Solution 14 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r11=0
r12=0
r13=0
r14=0
r15=0
m2*r316
r21=---------
a33
r22=0
i*m2*r316
r23=-----------
a33
r24=0
r27=0
r28=0
r210=0
i*a33*r26 - i*n3*r316
r212=-----------------------
a33
r213=0
r214=0
r215=0
- i*a33*r26 + i*n3*r316
r216=--------------------------
a33
r217=0
r218=0
r219=0
r220=0
- 2*i*c13*r316 + m2*r425
r31=---------------------------
a33
r33=0
2*c13*r316 + i*m2*r425
r34=------------------------
a33
r35=0
r36=0
r37=r32
r38=0
r39=0
a33*r321 - i*a33*r323 + n3*r425
r310=---------------------------------
a33
r311=0
2*i*a33*c13*r26 - 2*i*c13*n3*r316
r312=-----------------------------------
a33*m2
r313=0
r314=0
2*i*a33*c13*r26 - 2*i*c13*n3*r316
r315=-----------------------------------
a33*m2
r317=0
r318=0
r319=0
r320=0
r322=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= - r323
r337=0
r338=r321
r339=0
r340=0
r341=0
r342=0
r343=0
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*i*c13*r425
r41=-----------------
a33
r43=0
r44=0
2*c13*r425
r45=------------
a33
r46=0
r47=0
r48=0
r49=r42
r410=0
r411=0
r412=0
r413=0
2 2 2
4*a33*c13 *r26 - 2*a33*c13*m2*r323 + a33*m2 *r474 - 4*c13 *n3*r316
r415=--------------------------------------------------------------------
2
a33*m2
r416=0
2 2
- 4*a33*c13 *r26 + 2*a33*c13*m2*r323 + 4*c13 *n3*r316
r417=--------------------------------------------------------
2
a33*m2
r418=0
r419=0
r420=0
r421=0
r423=0
r424=0
r426=0
r427=0
r428=0
r429=0
r431=0
r432=0
r433=0
r435=0
r436=r474
r437=0
r438=0
2 2
4*i*a33*c13 *r26 - 2*i*a33*c13*m2*r323 - 4*i*c13 *n3*r316
r439=-----------------------------------------------------------
2
a33*m2
r440=0
r441=0
r442=0
r444=0
r445=0
r446=0
r447=0
r448=0
r449=0
r450=0
r451=0
r453=0
r454=0
r455=0
r456=0
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
r464=0
r465=0
r466=0
r467=0
r468=0
r469=0
r470=0
2 2
- 4*i*a33*c13 *r26 + 2*i*a33*c13*m2*r323 + 4*i*c13 *n3*r316
r471=--------------------------------------------------------------
2
a33*m2
r472=0
r473=0
r475=0
r476=0
r477=0
r478=0
r479=0
r480=0
r481=0
r482=0
r483=0
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=0
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4107=0
r4108=0
r4109=0
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
m3=0
m1=i*m2
n2=0
n1=0
c33=0
c23= - i*c13
c22=0
c12=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 2
- 4*a33*c13 *r26 + 2*a33*c13*m2*r323 + 4*c13 *n3*r316
r422=--------------------------------------------------------
2
a33*m2
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:
r26, r40, r30, r20, r10, r42, r32, r425, r316, r323,
r321, r474, m2, n3, c13, 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,
c12,
c22,
i*c13 + c23,
c33,
n1,
n2,
m1 - i*m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2
HAM=u3 *a33 + u3*n3 + 2*v1*v3*c13 + i*v1*m2 - 2*i*v2*v3*c13 + v2*m2
has apart from the Hamiltonian and Casimirs the following 12 first integrals:
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
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
FI=2*i*u1*v2*v3 *c13 - u1*v2*v3*m2 - 2*i*u2*v1*v3 *c13 + u2*v1*v3*m2
2 2 3 2
+ 2*u3*v1 *v3*c13 + 2*u3*v2 *v3*c13 - 2*u3*v3 *c13 - i*u3*v3 *m2
= a product of the elements of: {2*i,
v3,
i*u1*v2*m2 - i*u2*v1*m2 2
u1*v2*v3*c13 + ------------ - u2*v1*v3*c13 + --------------- - i*u3*v1 *c13
2 2
2 2 - u3*v3*m2
- i*u3*v2 *c13 + i*u3*v3 *c13 + -------------}
2
{HAM,FI} = 0
2 2 2 2 2
FI=4*i*u1*v2*v3 *c13 *n3 + i*u1*v2*m2 *n3 - 4*i*u2*v1*v3 *c13 *n3
2 2 2 2 2
- i*u2*v1*m2 *n3 + u3 *v3*a33*m2 + 4*u3*v1 *v3*c13 *n3
2 2 2 2
- 2*i*u3*v1 *c13*m2*n3 + 4*u3*v2 *v3*c13 *n3 - 2*i*u3*v2 *c13*m2*n3
3 2 2 2 3 2 2
- 4*u3*v3 *c13 *n3 + 2*v1*v3 *c13*m2 + i*v1*v3*m2 - 2*i*v2*v3 *c13*m2
3
+ v2*v3*m2
which the program can not factorize further.
