Solution 3 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
2
- a22*b33*n1*n3*r15 - b33*n1 *n3*r29
r10=---------------------------------------
2 2 2 2
a22 *n1 - a22 *n3
r11=0
1 2 1 2 3
- ---*a22*b33*n1 *r15 - ---*a22*b33*n3 *r15 - b33*n1 *r29
2 2
r12=------------------------------------------------------------
2 2 2 2
a22 *n1 - a22 *n3
n3*r15
r13=--------
n1
r14=0
1 2 1 2 2
- ---*a22*b33 *n1*r15 - ---*b33 *n1 *r29
4 4
r20=-------------------------------------------
2 2 2 2
a22 *n1 - a22 *n3
r21=0
r23=0
r24=0
1 2 1 2 2
- ---*a22*b33 *n1*r15 - ---*b33 *n1 *r29
4 4
r25=-------------------------------------------
2 2 2 2
a22 *n1 - a22 *n3
2
a22*b33*n1*r15 + b33*n1 *r29
r26=------------------------------
2 2
a22*n1 - a22*n3
r27=0
- a22*b33*n3*r15 - b33*n1*n3*r29
r28=-----------------------------------
2 2
a22*n1 - a22*n3
r210=0
r212=0
r213=0
2
a22*n1*r15 + n1 *r29
r214=----------------------
2 2
n1 - n3
r215=0
r216=0
2
- a22*b33*n1*r15 - b33*n1 *r29
r217=---------------------------------
2 2
a22*n1 - a22*n3
- 2*a22*n3*r15 - 2*n1*n3*r29
r218=-------------------------------
2 2
n1 - n3
r219=0
r30=0
r31=0
r32=0
r33=0
1
- ---*b33*n1*r425
2
r34=--------------------
2
a22
r35=0
1
- ---*b33*n1*r425
2
r36=--------------------
2
a22
r37=0
r38=0
1
- ---*b33*n1*r425
2
r39=--------------------
2
a22
- n3*r425
r310=------------
a22
r311=0
- n3*r425
r312=------------
a22
1
- ---*b33*n1*r488
2
r313=--------------------
2
a22
r314=0
- n3*r425
r315=------------
a22
- n3*r488
r316=------------
a22
r317=0
a22*b33*r15 + b33*n1*r29
r318=--------------------------
2 2
n1 - n3
2
2*a22 *n3*r15 + 2*a22*n1*n3*r29
r319=---------------------------------
3 2
n1 - n1*n3
r320=0
r321=0
r322=0
r323=0
1
- ---*b33*n1*r488
2
r324=--------------------
2
a22
r325=0
r326=0
- n3*r488
r327=------------
a22
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
- n1*r425
r335=------------
a22
r336=0
- n1*r425
r337=------------
a22
r338=0
r339=0
1
- a22*n1*r425 - ---*b33*n1*r488
2
r340=----------------------------------
2
a22
- n1*r488
r341=------------
a22
r342=0
- n3*r488
r343=------------
a22
2
2*a22 *r15 + 2*a22*n1*r29
r344=---------------------------
2 2
n1 - n3
r345=0
- n1*r488
r346=------------
a22
r347=0
r348=0
r349=0
r350=0
r351=0
- n1*r488
r352=------------
a22
r353=0
r354=0
r355=0
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
r426=0
r427=r425
r428=0
r429=0
r430=r425
r431=r488
r432=0
r433=0
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=r488
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
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
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
r4108=0
r4109=0
r4110=0
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
- b33*n3
m3=-----------
a22
m2=0
1
- ---*b33*n1
2
m1=---------------
a22
n2=0
1 2
- ---*b33
4
c33=-------------
a22
c23=0
c22=0
c13=0
c12=0
c11=0
b32=0
b31=0
b23=0
b22=0
b21=0
b13=0
b12=0
b11=0
a33=0
a23=0
a13=0
a12=0
a11=a22
3 2
- a22 *r15 - a22 *n1*r29
r434=---------------------------
3 2
n1 - n1*n3
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:
r29, r15, r425, r488, n3, b33, n1, 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,a22}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11 - a22,
a12,
a13,
a23,
