Solution 5 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
r13=0
r14=0
r20=0
r21=0
r22=0
r23=0
r24=0
r26=0
r27=0
r28=0
3 3 3
- 18*a22 *r15 + 4*n1 *r4119 - n1 *r464
r29=-----------------------------------------
2
6*a22 *n1
r210=0
r212=0
r213=0
3 3 3
- 6*a22 *r15 + 4*n1 *r4119 - n1 *r464
r214=----------------------------------------
2
6*a22 *n1
r215=0
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
r317=0
r318=0
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
r326=0
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- n1*r4102
r341=-------------
a22
r342=0
r343=0
3*n1*r4119 - n1*r464
r344=----------------------
a22
r345=0
- n1*r4102
r346=-------------
a22
r347=0
r348=0
n1*r4119 - n1*r464
r349=--------------------
3*a22
r350=0
r351=0
- n1*r4102
r352=-------------
a22
r353=0
r354=0
- n1*r4119
r355=-------------
a22
r40=0
r41=0
r42=0
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
r49=0
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
- r446
r427=---------
2
r428=0
r429=0
- r446
r430=---------
2
r431=3*r4102
r432=0
r433=0
- 6*r4119 + 3*r464
r434=---------------------
2
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
r447=0
r448=0
r449=0
r450=0
r451=0
r452=3*r4102
r453=0
r454=0
- r446
r455=---------
2
r456=0
r458=0
r459=0
- r446
r460=---------
2
r461=r4102
r462=0
r463=0
r465=0
r466=r4102
r467=0
r468=0
2*r4119 + r464
r469=----------------
6
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=r446
r484=0
r485=0
r486=0
r487=0
r488=3*r4102
r489=0
r490=0
r491=0
r492=0
r493=0
r494=r446
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4103=0
r4104=0
- r446
r4105=---------
2
r4106=0
- r446
r4107=---------
2
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4114=3*r4119
r4115=0
r4117=0
r4118=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
m3=0
m2=0
m1=0
n3=0
n2=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= - a22
a23=0
a13=0
a12=0
a11=2*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:
r15, r4119, r4102, r446, r464, n1, 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.
{n1,a22}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11 - 2*a22,
a12,
a13,
a23,
a22 + a33,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
b33,
c11,
c12,
c13,
c22,
c23,
c33,
n2,
n3,
m1,
m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=2*u1 *a22 + u1*n1 + u2 *a22 - u3 *a22
has apart from the Hamiltonian and Casimirs the following 5 first integrals:
2 2 4 2 2 2 2 2 2
FI= - 2*u1*u2 *a22*n1 - 6*u1*u3 *a22*n1 + u2 *a22 + 6*u2 *u3 *a22 - u2 *n1
4 2 2 2
+ 9*u3 *a22 - u3 *n1
which the program can not factorize further.
{HAM,FI} = 0
2 2 2 2 2 2 2 2
FI= - u1 *v2 - u1 *v3 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2 *v1 - u2 *v3
2 2 2 2
+ 2*u2*u3*v2*v3 - u3 *v1 - u3 *v2
which the program can not factorize further.
{HAM,FI} = 0
2 2 2
FI= - u1 *v1*n1 + u1*u2 *v1*a22 - u1*u2*v2*n1 + 3*u1*u3 *v1*a22 - u1*u3*v3*n1
3 2 2 3
+ u2 *v2*a22 + u2 *u3*v3*a22 + 3*u2*u3 *v2*a22 + 3*u3 *v3*a22
= a product of the elements of: {u1*v1 + u2*v2 + u3*v3,
2 2
- u1*n1 + u2 *a22 + 3*u3 *a22}
{HAM,FI} = 0
3 2 2 2 2 2 2 2
FI= - 3*u1 *a22*n1 + 3*u1 *u2 *a22 + 9*u1 *u3 *a22 + u1*u2 *a22*n1
2 4 2 2 2 4 2 2 2
+ 9*u1*u3 *a22*n1 + u2 *a22 + 2*u2 *n1 - 9*u3 *a22 + 2*u3 *n1
= a product of the elements of: {3,
2
3 2 2 2 2 2 2 u1*u2 *a22*n1
- u1 *a22*n1 + u1 *u2 *a22 + 3*u1 *u3 *a22 + ---------------
3
4 2 2 2 2 2
2 u2 *a22 2*u2 *n1 4 2 2*u3 *n1
+ 3*u1*u3 *a22*n1 + ---------- + ----------- - 3*u3 *a22 + -----------}
3 3 3
{HAM,FI} = 0
2 2
FI=u1*n1 - u2 *a22 - 3*u3 *a22
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u1**2*a22 + u1*n1 + u2**2*a22 - u3**2*a22$
FI= - 2*u1*u2**2*a22*n1 - 6*u1*u3**2*a22*n1 + u2**4*a22**2 + 6*u2**2*u3**2*a22**
2 - u2**2*n1**2 + 9*u3**4*a22**2 - u3**2*n1**2$
FI= - u1**2*v2**2 - u1**2*v3**2 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2**2*v1**2 -
u2**2*v3**2 + 2*u2*u3*v2*v3 - u3**2*v1**2 - u3**2*v2**2$
FI= - u1**2*v1*n1 + u1*u2**2*v1*a22 - u1*u2*v2*n1 + 3*u1*u3**2*v1*a22 - u1*u3*v3
*n1 + u2**3*v2*a22 + u2**2*u3*v3*a22 + 3*u2*u3**2*v2*a22 + 3*u3**3*v3*a22$
FI= - 3*u1**3*a22*n1 + 3*u1**2*u2**2*a22**2 + 9*u1**2*u3**2*a22**2 + u1*u2**2*
a22*n1 + 9*u1*u3**2*a22*n1 + u2**4*a22**2 + 2*u2**2*n1**2 - 9*u3**4*a22**2 + 2*
u3**2*n1**2$
FI=u1*n1 - u2**2*a22 - 3*u3**2*a22$