Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
1
---*m2*n2*n3*r464
2
r10=-------------------
3
a22
1 2 1 2 - 1 2
---*a22*m2*n2 *r464 - ---*a22*m2*n3 *r464 + ------*i*b31*n2 *n3*r464
2 4 4
r11=----------------------------------------------------------------------
4
a22
1 2 1 2 1 2
---*i*a22*m2*n2 *r464 + ---*i*a22*m2*n3 *r464 + ---*b31*n2 *n3*r464
2 4 4
r12=---------------------------------------------------------------------
4
a22
1 3
- ---*n3 *r464
4
r13=-----------------
3
a22
1 2
- ---*n2*n3 *r464
4
r14=--------------------
3
a22
- 1 2
------*i*n2*n3 *r464
4
r15=----------------------
3
a22
1 2 2 1 2 2
- ---*a22 *m2 *r464 - ---*b31 *n2 *r464
4 4
r20=------------------------------------------
4
a22
- 1
------*i*b31*m2*n2*r464
2
r21=-------------------------
3
a22
1
- ---*b31*m2*n2*r464
2
r23=-----------------------
3
a22
- 1 2
------*i*m2 *r464
2
r24=-------------------
2
a22
1 2
- ---*m2 *r464
2
r25=-----------------
2
a22
1 - 1
---*a22*m2*n2*r464 + ------*i*b31*n2*n3*r464
2 4
r26=----------------------------------------------
3
a22
1 1 2 - 1 2
---*a22*m2*n3*r464 + ---*i*b31*n2 *r464 + ------*i*b31*n3 *r464
2 2 4
r27=-----------------------------------------------------------------
3
a22
- 1 1 2 1 2
------*i*a22*m2*n3*r464 - ---*b31*n2 *r464 - ---*b31*n3 *r464
2 2 4
r28=---------------------------------------------------------------
3
a22
1 2
- ---*n3 *r464
4
r29=-----------------
2
a22
1 - 1 2
- ---*a22*m2*n3*r464 + ------*i*b31*n2 *r464
2 2
r210=-----------------------------------------------
3
a22
- 3 1
------*i*a22*m2*n2*r464 - ---*b31*n2*n3*r464
2 4
r212=----------------------------------------------
3
a22
r213=0
r214=0
1 1 2
---*i*a22*m2*n3*r464 + ---*b31*n2 *r464
2 2
r215=-----------------------------------------
3
a22
3 1
---*i*a22*m2*n2*r464 + ---*b31*n2*n3*r464
2 4
r216=-------------------------------------------
3
a22
r217=0
r218=0
r219=0
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=0
1 2
---*b31 *n2*r464
2
r311=------------------
3
a22
1
---*i*b31*m2*r464
2
r312=-------------------
2
a22
- 1 2
------*i*b31 *n2*r464
2
r313=-----------------------
3
a22
b31*m2*r464
r314=-------------
2
a22
- 1
------*i*b31*m2*r464
2
r315=----------------------
2
a22
- i*b31*n2*r464
r316=------------------
2
a22
1
a22*m2*r464 + ---*i*b31*n3*r464
2
r317=---------------------------------
2
a22
1
- i*a22*m2*r464 + ---*b31*n3*r464
