Solution 24 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
r11=0
r12=0
r13=0
r14=0
r15=0
a33*m2*r316 - i*m2*n3*r453
r21=----------------------------
2
a33
r22=0
- i*a33*m2*r316 - m2*n3*r453
r23=-------------------------------
2
a33
r24=0
r27=0
r28=0
r210=0
2 2
- i*a33 *r26 + i*a33*n3*r316 + n3 *r453
r212=------------------------------------------
2
a33
r213=0
r214=0
r215=0
2 2
i*a33 *r26 - i*a33*n3*r316 - n3 *r453
r216=---------------------------------------
2
a33
r217=0
r218=0
r219=0
r220=0
2*m2*r425 - i*m2*r448
r31=-----------------------
2*a33
r33=0
- 2*i*m2*r425 - m2*r448
r34=--------------------------
2*a33
2*a33*c12*r316 - 2*i*c12*n3*r453
r35=----------------------------------
2
a33
r36=0
2
a33 *r32 - 2*i*a33*c12*r316 - 2*c12*n3*r453
r37=---------------------------------------------
2
a33
r38=0
r39=0
2*i*a33*r323 + 2*n3*r425 - i*n3*r448
r310=--------------------------------------
2*a33
i*m2*r453
r311=-----------
a33
r312=0
r313=0
r314=0
r315=0
r317=0
r318=0
r319=0
- 4*i*m2*r496
r320=----------------
a33
- i*m2*r453
r325=--------------
a33
r326=0
n3*r453
r328=---------
a33
r329=0
2*i*n3*r496
r330=-------------
a33
r332=0
r333=0
r334=0
- 4*m2*r496
r335=--------------
a33
r336= - r323
- m2*r453
r337=------------
a33
r338=0
i*m2*r453
r339=-----------
a33
- m2*r453
r340=------------
a33
r341=0
- n3*r453
r342=------------
a33
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
2*i*n3*r496
r350=-------------
a33
r351=0
r352=0
r353=0
r354=0
r355=0
r41=0
r43=0
r44=0
r45=0
2*c12*r425 - i*c12*r448
r46=-------------------------
a33
r47=0
r48=0
a33*r42 - 2*i*c12*r425 - c12*r448
r49=-----------------------------------
a33
r410=0
r411=0
r412=0
r413=0
i*a33*r439 - 8*c12*r496
r415=-------------------------
a33
r416=0
- i*a33*r420 - 2*c12*r453
r417=----------------------------
a33
r418=0
r419=0
r421=0
- i*a33*r420 - 2*c12*r453
r422=----------------------------
a33
r423=0
r424=0
r426=0
r427=0
r428=0
r429=0
r431=i*r453
r432=0
r433=0
r435=0
r442=0
r444= - r420
r445=0
r450=0
r451=0
r454=0
r455=0
r458=0
- i*r448
r460=-----------
2
r461=3*i*r496
r463=0
r464=0
r465=0
r467=r496
r468=0
r469=0
r470=0
- a33*r439 - 8*i*c12*r496
r471=----------------------------
a33
r472=0
a33*r420 - 2*i*c12*r453
r473=-------------------------
a33
- 8*c12*r496
r474=---------------
a33
r475=0
- 2*c12*r453
r476=---------------
a33
r477=0
2*a33*r420 - 2*i*c12*r453
r478=---------------------------
a33
- 2*c12*r453
r479=---------------
a33
r480=0
r481= - r448
r482=0
r483= - i*r448
r484=0
r485=0
r486=0
r487= - r453
r488=0
r489=0
r490=0
r493=0
r495=0
r498=0
r499=0
r4100=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
- i*r448
r4107=-----------
2
r4108=0
r4109=0
r4110= - i*r448
r4111=3*i*r496
r4112=0
r4113=0
r4114=0
r4115=0
r4117=2*r496
r4118=0
r4119=0
r4120=0
r4121= - r496
r4122=0
r4123=0
r4124=0
r4125=0
m3=0
m1= - i*m2
n2=0
n1=0
c33=i*c12
c23=0
c22=2*i*c12
c13=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:
r40, r30, r20, r10, r32, r42, r26, r323, r316, r425,
r439, r420, r453, r448, r496, m2, n3, c12, 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}
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,
c13,
- 2*i*c12 + c22,
c23,
- i*c12 + c33,
n1,
n2,
m1 + i*m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u3 *a33 + u3*n3 + 2*v1*v2*c12 - i*v1*m2 + 2*i*v2 *c12 + v2*m2 + i*v3 *c12
has apart from the Hamiltonian and Casimirs the following 15 first integrals:
3 2 2 2
FI= - u1 *v2*a33 + 2*u1 *u2*v1*a33 + 3*i*u1 *u3*v3*a33 + 2*i*u1 *v3*n3
2 2 2
+ u1*u2*u3*v3*a33 - 8*u1*v1*v3 *c12 - 8*i*u1*v2*v3 *c12 - 4*u1*v3 *m2
3 2 2 2
+ u2 *v1*a33 + 3*i*u2 *u3*v3*a33 + 2*i*u2 *v3*n3 - 4*i*u2*v3 *m2
3
- 8*u3*v3 *c12
which the program can not factorize further.