a33,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
c11,
c12,
c13,
c22,
c23,
a22*c33 + 1/4*b33**2,
n2,
a22*m1 + 1/2*b33*n1,
m2,
a22*m3 + b33*n3}$
The system of equations related to the Hamiltonian HAM:
1
- ---*v1*b33*n1
2 2 2
HAM=u1 *a22 + u1*n1 + u2 *a22 + u3*v3*b33 + u3*n3 + ------------------
a22
1 2 2
- ---*v3 *b33
4 - v3*b33*n3
+ ----------------- + --------------
a22 a22
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 2 2
FI= - u1 *v1*a22*n1 - u1*u2*v2*a22*n1 + u1*u3 *v1*a22 - u1*u3*v1*a22*n3
1 2 2 2
- u1*u3*v3*a22*n1 - ---*u1*v1 *b33*n1 + u2*u3 *v2*a22 - u2*u3*v2*a22*n3
2
1 3 2 2 1
- ---*u2*v1*v2*b33*n1 + u3 *v3*a22 - u3 *v3*a22*n3 - ---*u3*v1*v3*b33*n1
2 2
= a product of the elements of: {u1*v1 + u2*v2 + u3*v3,
2 2 1
- u1*a22*n1 + u3 *a22 - u3*a22*n3 - ---*v1*b33*n1}
2
{HAM,FI} = 0
2 2 2 2 2 2
FI= - u1*v1 *a22*n1 - u1*v2 *a22*n1 - u1*v3 *a22*n1 + u3 *v1 *a22
2 2 2 2 2 2 2 2
+ u3 *v2 *a22 + u3 *v3 *a22 - u3*v1 *a22*n3 - u3*v2 *a22*n3
2 1 3 1 2 1 2
- u3*v3 *a22*n3 - ---*v1 *b33*n1 - ---*v1*v2 *b33*n1 - ---*v1*v3 *b33*n1
2 2 2
2 2 2
= a product of the elements of: {v1 + v2 + v3 ,
2 2 1
- u1*a22*n1 + u3 *a22 - u3*a22*n3 - ---*v1*b33*n1}
2
{HAM,FI} = 0
2 3 2 2
FI=2*u1*u3 *a22 *n1 - 2*u1*u3*a22 *n1*n3 - u1*v1*a22*b33*n1
3 2 2 2 2 4 4 3 3
+ u1*(a22*n1 - a22*n1*n3 ) + u2 *a22 *n1 - u3 *a22 + 2*u3 *a22 *n3
2 2 2
+ u3 *v1*a22 *b33*n1 - u3*v1*a22*b33*n1*n3 + u3*v3*a22*b33*n1
2 3 1 2 2 2
+ u3*(a22*n1 *n3 - a22*n3 ) - ---*v1 *b33 *n1
4
1 3 1 2 1 2 2 2 2
+ v1*( - ---*b33*n1 - ---*b33*n1*n3 ) - ---*v3 *b33 *n1 - v3*b33*n1 *n3
2 2 4
which the program can not factorize further.
{HAM,FI} = 0
2 3 2 2 2 2 2
FI=2*u1*u3 *a22 *n1 - 2*u1*u3*a22 *n1*n3 - u1*v1*a22*b33*n1 + u2 *a22 *n1
4 4 3 3 2 2 2 2 2 2 2
- u3 *a22 + 2*u3 *a22 *n3 + u3 *v1*a22 *b33*n1 + u3 *(a22 *n1 - a22 *n3 )
2 1 2 2 2 3
- u3*v1*a22*b33*n1*n3 + u3*v3*a22*b33*n1 - ---*v1 *b33 *n1 - v1*b33*n1
4
1 2 2 2 2
- ---*v3 *b33 *n1 - v3*b33*n1 *n3
4
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a22 + u1*n1 + u2**2*a22 + u3*v3*b33 + u3*n3 + ( - 1/2*v1*b33*n1)/a22 +
( - 1/4*v3**2*b33**2)/a22 + ( - v3*b33*n3)/a22$
FI= - u1**2*v1*a22*n1 - u1*u2*v2*a22*n1 + u1*u3**2*v1*a22**2 - u1*u3*v1*a22*n3 -
u1*u3*v3*a22*n1 - 1/2*u1*v1**2*b33*n1 + u2*u3**2*v2*a22**2 - u2*u3*v2*a22*n3 -
1/2*u2*v1*v2*b33*n1 + u3**3*v3*a22**2 - u3**2*v3*a22*n3 - 1/2*u3*v1*v3*b33*n1$
FI= - u1*v1**2*a22*n1 - u1*v2**2*a22*n1 - u1*v3**2*a22*n1 + u3**2*v1**2*a22**2 +
u3**2*v2**2*a22**2 + u3**2*v3**2*a22**2 - u3*v1**2*a22*n3 - u3*v2**2*a22*n3 -
u3*v3**2*a22*n3 - 1/2*v1**3*b33*n1 - 1/2*v1*v2**2*b33*n1 - 1/2*v1*v3**2*b33*n1$
FI=2*u1*u3**2*a22**3*n1 - 2*u1*u3*a22**2*n1*n3 - u1*v1*a22*b33*n1**2 + u1*(a22*
n1**3 - a22*n1*n3**2) + u2**2*a22**2*n1**2 - u3**4*a22**4 + 2*u3**3*a22**3*n3 +
u3**2*v1*a22**2*b33*n1 - u3*v1*a22*b33*n1*n3 + u3*v3*a22*b33*n1**2 + u3*(a22*n1
**2*n3 - a22*n3**3) - 1/4*v1**2*b33**2*n1**2 + v1*( - 1/2*b33*n1**3 - 1/2*b33*n1
*n3**2) - 1/4*v3**2*b33**2*n1**2 - v3*b33*n1**2*n3$
FI=2*u1*u3**2*a22**3*n1 - 2*u1*u3*a22**2*n1*n3 - u1*v1*a22*b33*n1**2 + u2**2*a22
**2*n1**2 - u3**4*a22**4 + 2*u3**3*a22**3*n3 + u3**2*v1*a22**2*b33*n1 + u3**2*(
a22**2*n1**2 - a22**2*n3**2) - u3*v1*a22*b33*n1*n3 + u3*v3*a22*b33*n1**2 - 1/4*
v1**2*b33**2*n1**2 - v1*b33*n1**3 - 1/4*v3**2*b33**2*n1**2 - v3*b33*n1**2*n3$