2
r318=------------------------------------
2
a22
n3*r464
r319=---------
a22
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
- 1
------*i*b31*n3*r464
2
r326=----------------------
2
a22
- 1
------*i*b31*n2*r464
2
r327=----------------------
2
a22
3
---*b31*n2*r464
2
r328=-----------------
2
a22
n2*r464
r329=---------
a22
- 1
------*i*b31*n2*r464
2
r330=----------------------
2
a22
m2*r464
r331=---------
a22
r332=0
n3*r464
r333=---------
a22
n2*r464
r334=---------
a22
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
1
- ---*b31*n3*r464
2
r341=--------------------
2
a22
3
- ---*b31*n2*r464
2
r342=--------------------
2
a22
- 1
------*i*b31*n2*r464
2
r343=----------------------
2
a22
i*n2*r464
r344=-----------
a22
r345=0
- i*m2*r464
r346=--------------
a22
m2*r464
r347=---------
a22
r348=0
i*n2*r464
r349=-----------
a22
- 1
------*i*b31*n2*r464
2
r350=----------------------
2
a22
r351=0
- i*m2*r464
r352=--------------
a22
n3*r464
r353=---------
a22
n2*r464
r354=---------
a22
i*n2*r464
r355=-----------
a22
r40=0
r41=0
r42=0
r43=0
r45=0
r46=0
r47=0
r48=0
r49=0
r410=0
r411=0
r412=0
r413=0
r414=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
r425=0
r426=0
1 2
- ---*b31 *r464
4
r427=------------------
2
a22
r428=0
1 2
---*i*b31 *r464
2
r429=-----------------
2
a22
1 2
---*b31 *r464
4
r430=---------------
2
a22
r431=0
i*b31*r464
r432=------------
a22
b31*r464
r433=----------
a22
r434=r464
r435=0
r436=0
r437=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r446=0
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
r453=0
r454=0
r455=0
r456=0
r458=0
r459=0
r460=0
r461=0
i*b31*r464
r462=------------
a22
r463=0
r465=0
r466=0
r467=0
r468=0
r469=0
r470=0
r471=0
r472=0
r473=0
r474=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
b31*r464
r497=----------
a22
i*b31*r464
r498=------------
a22
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
b31*r464
r4113=----------
a22
r4114=r464
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
- i*b31*n2
m3=-------------
a22
m1= - i*m2
c33=0
c23=0
c22=0
c13=0
c12=0
c11=0
b33=0
b32=i*b31
b23=0
b22=0
b21=0
b13=0
b12=0
b11=0
a33=2*a22
a23=0
a13=0
a12=0
a11=a22
n1=i*n2
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:
n2, r464, m2, n3, b31, a22
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.
{n1,r464,a11,a22}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{n1 - i*n2,