{HAM,FI} = {2,
- u1*v1 - u2*v2 - u3*v3,
a33,
2
2 u1 *n3
u1 *u3*a33 + -------- - i*u1*v1*v3*c12 + u1*v2*v3*c12
2
2
- i*u1*v3*m2 2 - u2 *n3
+ --------------- - u2 *u3*a33 + ----------- + 3*u2*v1*v3*c12
2 2
3*u2*v3*m2
+ 3*i*u2*v2*v3*c12 + ------------}
2
2 2 2 2
FI= - 2*i*u1 *v1 *a33 - i*u1 *v2 *a33 - 2*i*u1*u3*v1*v3*a33 - 2*u1*u3*v2*v3*a33
2 2 2 2 2
- i*u2 *v1 *a33 + 2*u2*u3*v1*v3*a33 - i*u3*v3 *n3 - 2*v1 *v3 *c12
2 2 2
- 2*i*v1*v2*v3 *c12 - v1*v3 *m2 - i*v2*v3 *m2
= a product of the elements of: { - 2*i,
2 2
2 2 u1 *v2 *a33
u1 *v1 *a33 + ------------- + u1*u3*v1*v3*a33 - i*u1*u3*v2*v3*a33
2
2 2 2
u2 *v1 *a33 u3*v3 *n3 2 2
+ ------------- + i*u2*u3*v1*v3*a33 + ----------- - i*v1 *v3 *c12
2 2
2 2
2 - i*v1*v3 *m2 v2*v3 *m2
+ v1*v2*v3 *c12 + ---------------- + -----------}
2 2
{HAM,FI} = { - 4*i,
u1*v1 + u2*v2 + u3*v3,
a33,
u1*v2*n3 u2*v1*n3 2
u1*u3*v2*a33 + ---------- + u2*u3*v1*a33 + ---------- - v1 *v3*c12
2 2
- v1*v3*m2
- i*v1*v2*v3*c12 + -------------}
2
2 2 3
FI= - u1*u3 *v2*a33 - u1*u3*v2*a33*n3 - 2*u1*v1 *a33*c12
2 2 2
- 2*i*u1*v1 *v2*a33*c12 - u1*v1 *a33*m2 - 2*u1*v1*v2 *a33*c12
3 2 2
+ i*u1*v1*v2*a33*m2 - 2*i*u1*v2 *a33*c12 - u1*v2 *a33*m2 - u1*v2*n3
2 2 2 2
+ u2*u3 *v1*a33 + u2*u3*v1*a33*n3 - i*u2*v1 *a33*m2 + u2*v1*n3
3 2 2 2
+ i*u3 *v3*a33 - 2*u3*v1 *v3*a33*c12 - 2*u3*v2 *v3*a33*c12
2
+ i*u3*v2*v3*a33*m2 - 2*v1 *v3*c12*n3 - 2*i*v1*v2*v3*c12*n3 - v1*v3*m2*n3
- i*v2*v3*m2*n3
which the program can not factorize further.
{HAM,FI} = {4*i,
u1*v1 + u2*v2 + u3*v3,
2 2 i*v2*m2
v1 *c12 + 2*i*v1*v2*c12 - v2 *c12 + ---------,
2
a33,
n3
u3*a33 + ----}
2
2 3 3 2 2
FI=2*u1*v1 *v2 + u1*v2 - u2*v1 - i*u3*v1 *v3 + u3*v1*v2*v3 - i*u3*v2 *v3
= a product of the elements of: {2,
3 3 2
2 u1*v2 - u2*v1 - i*u3*v1 *v3 u3*v1*v2*v3
u1*v1 *v2 + -------- + ----------- + ---------------- + -------------
2 2 2 2
2
- i*u3*v2 *v3
+ ----------------}
2
{HAM,FI} = {2,
- v1 + v2,
v1 + v2,
u1*v1 + u2*v2 + u3*v3,
n3
u3*a33 + ----}
2
2 2 3
FI= - u1*v2*v3 + u2*v1*v3 + i*u3*v3
= a product of the elements of: { - v3,
v3,
u1*v2 - u2*v1 - i*u3*v3}
{HAM,FI} = 0
2 2 2 2 2 2 2
FI=u3 *v3 *a33 + u3*v3 *n3 - 2*i*v1 *v3 *c12 + 2*v1*v2*v3 *c12 - i*v1*v3 *m2
2
+ v2*v3 *m2
= a product of the elements of: {v3,
v3,
2 2
u3 *a33 + u3*n3 - 2*i*v1 *c12 + 2*v1*v2*c12 - i*v1*m2 + v2*m2}
{HAM,FI} = 0
2 2
FI= - i*u1*v2*n3 + i*u2*v1*n3 + u3 *v3*a33 - 2*i*v1 *v3*c12 + 2*v1*v2*v3*c12
- i*v1*v3*m2 + v2*v3*m2
which the program can not factorize further.