{HAM,FI} = 0
2 2 2 3 2 3
FI=u3 *v3 *a33 + u3*v3 *n3 + 2*v1*v3 *c13 + i*v1*v3 *m2 - 2*i*v2*v3 *c13
2
+ v2*v3 *m2
= a product of the elements of: {v3,
v3,
2
u3 *a33 + u3*n3 + 2*v1*v3*c13 + i*v1*m2 - 2*i*v2*v3*c13 + v2*m2}
{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 2 2 2 2
FI= - 4*i*u1*v2*v3 *c13 - i*u1*v2*m2 + 4*i*u2*v1*v3 *c13 + i*u2*v1*m2
2 2 2 2 2
- 4*u3*v1 *v3*c13 + 2*i*u3*v1 *c13*m2 - 4*u3*v2 *v3*c13
2 3 2 2
+ 2*i*u3*v2 *c13*m2 + 4*u3*v3 *c13 + u3*v3*m2
= a product of the elements of: { - 4*i,
- i*m2
v3*c13 + ---------,
2
i*u1*v2*m2 - i*u2*v1*m2 2
u1*v2*v3*c13 + ------------ - u2*v1*v3*c13 + --------------- - i*u3*v1 *c13
2 2
2 2 - u3*v3*m2
- i*u3*v2 *c13 + i*u3*v3 *c13 + -------------}
2
{HAM,FI} = 0
And again in machine readable form:
HAM=u3**2*a33 + u3*n3 + 2*v1*v3*c13 + i*v1*m2 - 2*i*v2*v3*c13 + v2*m2$
FI=u1*v1*v3**2 + u2*v2*v3**2 + u3*v3**3$
FI=u1*v1*v3 + u2*v2*v3 + u3*v3**2$
FI=2*i*u1*v2*v3**2*c13 - u1*v2*v3*m2 - 2*i*u2*v1*v3**2*c13 + u2*v1*v3*m2 + 2*u3*
v1**2*v3*c13 + 2*u3*v2**2*v3*c13 - 2*u3*v3**3*c13 - i*u3*v3**2*m2$
FI=4*i*u1*v2*v3**2*c13**2*n3 + i*u1*v2*m2**2*n3 - 4*i*u2*v1*v3**2*c13**2*n3 - i*
u2*v1*m2**2*n3 + u3**2*v3*a33*m2**2 + 4*u3*v1**2*v3*c13**2*n3 - 2*i*u3*v1**2*c13
*m2*n3 + 4*u3*v2**2*v3*c13**2*n3 - 2*i*u3*v2**2*c13*m2*n3 - 4*u3*v3**3*c13**2*n3
+ 2*v1*v3**2*c13*m2**2 + i*v1*v3*m2**3 - 2*i*v2*v3**2*c13*m2**2 + v2*v3*m2**3$
FI=u3**2*v3**2*a33 + u3*v3**2*n3 + 2*v1*v3**3*c13 + i*v1*v3**2*m2 - 2*i*v2*v3**3
*c13 + v2*v3**2*m2$
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= - 4*i*u1*v2*v3**2*c13**2 - i*u1*v2*m2**2 + 4*i*u2*v1*v3**2*c13**2 + i*u2*v1*
m2**2 - 4*u3*v1**2*v3*c13**2 + 2*i*u3*v1**2*c13*m2 - 4*u3*v2**2*v3*c13**2 + 2*i*
u3*v2**2*c13*m2 + 4*u3*v3**3*c13**2 + u3*v3*m2**2$