a11 - a22,
a12,
a13,
a23,
- 2*a22 + a33,
b11,
b12,
b13,
b21,
b22,
b23,
- i*b31 + b32,
b33,
c11,
c12,
c13,
c22,
c23,
c33,
m1 + i*m2,
a22*m3 + i*b31*n2}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a22 + i*u1*n2 + u2 *a22 + u2*n2 + 2*u3 *a22 + u3*v1*b31 + i*u3*v2*b31
- i*v3*b31*n2
+ u3*n3 - i*v1*m2 + v2*m2 + ----------------
a22
has apart from the Hamiltonian and Casimirs only the following first integral:
3 3 2 3 2 2 4 2 3
FI=i*u1 *a22 *n2 + u1 *u2*a22 *n2 + u1 *u3 *a22 + u1 *u3*v1*a22 *b31
2 3 2 3 - 1 2 2
+ u1 *u3*a22 *n3 - i*u1 *v1*a22 *m2 + ------*i*u1 *v3*a22 *b31*n2
2
2 3 3 3
+ i*u1*u2 *a22 *n2 + i*u1*u2*u3*v1*a22 *b31 + u1*u2*u3*v2*a22 *b31
3 3 2 3
+ u1*u2*v1*a22 *m2 - i*u1*u2*v2*a22 *m2 + i*u1*u3 *a22 *n2
- 1 2 3 2
+ ------*i*u1*u3*v1*a22 *b31*n2 - ---*u1*u3*v2*a22 *b31*n2
2 2
1 2 3 2 1
- ---*u1*u3*v3*a22 *b31*n3 + u1*v2*(---*i*a22 *m2*n2 + ---*a22*b31*n2*n3)
2 2 4
1 2 1 2 - 1 2
+ u1*v3*(---*i*a22 *m2*n3 + ---*a22*b31*n2 ) + ------*i*u1*a22*n2*n3
2 2 4
3 3 2 2 4 2 3 2 3
+ u2 *a22 *n2 + u2 *u3 *a22 + i*u2 *u3*v2*a22 *b31 + u2 *u3*a22 *n3
2 3 - 1 2 2 2 3
+ u2 *v2*a22 *m2 + ------*i*u2 *v3*a22 *b31*n2 + u2*u3 *a22 *n2
2
3 2 - 1 2
+ ---*u2*u3*v1*a22 *b31*n2 + ------*i*u2*u3*v2*a22 *b31*n2
2 2
- 1 2
+ ------*i*u2*u3*v3*a22 *b31*n3
2
- 3 2 1
+ u2*v1*(------*i*a22 *m2*n2 - ---*a22*b31*n2*n3)
2 4
1 2 - 1 2 1 2
+ u2*v3*( - ---*a22 *m2*n3 + ------*i*a22*b31*n2 ) - ---*u2*a22*n2*n3
2 2 4
4 4 3 3 3 3 3 3
+ u3 *a22 + u3 *v1*a22 *b31 + i*u3 *v2*a22 *b31 + u3 *a22 *n3
1 2 2 2 2 1 2 2 2
+ ---*u3 *v1 *a22 *b31 + ---*i*u3 *v1*v2*a22 *b31
4 2
2 3 1 2 1 2 2 2 2
+ u3 *v1*( - i*a22 *m2 + ---*a22 *b31*n3) - ---*u3 *v2 *a22 *b31
2 4
2 3 1 2 2 2
+ u3 *v2*(a22 *m2 + ---*i*a22 *b31*n3) - i*u3 *v3*a22 *b31*n2
2
1 2 2 2 - 1 2 2 2
- ---*u3 *a22 *n3 + ------*i*u3*v1 *a22 *b31*m2 + u3*v1*v2*a22 *b31*m2
4 2
- 1 2
+ ------*i*u3*v1*v3*a22*b31 *n2
2
- 1 2 1 2 1 2
+ u3*v1*(------*i*a22 *m2*n3 - ---*a22*b31*n2 - ---*a22*b31*n3 )
2 2 4
1 2 2 1 2
+ ---*i*u3*v2 *a22 *b31*m2 + ---*u3*v2*v3*a22*b31 *n2
2 2
1 2 1 2 - 1 2
+ u3*v2*(---*a22 *m2*n3 + ---*i*a22*b31*n2 + ------*i*a22*b31*n3 )
2 2 4
1 2 - 1 1 3
+ u3*v3*(---*a22 *m2*n2 + ------*i*a22*b31*n2*n3) - ---*u3*a22*n3
2 4 4
1 2 2 2 - 1 2 2 1
- ---*v1 *a22 *m2 + ------*i*v1*v2*a22 *m2 - ---*v1*v3*a22*b31*m2*n2
2 2 2
1 2 1 2 1 2
+ v1*(---*i*a22*m2*n2 + ---*i*a22*m2*n3 + ---*b31*n2 *n3)
