Solution 18 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r11=0
r12=0
r13=0
r14=0
r15=0
i*a33*m1*r316 + m1*n3*r453
r21=----------------------------
2
a33
r22=0
a33*m1*r316 - i*m1*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*i*m1*r425 - 2*i*m1*r446 - i*m1*r494
r31=---------------------------------------
2*a33
r33=0
2*m1*r425 - 2*m1*r446 - m1*r494
r34=---------------------------------
2*a33
r36=0
2
a33 *r32 - 2*i*a33*c12*r316 - 2*c12*n3*r453
r37=---------------------------------------------
2
a33
r38=0
r39=0
2*a33*r321 + 2*i*a33*r323 + 2*n3*r425 - 2*n3*r446 - n3*r494
r310=-------------------------------------------------------------
2*a33
- m1*r453
r311=------------
a33
r312=0
i*m1*r453
r313=-----------
a33
r314=0
r315=0
r317=0
r318=0
r319=0
- 4*i*m1*r4116
r320=-----------------
a33
r322=0
i*m1*r453
r324=-----------
a33
m1*r453
r325=---------
a33
r326=0
r327=0
n3*r453
r328=---------
a33
r329=0
2*n3*r4116
r330=------------
a33
r331=0
r332=0
r333=0
r334=0
- 4*m1*r4116
r335=---------------
a33
r336= - r323
- i*m1*r453
r337=--------------
a33
r338=r321
- m1*r453
r339=------------
a33
r340=0
r341=0
- n3*r453
r342=------------
a33
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
2*n3*r4116
r350=------------
a33
r351=0
r352=0
r353=0
r354=0
r355=0
r41=0
r43=0
r44=0
r45=0
r47=0
r48=0
a33*r42 - 2*i*c12*r425 + 2*i*c12*r446 + i*c12*r494
r49=----------------------------------------------------
a33
r410=0
r411=0
r412=0
r413=0
- i*a33*r471 + a33*r474 - 8*i*c12*r4116
r415=------------------------------------------
a33
r416=0
a33*r422 - 2*c12*r453
r417=-----------------------
a33
r418=0
r419=0
r421=0
r423=0
r424=0
r426=0
r427=0
r428=0
r429=0
r432=0
r433=0
r435=0
a33*r474 - 8*i*c12*r4116
r436=--------------------------
a33
r437=0
- 2*c12*r453
r438=---------------
a33
- a33*r471 - 8*c12*r4116
r439=---------------------------
a33
r440=0
r441= - i*r422
r442=0
- i*a33*r422 - 2*i*c12*r453
r444=------------------------------
a33
r445=0
r447=0
r448= - i*r446 - i*r494
r449=0
r450=0
r451=0
r454=0
r455=0
r456=0
r458=0
r459= - i*r446
- r494
r460=---------
2
r461=4*r4116
r462=0
r463=0
r464=0
r465=0
r466=r4116
r467= - i*r4116
r468=0
r469=0
r470=0
r472=0
r473=i*r422
r475=0
- 4*c12*r453
r476=---------------
a33
r477=0
i*a33*r422 + 2*i*c12*r453
r478=---------------------------
a33
- 2*c12*r453
r479=---------------
a33
r480=0
r481=i*r446 + i*r494
r482=0
r483=r446
r484=0
r485=0
r486=0
r487= - r453
r488=0
r489=0
r490=0
r491=0
r492=i*r446
r493=0
r495= - i*r446
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=i*r4116
r4102=r4116
r4103=0
r4104=0
r4105=0
r4106=0
- r494
r4107=---------
2
r4108=0
r4109=i*r446
r4110=0
r4111=4*r4116
r4112=0
r4113=0
r4114=0
r4115=0
r4118=0
r4119=0
r4120=0
r4121=i*r4116
r4122=r4116
r4123=0
r4124=0
r4125=0
m3=0
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
2*i*c12*r453
r420=--------------
a33
2*a33*c12*r316 - 2*i*c12*n3*r453
r35=----------------------------------
2
a33
