 - a33*m2**2) + 2*v3*m2*n2*n3$