2 4 4
- 1
+ ------*i*v2*v3*a22*b31*m2*n2
2
1 2 1 2 - 1 2
+ v2*(---*a22*m2*n2 - ---*a22*m2*n3 + ------*i*b31*n2 *n3)
2 4 4
2 1 2 2 1 2 2 1
+ v3 *( - ---*a22 *m2 - ---*b31 *n2 ) + ---*v3*a22*m2*n2*n3
4 4 2
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a22 + i*u1*n2 + u2**2*a22 + u2*n2 + 2*u3**2*a22 + u3*v1*b31 + i*u3*v2*
b31 + u3*n3 - i*v1*m2 + v2*m2 + ( - i*v3*b31*n2)/a22$
FI=i*u1**3*a22**3*n2 + u1**2*u2*a22**3*n2 + u1**2*u3**2*a22**4 + u1**2*u3*v1*a22
**3*b31 + u1**2*u3*a22**3*n3 - i*u1**2*v1*a22**3*m2 + ( - 1)/2*i*u1**2*v3*a22**2
*b31*n2 + i*u1*u2**2*a22**3*n2 + i*u1*u2*u3*v1*a22**3*b31 + u1*u2*u3*v2*a22**3*
b31 + u1*u2*v1*a22**3*m2 - i*u1*u2*v2*a22**3*m2 + i*u1*u3**2*a22**3*n2 + ( - 1)/
2*i*u1*u3*v1*a22**2*b31*n2 - 3/2*u1*u3*v2*a22**2*b31*n2 - 1/2*u1*u3*v3*a22**2*
b31*n3 + u1*v2*(3/2*i*a22**2*m2*n2 + 1/4*a22*b31*n2*n3) + u1*v3*(1/2*i*a22**2*m2
*n3 + 1/2*a22*b31*n2**2) + ( - 1)/4*i*u1*a22*n2*n3**2 + u2**3*a22**3*n2 + u2**2*
u3**2*a22**4 + i*u2**2*u3*v2*a22**3*b31 + u2**2*u3*a22**3*n3 + u2**2*v2*a22**3*
m2 + ( - 1)/2*i*u2**2*v3*a22**2*b31*n2 + u2*u3**2*a22**3*n2 + 3/2*u2*u3*v1*a22**
2*b31*n2 + ( - 1)/2*i*u2*u3*v2*a22**2*b31*n2 + ( - 1)/2*i*u2*u3*v3*a22**2*b31*n3
+ u2*v1*(( - 3)/2*i*a22**2*m2*n2 - 1/4*a22*b31*n2*n3) + u2*v3*( - 1/2*a22**2*m2
*n3 + ( - 1)/2*i*a22*b31*n2**2) - 1/4*u2*a22*n2*n3**2 + u3**4*a22**4 + u3**3*v1*
a22**3*b31 + i*u3**3*v2*a22**3*b31 + u3**3*a22**3*n3 + 1/4*u3**2*v1**2*a22**2*
b31**2 + 1/2*i*u3**2*v1*v2*a22**2*b31**2 + u3**2*v1*( - i*a22**3*m2 + 1/2*a22**2
*b31*n3) - 1/4*u3**2*v2**2*a22**2*b31**2 + u3**2*v2*(a22**3*m2 + 1/2*i*a22**2*
b31*n3) - i*u3**2*v3*a22**2*b31*n2 - 1/4*u3**2*a22**2*n3**2 + ( - 1)/2*i*u3*v1**
2*a22**2*b31*m2 + u3*v1*v2*a22**2*b31*m2 + ( - 1)/2*i*u3*v1*v3*a22*b31**2*n2 +
u3*v1*(( - 1)/2*i*a22**2*m2*n3 - 1/2*a22*b31*n2**2 - 1/4*a22*b31*n3**2) + 1/2*i*
u3*v2**2*a22**2*b31*m2 + 1/2*u3*v2*v3*a22*b31**2*n2 + u3*v2*(1/2*a22**2*m2*n3 +
1/2*i*a22*b31*n2**2 + ( - 1)/4*i*a22*b31*n3**2) + u3*v3*(1/2*a22**2*m2*n2 + ( -
1)/4*i*a22*b31*n2*n3) - 1/4*u3*a22*n3**3 - 1/2*v1**2*a22**2*m2**2 + ( - 1)/2*i*
v1*v2*a22**2*m2**2 - 1/2*v1*v3*a22*b31*m2*n2 + v1*(1/2*i*a22*m2*n2**2 + 1/4*i*
a22*m2*n3**2 + 1/4*b31*n2**2*n3) + ( - 1)/2*i*v2*v3*a22*b31*m2*n2 + v2*(1/2*a22*
m2*n2**2 - 1/4*a22*m2*n3**2 + ( - 1)/4*i*b31*n2**2*n3) + v3**2*( - 1/4*a22**2*m2
**2 - 1/4*b31**2*n2**2) + 1/2*v3*a22*m2*n2*n3$