2*c12*r425 - 2*c12*r446 - c12*r494
r46=------------------------------------
a33
m2=i*m1
r431=i*r453
r4117= - i*r4116
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:
r4116, r453, m1, r425, r316, r40, r30, r20, r10, r42,
r32, r26, r323, r471, r321, r422, r474, r494, r446, 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:
{ - i*m1 + m2,
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,
m3}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u3 *a33 + u3*n3 + 2*v1*v2*c12 + v1*m1 + 2*i*v2 *c12 + i*v2*m1 + i*v3 *c12
has apart from the Hamiltonian and Casimirs the following 18 first integrals:
2 2 2
FI=i*u1 *v1*v2*a33 - i*u1*u2*v1 *a33 + i*u1*u2*v2 *a33 + u1*u3*v1*v3*a33
2
+ i*u1*u3*v2*v3*a33 - i*u2 *v1*v2*a33 - i*u2*u3*v1*v3*a33 + u2*u3*v2*v3*a33
2 2 2 2 2 2
- u3*v3 *n3 + 2*i*v1 *v3 *c12 - 2*v1*v2*v3 *c12 - v1*v3 *m1 - i*v2*v3 *m1
= a product of the elements of: {i,
2 2 2
u1 *v1*v2*a33 - u1*u2*v1 *a33 + u1*u2*v2 *a33 - i*u1*u3*v1*v3*a33
2
+ u1*u3*v2*v3*a33 - u2 *v1*v2*a33 - u2*u3*v1*v3*a33 - i*u2*u3*v2*v3*a33
2 2 2 2 2 2
+ i*u3*v3 *n3 + 2*v1 *v3 *c12 + 2*i*v1*v2*v3 *c12 + i*v1*v3 *m1 - v2*v3 *m1}
{HAM,FI} = 0
2 2 2 2
FI= - u1 *v2 *a33 + 2*u1*u2*v1*v2*a33 + 2*i*u1*u3*v2*v3*a33 - u2 *v1 *a33
2 2 2 2
- 2*i*u2*u3*v1*v3*a33 - u3*v3 *n3 + 2*i*v1 *v3 *c12 - 2*v1*v2*v3 *c12
2 2
- v1*v3 *m1 - i*v2*v3 *m1
which the program can not factorize further.
{HAM,FI} = 0
2 2 3
FI=u1*v1*v3 + u2*v2*v3 + u3*v3
= a product of the elements of: {v3,
v3,
u1*v1 + u2*v2 + u3*v3}
{HAM,FI} = 0
2 3 3 2 2 2
FI=i*u1*v1 *v2 + i*u1*v2 - i*u2*v1 - i*u2*v1*v2 + u3*v1 *v3 + u3*v2 *v3
= a product of the elements of: {i,
v1 - i*v2,
v1 + i*v2,
u1*v2 - u2*v1 - i*u3*v3}
{HAM,FI} = 0
2
FI=u1*v1*v3 + u2*v2*v3 + u3*v3
= a product of the elements of: {v3,u1*v1 + u2*v2 + u3*v3}
{HAM,FI} = 0
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
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
FI=v1 *v3 + v2 *v3
= a product of the elements of: {v3,v1 - i*v2,v1 + i*v2}
{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
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
2 2
FI= - i*u1*v2*n3 + i*u2*v1*n3 + u3 *v3*a33 - 2*i*v1 *v3*c12 + 2*v1*v2*v3*c12
+ v1*v3*m1 + i*v2*v3*m1
which the program can not factorize further.
{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 + v1*v3 *m1
2
+ i*v2*v3 *m1
= a product of the elements of: {v3,
v3,
2 2
u3 *a33 + u3*n3 - 2*i*v1 *c12 + 2*v1*v2*c12 + v1*m1 + i*v2*m1}
{HAM,FI} = 0
2 2 3
FI= - u1*u3 *v2*a33 - u1*u3*v2*a33*n3 - 2*u1*v1 *a33*c12
2 2
+ 2*i*u1*v1 *v2*a33*c12 - 4*u1*v1*v2 *a33*c12 - u1*v1*v2*a33*m1
2 2 2 2
- i*u1*v2 *a33*m1 - u1*v2*n3 + u2*u3 *v1*a33 + u2*u3*v1*a33*n3
3 2 2
- 2*i*u2*v1 *a33*c12 + u2*v1 *a33*m1 + i*u2*v1*v2*a33*m1 + u2*v1*n3
3 3 2
- 2*u2*v2 *a33*c12 + i*u3 *v3*a33 + 2*i*u3*v1*v2*v3*a33*c12
2
+ i*u3*v1*v3*a33*m1 - 2*u3*v2 *v3*a33*c12 - u3*v2*v3*a33*m1
2
- 2*v1 *v3*c12*n3 - 2*i*v1*v2*v3*c12*n3 - i*v1*v3*m1*n3 + v2*v3*m1*n3
which the program can not factorize further.