{HAM,FI} = 0
2
FI= - u1*v2*v3 + u2*v1*v3 + i*u3*v3
= a product of the elements of: { - v3,u1*v2 - u2*v1 - i*u3*v3}
{HAM,FI} = 0
FI=i*u1*v2 - i*u2*v1 + u3*v3
= a product of the elements of: {i,u1*v2 - u2*v1 - i*u3*v3}
{HAM,FI} = 0
2 2 2 2
FI=v1 *v3 + v2 *v3
= a product of the elements of: {v3,
v3,
v1 - i*v2,
v1 + i*v2}
{HAM,FI} = 0
2 2
FI=v1 *v3 + v2 *v3
= a product of the elements of: {v3,v1 - i*v2,v1 + i*v2}
{HAM,FI} = 0
FI=v3
which the program can not factorize further.
{HAM,FI} = 0
2
FI=v3
= a product of the elements of: {v3,v3}
{HAM,FI} = 0
3
FI=v3
= a product of the elements of: {v3,v3,v3}
{HAM,FI} = 0
4
FI=v3
= a product of the elements of: {v3,v3,v3,v3}
{HAM,FI} = 0
And again in machine readable form:
HAM=u3**2*a33 + u3*n3 + 2*v1*v2*c12 - i*v1*m2 + 2*i*v2**2*c12 + v2*m2 + i*v3**2*
c12$
FI= - u1**3*v2*a33 + 2*u1**2*u2*v1*a33 + 3*i*u1**2*u3*v3*a33 + 2*i*u1**2*v3*n3 +
u1*u2*u3*v3*a33 - 8*u1*v1*v3**2*c12 - 8*i*u1*v2*v3**2*c12 - 4*u1*v3**2*m2 + u2
**3*v1*a33 + 3*i*u2**2*u3*v3*a33 + 2*i*u2**2*v3*n3 - 4*i*u2*v3**2*m2 - 8*u3*v3**
3*c12$
FI= - 2*i*u1**2*v1**2*a33 - i*u1**2*v2**2*a33 - 2*i*u1*u3*v1*v3*a33 - 2*u1*u3*v2
*v3*a33 - i*u2**2*v1**2*a33 + 2*u2*u3*v1*v3*a33 - i*u3*v3**2*n3 - 2*v1**2*v3**2*
c12 - 2*i*v1*v2*v3**2*c12 - v1*v3**2*m2 - i*v2*v3**2*m2$
FI= - u1*u3**2*v2*a33**2 - u1*u3*v2*a33*n3 - 2*u1*v1**3*a33*c12 - 2*i*u1*v1**2*
v2*a33*c12 - u1*v1**2*a33*m2 - 2*u1*v1*v2**2*a33*c12 + i*u1*v1*v2*a33*m2 - 2*i*
u1*v2**3*a33*c12 - u1*v2**2*a33*m2 - u1*v2*n3**2 + u2*u3**2*v1*a33**2 + u2*u3*v1
*a33*n3 - i*u2*v1**2*a33*m2 + u2*v1*n3**2 + i*u3**3*v3*a33**2 - 2*u3*v1**2*v3*
a33*c12 - 2*u3*v2**2*v3*a33*c12 + i*u3*v2*v3*a33*m2 - 2*v1**2*v3*c12*n3 - 2*i*v1
*v2*v3*c12*n3 - v1*v3*m2*n3 - i*v2*v3*m2*n3$
FI=2*u1*v1**2*v2 + u1*v2**3 - u2*v1**3 - i*u3*v1**2*v3 + u3*v1*v2*v3 - i*u3*v2**
2*v3$
FI= - u1*v2*v3**2 + u2*v1*v3**2 + i*u3*v3**3$
FI=u3**2*v3**2*a33 + u3*v3**2*n3 - 2*i*v1**2*v3**2*c12 + 2*v1*v2*v3**2*c12 - i*
v1*v3**2*m2 + v2*v3**2*m2$
FI= - i*u1*v2*n3 + i*u2*v1*n3 + u3**2*v3*a33 - 2*i*v1**2*v3*c12 + 2*v1*v2*v3*c12
- i*v1*v3*m2 + v2*v3*m2$
FI= - u1*v2*v3 + u2*v1*v3 + i*u3*v3**2$
FI=i*u1*v2 - i*u2*v1 + u3*v3$
FI=v1**2*v3**2 + v2**2*v3**2$
FI=v1**2*v3 + v2**2*v3$
FI=v3$
FI=v3**2$
FI=v3**3$
FI=v3**4$