Solution 29 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
- m3*r216
r10=------------
b21
r11=0
r12=0
r14=0
r15=0
2
- 2*b33*c33*r216 + m3 *r448
r20=------------------------------
2*b21*b33
r21=0
r22=0
r23=0
r24=0
r27=0
r28=0
r210=0
r212= - r216
r213=0
r214=0
r215=0
r217=0
r218=0
r219=0
r220=0
- b33*m3*r420 + c33*m3*r448
r30=------------------------------
b21*b33
r31=0
- m3*r420
r32=------------
b21
r33=0
r34=0
r35=0
r36=0
- m3*r420
r37=------------
b21
r38=0
r39=0
r311=0
b21*r310 - c33*r328 - m3*r448
r312=-------------------------------
b21
r313=0
r314=0
b21*r310 - c33*r328 - m3*r448
r315=-------------------------------
b21
b33*r328 + m3*r453
r316=--------------------
b21
r317=0
r318=0
r320=0
m3*r448
r323=---------
b33
r325=0
r326=0
r329=0
r330=0
r332=0
r333=0
r334=0
r335=0
- m3*r448
r336=------------
b33
r337=0
r338=0
r339=0
r340=0
r341=0
r342= - r328
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
2
- 2*b33*c33*r420 + c33 *r448
r40=-------------------------------
2*b21*b33
r41=0
- c33*r420
r42=-------------
b21
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
- c33*r420
r49=-------------
b21
r410=0
r411=0
r412=0
r413=0
- b33*r420 + c33*r448
r415=------------------------
b21
r416=0
- b33*r420
r417=-------------
b21
r418=0
r419=0
r421=0
- b33*r420
r422=-------------
b21
r423=0
r424=0
b33*r448 + 2*c33*r453
r425=-----------------------
2*b21
r426=0
r427=0
r428=0
r429=0
b33*r453
r431=----------
b21
r432=0
r433=0
r435=0
- b33*r420 + c33*r448
r439=------------------------
b33
r442=0
r444= - r420
r445=0
r450=0
r451=0
r454=0
r455=0
r458=0
b21*r448
r460=----------
2*b33
r461=0
r463=0
r464=0
r465=0
r467=0
r468=0
r469=0
r470=0
b33*r420 - c33*r448
r471=---------------------
b33
r472=0
r473=r420
r474=0
r475=0
r476=0
r477=0
r478=2*r420
r479=0
r480=0
r481= - r448
r482=0
b21*r448
r483=----------
b33
r484=0
r485=0
r486=0
r487= - r453
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
b21*r448
r4107=----------
2*b33
r4108=0
r4109=0
b21*r448
r4110=----------
b33
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
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*r216 + m3*r328
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:
r216, r13, r319, r310, r420, r328, r448, r453, m3, c33,
n3, b21, b33, 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,b33,b21}
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 8 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 2 2 2 2
FI=2*u1 *v1 *b21 + u1 *v2 *b21 + 2*u1*u3*v1*v3*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
= a product of the elements of: {2,
2 2 2
2 2 2 u1 *v2 *b21 2
u1 *v1 *b21 + -------------- + u1*u3*v1*v3*b21 - u1*u3*v2*v3*b21*b33
2
2 2 2
2 u2 *v1 *b21
- u1*v2*v3 *b21*c33 - u1*v2*v3*b21*m3 + -------------- + u2*u3*v1*v3*b21*b33
2
2 2 2
2 u3 *v3 *b33 2
+ u2*v1*v3 *b21*c33 + u2*v1*v3*b21*m3 + -------------- - u3*v1 *b33*m3
2
4 2 2 2
2 3 v3 *c33 3 v3 *m3
- u3*v2 *b33*m3 + u3*v3 *b33*c33 + ---------- + v3 *c33*m3 + ---------}
2 2
{HAM,FI} = {4,
b21,
b21,
u1*v1 + u2*v2 + u3*v3,
u1*v2*v3*b33 u1*v2*n3
u1*u3*v2*a33 + -------------- + ---------- + u2*u3*v1*a33
2 2
2
u2*v1*v3*b33 u2*v1*n3 - u3*v1 *b21 u3*v1*v2*b33
+ -------------- + ---------- + --------------- + --------------
2 2 2 2
v1*v2*m3
+ v1*v2*v3*c33 + ----------}
2
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 3 2 3 2
FI=2*u1*v1 *v2*b21 + u1*v2 *b21 + u1*v2*v3 *b21 - u2*v1 *b21 - u2*v1*v3 *b21
2 2 3 2 2
- u3*v1 *v3*b33 + u3*v1*v2*v3*b21 - 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
= a product of the elements of: {2,
3 2 3
2 u1*v2 *b21 u1*v2*v3 *b21 - u2*v1 *b21
u1*v1 *v2*b21 + ------------ + --------------- + ---------------
2 2 2
2 2
- u2*v1*v3 *b21 - u3*v1 *v3*b33 u3*v1*v2*v3*b21
+ ------------------ + ------------------ + -----------------
2 2 2
2 3 2 2 2
- u3*v2 *v3*b33 - u3*v3 *b33 - v1 *v3 *c33 - v1 *v3*m3
+ ------------------ + --------------- + ---------------- + --------------
2 2 2 2
2 2 2 4 3
- v2 *v3 *c33 - v2 *v3*m3 - v3 *c33 - v3 *m3
+ ---------------- + -------------- + ------------ + -----------}
2 2 2 2
{HAM,FI} = {2,
- u1*v1 - u2*v2 - u3*v3,
b21,
2 2
2 2 v1 *v3*b33 v1 *n3
u3*v1 *a33 - u3*v2 *a33 + ------------ + -------- + v1*v2*v3*b21
2 2
2 2
- v2 *v3*b33 - v2 *n3
+ --------------- + -----------}
2 2
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
3
FI=u3
= a product of the elements of: {u3,u3,u3}
{HAM,FI} = 0
FI=u3
which the program can not factorize further.
{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=2*u1**2*v1**2*b21**2 + u1**2*v2**2*b21**2 + 2*u1*u3*v1*v3*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*v2*b21 + u2*u3*v1*b21 + u3**2*v3*b33 - u3*v1**2*c33 - u3*v2**2*c33 +
u3*v3*m3$
FI=2*u1*v1**2*v2*b21 + u1*v2**3*b21 + u1*v2*v3**2*b21 - u2*v1**3*b21 - u2*v1*v3
**2*b21 - u3*v1**2*v3*b33 + u3*v1*v2*v3*b21 - 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=u3*v1**2 + u3*v2**2 + u3*v3**2$
FI=u3**3$
FI=u3$
FI=u1*v2*b21 - u2*v1*b21 - u3*v3*b33 - v3**2*c33 - v3*m3$