{HAM,FI} = 0
3 3 2 2 2
FI=u1 *v1*a33 + i*u1 *v2*a33 - i*u1 *u2*v1*a33 + u1 *u2*v2*a33 + 4*u1 *u3*v3*a33
2 2 2 2 3
+ 2*u1 *v3*n3 + u1*u2 *v1*a33 + i*u1*u2 *v2*a33 - 4*u1*v3 *m1 - i*u2 *v1*a33
3 2 2 2
+ u2 *v2*a33 + 4*u2 *u3*v3*a33 + 2*u2 *v3*n3 - 8*u2*v1*v3 *c12
2 2 3
- 8*i*u2*v2*v3 *c12 - 4*i*u2*v3 *m1 - 8*i*u3*v3 *c12
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u3**2*a33 + u3*n3 + 2*v1*v2*c12 + v1*m1 + 2*i*v2**2*c12 + i*v2*m1 + i*v3**2*
c12$
FI=i*u1**2*v1*v2*a33 - i*u1*u2*v1**2*a33 + i*u1*u2*v2**2*a33 + u1*u3*v1*v3*a33 +
i*u1*u3*v2*v3*a33 - i*u2**2*v1*v2*a33 - i*u2*u3*v1*v3*a33 + u2*u3*v2*v3*a33 -
u3*v3**2*n3 + 2*i*v1**2*v3**2*c12 - 2*v1*v2*v3**2*c12 - v1*v3**2*m1 - i*v2*v3**2
*m1$
FI= - u1**2*v2**2*a33 + 2*u1*u2*v1*v2*a33 + 2*i*u1*u3*v2*v3*a33 - u2**2*v1**2*
a33 - 2*i*u2*u3*v1*v3*a33 - u3*v3**2*n3 + 2*i*v1**2*v3**2*c12 - 2*v1*v2*v3**2*
c12 - v1*v3**2*m1 - i*v2*v3**2*m1$
FI=u1*v1*v3**2 + u2*v2*v3**2 + u3*v3**3$
FI=i*u1*v1**2*v2 + i*u1*v2**3 - i*u2*v1**3 - i*u2*v1*v2**2 + u3*v1**2*v3 + u3*v2
**2*v3$
FI=u1*v1*v3 + u2*v2*v3 + u3*v3**2$
FI=u1*v2*v3**2 - u2*v1*v3**2 - i*u3*v3**3$
FI= - u1*v2*v3 + u2*v1*v3 + i*u3*v3**2$
FI=i*u1*v2 - i*u2*v1 + u3*v3$
FI=v1**2*v3 + v2**2*v3$
FI=v1**2*v3**2 + v2**2*v3**2$
FI=v3$
FI=v3**2$
FI=v3**3$
FI=v3**4$
FI= - i*u1*v2*n3 + i*u2*v1*n3 + u3**2*v3*a33 - 2*i*v1**2*v3*c12 + 2*v1*v2*v3*c12
+ v1*v3*m1 + i*v2*v3*m1$
FI=u3**2*v3**2*a33 + u3*v3**2*n3 - 2*i*v1**2*v3**2*c12 + 2*v1*v2*v3**2*c12 + v1*
v3**2*m1 + i*v2*v3**2*m1$
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 - 4*u1*v1*v2**2*a33*c12 - u1*v1*v2*a33*m1 - i*u1*v2**2*a33*m1 - u1*v2
*n3**2 + u2*u3**2*v1*a33**2 + u2*u3*v1*a33*n3 - 2*i*u2*v1**3*a33*c12 + u2*v1**2*
a33*m1 + i*u2*v1*v2*a33*m1 + u2*v1*n3**2 - 2*u2*v2**3*a33*c12 + i*u3**3*v3*a33**
2 + 2*i*u3*v1*v2*v3*a33*c12 + i*u3*v1*v3*a33*m1 - 2*u3*v2**2*v3*a33*c12 - u3*v2*
v3*a33*m1 - 2*v1**2*v3*c12*n3 - 2*i*v1*v2*v3*c12*n3 - i*v1*v3*m1*n3 + v2*v3*m1*
n3$
FI=u1**3*v1*a33 + i*u1**3*v2*a33 - i*u1**2*u2*v1*a33 + u1**2*u2*v2*a33 + 4*u1**2
*u3*v3*a33 + 2*u1**2*v3*n3 + u1*u2**2*v1*a33 + i*u1*u2**2*v2*a33 - 4*u1*v3**2*m1
- i*u2**3*v1*a33 + u2**3*v2*a33 + 4*u2**2*u3*v3*a33 + 2*u2**2*v3*n3 - 8*u2*v1*
v3**2*c12 - 8*i*u2*v2*v3**2*c12 - 4*i*u2*v3**2*m1 - 8*i*u3*v3**3*c12$