Solution 24 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
m3*r212
r10=---------
b21
r11=0
r12=0
r14=0
r15=0
2
2*b21*c33*r212 - m3 *r494
r20=---------------------------
2
2*b21
r21=0
r22=0
r23=0
r24=0
r27=0
r28=0
r210=0
r213=0
r214=0
r215=0
r216= - r212
r217=0
r218=0
r219=0
r220=0
b21*m3*r441 - c33*m3*r494
r30=---------------------------
2
b21
r31=0
m3*r441
r32=---------
b21
r33=0
r34=0
r35=0
r36=0
m3*r441
r37=---------
b21
r38=0
r39=0
r311=0
2 2 2
b21 *b33*r310 - b21 *m3*r446 + b21*b33*c33*r342 + b33 *m3*r494
r312=----------------------------------------------------------------
2
b21 *b33
r313=0
r314=0
2 2 2
b21 *b33*r310 - b21 *m3*r446 + b21*b33*c33*r342 + b33 *m3*r494
r315=----------------------------------------------------------------
2
b21 *b33
2
b21*b33*r327 + b21*m3*r431 - b33 *r342
r316=----------------------------------------
b21*b33
r317=0
r318=0
r320=0
m3*r446
r321=---------
b33
r322=0
- m3*r494
r323=------------
b21
r324=0
r325=0
r326=0
r328= - r342
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
r335=0
m3*r494
r336=---------
b21
r337=0
m3*r446
r338=---------
b33
r339=0
r340=0
r341=0
r343=r327
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
2
2*b21*c33*r441 - c33 *r494
r40=----------------------------
2
2*b21
r41=0
c33*r441
r42=----------
b21
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
c33*r441
r49=----------
b21
r410=0
r411=0
r412=0
r413=0
2 2 2
b21 *c33*r446 + b21*b33 *r441 - b33 *c33*r494
r415=-----------------------------------------------
2
b21 *b33
r416=0
b33*r441
r417=----------
b21
r418=0
r419=0
r420=0
r421=0
r423=0
r424=0
2 2 3
2*b21 *b33*r446 + 2*b21 *c33*r431 - b33 *r494
r425=-----------------------------------------------
2
2*b21 *b33
r426=0
r427=0
r428=0
r429=0
r432=0
r433=0
r435=0
c33*r446
r436=----------
b33
r437=0
r438=0
b21*r441 - c33*r494
r439=---------------------
b21
r440=0
r442=0
r444=r441
r445=0
r447=0
2 2
b21 *r446 - b33 *r494
r448=-----------------------
b21*b33
r449=0
r450=0
r451=0
b21*r431
r453=----------
b33
r454=0
r455=0
r456=0
r458=0
b21*r446
r459=----------
b33
- r494
r460=---------
2
r461=0
r462=0
r463=0
r464=0
r465=0
r466=0
r467=0
r468=0
r469=0
r470=0
- b21*r441 + c33*r494
r471=------------------------
b21
r472=0
r473= - r441
c33*r446
r474=----------
b33
r475=0
r476=0
r477=0
r478= - r441
r479=0
r480=0
2 2
- b21 *r446 + b33 *r494
r481=--------------------------
b21*b33
r482=0
r483=r446
r484=0
r485=0
r486=0
- b21*r431
r487=-------------
b33
r488=0
r489=0
r490=0
r491=0
- b21*r446
r492=-------------
b33
r493=0
b21*r446
r495=----------
b33
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
- r494
r4107=---------
2
r4108=0
- b21*r446
r4109=-------------
b33
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
m2=0
m1=0
n2=0
n1=0
c23=0
c22=0
c13=0
c12=0
c11=0
b32=0
b31=0
b23=0
b22=0
b13=0
b12= - b21
b11=0
a23=0
a22=0
a13=0
a12=0
a11=0
b33*r441
r422=----------
b21
b33*r212 - m3*r342
r26=--------------------
b21
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:
r212, r441, r319, r13, r310, r342, r327, r494, r446, r431,
c33, m3, n3, b33, b21, 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.
{a33,b21,b33}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11,
a12,
a13,
a22,
a23,
b11,
b12 + b21,
b13,
b22,
b23,
b31,
b32,
c11,
c12,
c13,
c22,
c23,
n1,
n2,
m1,
m2}$
The system of equations related to the Hamiltonian HAM:
2 2
HAM= - u1*v2*b21 + u2*v1*b21 + u3 *a33 + u3*v3*b33 + u3*n3 + v3 *c33 + v3*m3
has apart from the Hamiltonian and Casimirs the following 10 first integrals:
2 2 3 2 2 2
FI= - u1*u3 *v2*b21 + u2*u3 *v1*b21 + u3 *v3*b33 + u3 *v3 *c33 + u3 *v3*m3
= a product of the elements of: { - u3,
u3,
2
u1*v2*b21 - u2*v1*b21 - u3*v3*b33 - v3 *c33 - v3*m3}
{HAM,FI} = 0
2 2 2
FI= - u1 *v1*v2*b21 + u1*u2*v1 *b21 - u1*u2*v2 *b21 + u1*u3*v1*v3*b33
2 2
- u1*u3*v2*v3*b21 + u1*v1*v3 *c33 + u1*v1*v3*m3 + u2 *v1*v2*b21
2
+ u2*u3*v1*v3*b21 + u2*u3*v2*v3*b33 + u2*v2*v3 *c33 + u2*v2*v3*m3
2 2 2 2 3
+ u3 *v3 *b33 - u3*v1 *m3 - u3*v2 *m3 + u3*v3 *c33
which the program can not factorize further.
