Solution 6 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r10=0
1 2
- ---*m2*n3 *r464
4
r11=--------------------
3
a22
1 2
- ---*m1*n3 *r464
4
r12=--------------------
3
a22
1 3
- ---*n3 *r464
4
r13=-----------------
3
a22
r14=0
r15=0
1 2
---*m1 *r464
4
r20=--------------
2
a22
r21=0
r23=0
1
---*m1*m2*r464
2
r24=----------------
2
a22
1 2 1 2
---*m1 *r464 - ---*m2 *r464
4 4
r25=-----------------------------
2
a22
r26=0
1
---*m2*n3*r464
2
r27=----------------
2
a22
1
---*m1*n3*r464
2
r28=----------------
2
a22
1 2
- ---*n3 *r464
4
r29=-----------------
2
a22
1
- ---*m2*n3*r464
2
r210=-------------------
2
a22
r212=0
r213=0
r214=0
1
- ---*m1*n3*r464
2
r215=-------------------
2
a22
r216=0
r217=0
r218=0
r219=0
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=0
r311=0
r312=0
r313=0
r314=0
r315=0
r316=0
m2*r464
r317=---------
a22
m1*r464
r318=---------
a22
n3*r464
r319=---------
a22
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
r326=0
r327=0
r328=0
r329=0
r330=0
m2*r464
r331=---------
a22
r332=0
n3*r464
r333=---------
a22
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
m1*r464
r346=---------
a22
m2*r464
r347=---------
a22
r348=0
r349=0
r350=0
r351=0
m1*r464
r352=---------
a22
n3*r464
r353=---------
a22
r354=0
r355=0
r40=0
r41=0
r42=0
r43=0
r45=0
r46=0
r47=0
r48=0
r49=0
r410=0
r411=0
r412=0
r413=0
r414=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
r430=0
r431=0
r432=0
r433=0
r434=r464
r435=0
r436=0
r437=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r446=0
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
r453=0
r454=0
r455=0
r456=0
r458=0
r459=0
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
r4105=0
r4106=0
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
r4113=0
r4114=r464
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
m3=0
n2=0
n1=0
c33=0
c23=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
a33=2*a22
a23=0
a13=0
a12=0
a11=a22
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:
r464, m1, n3, m2, a22
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.
{a22,a11,r464}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11 - a22,
a12,
a13,
a23,
- 2*a22 + a33,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
b33,
c11,
c12,
c13,
c22,
c23,
c33,
n1,
n2,
m3}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a22 + u2 *a22 + 2*u3 *a22 + u3*n3 + v1*m1 + v2*m2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 3 2 2 2 2 2
FI=u1 *u3 *a22 + u1 *u3*a22 *n3 + u1 *v1*a22 *m1 + u1*u2*v1*a22 *m2
2 1 2 2 3 2 2
+ u1*u2*v2*a22 *m1 - ---*u1*v3*a22*m1*n3 + u2 *u3 *a22 + u2 *u3*a22 *n3
2
2 2 1 4 3 3 2
+ u2 *v2*a22 *m2 - ---*u2*v3*a22*m2*n3 + u3 *a22 + u3 *a22 *n3
2
2 2 2 2 1 2 2 1
+ u3 *v1*a22 *m1 + u3 *v2*a22 *m2 - ---*u3 *a22*n3 + ---*u3*v1*a22*m1*n3
4 2
1 1 3 2 1 2 1 2
+ ---*u3*v2*a22*m2*n3 - ---*u3*n3 + v1 *(---*a22*m1 - ---*a22*m2 )
2 4 4 4
1 1 2 1 2 1 2 2
+ ---*v1*v2*a22*m1*m2 - ---*v1*m1*n3 - ---*v2*m2*n3 + ---*v3 *a22*m1
2 4 4 4
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a22 + u2**2*a22 + 2*u3**2*a22 + u3*n3 + v1*m1 + v2*m2$
FI=u1**2*u3**2*a22**3 + u1**2*u3*a22**2*n3 + u1**2*v1*a22**2*m1 + u1*u2*v1*a22**
2*m2 + u1*u2*v2*a22**2*m1 - 1/2*u1*v3*a22*m1*n3 + u2**2*u3**2*a22**3 + u2**2*u3*
a22**2*n3 + u2**2*v2*a22**2*m2 - 1/2*u2*v3*a22*m2*n3 + u3**4*a22**3 + u3**3*a22
**2*n3 + u3**2*v1*a22**2*m1 + u3**2*v2*a22**2*m2 - 1/4*u3**2*a22*n3**2 + 1/2*u3*
v1*a22*m1*n3 + 1/2*u3*v2*a22*m2*n3 - 1/4*u3*n3**3 + v1**2*(1/4*a22*m1**2 - 1/4*
a22*m2**2) + 1/2*v1*v2*a22*m1*m2 - 1/4*v1*m1*n3**2 - 1/4*v2*m2*n3**2 + 1/4*v3**2
*a22*m1**2$