{HAM,FI} = 0
2 2 2 2
FI= - u1 *v2 *b21 + 2*u1*u2*v1*v2*b21 + 2*u1*u3*v2*v3*b21*b33
2 2 2 2
+ 2*u1*v2*v3 *b21*c33 + 2*u1*v2*v3*b21*m3 - u2 *v1 *b21
2
- 2*u2*u3*v1*v3*b21*b33 - 2*u2*v1*v3 *b21*c33 - 2*u2*v1*v3*b21*m3
2 2 2 2 2 3
- u3 *v3 *b33 + 2*u3*v1 *b33*m3 + 2*u3*v2 *b33*m3 - 2*u3*v3 *b33*c33
4 2 3 2 2
- v3 *c33 - 2*v3 *c33*m3 - v3 *m3
Stop of a subroutine.
Number of garbage collections exceeds max_gc_fac.
The factorization crashed!
which the program can not factorize further.
{HAM,FI} = 0
2
FI=u1*u3*v1 + u2*u3*v2 + u3 *v3
= a product of the elements of: {u3,u1*v1 + u2*v2 + u3*v3}
{HAM,FI} = 0
2 2 2
FI=u1*u3*v2*b21 - u2*u3*v1*b21 - u3 *v3*b33 + u3*v1 *c33 + u3*v2 *c33 - u3*v3*m3
= a product of the elements of: {u3,
2 2
u1*v2*b21 - u2*v1*b21 - u3*v3*b33 + v1 *c33 + v2 *c33 - v3*m3}
{HAM,FI} = 0
2 2 2
FI=u3*v1 + u3*v2 + u3*v3
2 2 2
= a product of the elements of: {v1 + v2 + v3 ,u3}
{HAM,FI} = 0
FI=u3
which the program can not factorize further.
{HAM,FI} = 0
3
FI=u3
= a product of the elements of: {u3,u3,u3}
{HAM,FI} = 0
2 3 2 3 2
FI= - u1*v1 *v2*b21 - u1*v2 *b21 - u1*v2*v3 *b21 + u2*v1 *b21 + u2*v1*v2 *b21
2 2 2 3 2 2
+ u2*v1*v3 *b21 + u3*v1 *v3*b33 + u3*v2 *v3*b33 + u3*v3 *b33 + v1 *v3 *c33
2 2 2 2 4 3
+ v1 *v3*m3 + v2 *v3 *c33 + v2 *v3*m3 + v3 *c33 + v3 *m3
2 2 2
= a product of the elements of: { - v1 - v2 - v3 ,
2
u1*v2*b21 - u2*v1*b21 - u3*v3*b33 - v3 *c33 - v3*m3}
{HAM,FI} = 0
2
FI= - u1*v2*b21 + u2*v1*b21 + u3*v3*b33 + v3 *c33 + v3*m3
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM= - u1*v2*b21 + u2*v1*b21 + u3**2*a33 + u3*v3*b33 + u3*n3 + v3**2*c33 + v3*m3
$
FI= - u1*u3**2*v2*b21 + u2*u3**2*v1*b21 + u3**3*v3*b33 + u3**2*v3**2*c33 + u3**2
*v3*m3$
FI= - u1**2*v1*v2*b21 + u1*u2*v1**2*b21 - u1*u2*v2**2*b21 + u1*u3*v1*v3*b33 - u1
*u3*v2*v3*b21 + u1*v1*v3**2*c33 + u1*v1*v3*m3 + u2**2*v1*v2*b21 + u2*u3*v1*v3*
b21 + u2*u3*v2*v3*b33 + u2*v2*v3**2*c33 + u2*v2*v3*m3 + u3**2*v3**2*b33 - u3*v1
**2*m3 - u3*v2**2*m3 + u3*v3**3*c33$
FI= - u1**2*v2**2*b21**2 + 2*u1*u2*v1*v2*b21**2 + 2*u1*u3*v2*v3*b21*b33 + 2*u1*
v2*v3**2*b21*c33 + 2*u1*v2*v3*b21*m3 - u2**2*v1**2*b21**2 - 2*u2*u3*v1*v3*b21*
b33 - 2*u2*v1*v3**2*b21*c33 - 2*u2*v1*v3*b21*m3 - u3**2*v3**2*b33**2 + 2*u3*v1**
2*b33*m3 + 2*u3*v2**2*b33*m3 - 2*u3*v3**3*b33*c33 - v3**4*c33**2 - 2*v3**3*c33*
m3 - v3**2*m3**2$
FI=u1*u3*v1 + u2*u3*v2 + u3**2*v3$
FI=u1*u3*v2*b21 - u2*u3*v1*b21 - u3**2*v3*b33 + u3*v1**2*c33 + u3*v2**2*c33 - u3
*v3*m3$
FI=u3*v1**2 + u3*v2**2 + u3*v3**2$
FI=u3$
FI=u3**3$
FI= - u1*v1**2*v2*b21 - u1*v2**3*b21 - u1*v2*v3**2*b21 + u2*v1**3*b21 + u2*v1*v2
**2*b21 + u2*v1*v3**2*b21 + u3*v1**2*v3*b33 + u3*v2**2*v3*b33 + u3*v3**3*b33 +
v1**2*v3**2*c33 + v1**2*v3*m3 + v2**2*v3**2*c33 + v2**2*v3*m3 + v3**4*c33 + v3**
3*m3$
FI= - u1*v2*b21 + u2*v1*b21 + u3*v3*b33 + v3**2*c33 + v3*m3$