Solution 5 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r10=0
1
---*b33*r14
2
r11=-------------
a22
r12=0
r13=0
r15=0
1 3 1 2 2
---*a22 *b33*r26 - ----*b33 *n2 *r464
4 16
r20=---------------------------------------
4
a22
r21=0
r23=0
r24=0
1 3 1 2 2
---*a22 *b33*r26 - ----*b33 *n2 *r464
4 16
r25=---------------------------------------
4
a22
r27=0
r28=0
- 2*a22*r14
r29=--------------
n2
r210=0
r212=0
r213=0
3 1 2
- a22 *r26 - ---*b33*n2 *r464
4
r214=--------------------------------
2
a22 *b33
r215=0
r216=0
r217=r26
r218=0
r219=0
r30=0
1 3
----*b33 *n2*r464
16
r31=-------------------
4
a22
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=0
1 1 2
- ---*a22*b33*n2*r488 - ---*b33 *n2*r464
4 4
r311=-------------------------------------------
3
a22
r312=0
r313=0
r314=0
r315=0
r316=0
3 1 2
2*a22 *r26 - ---*b33*n2 *r464
4
r317=-------------------------------
2
a22 *n2
r318=0
r319=0
1 2
---*b33 *n2*r464
8
r320=------------------
3
a22
r321=0
1
- ---*b33*n2*r488
4
r322=--------------------
2
a22
r323=0
r324=0
r325=0
1 1
- ---*a22*n2*r488 - ---*b33*n2*r464
2 2
r326=--------------------------------------
2
a22
r327=0
r328=0
3 1 2
4*a22 *r26 + ---*b33*n2 *r464
2
r329=-------------------------------
a22*b33*n2
r330=0
1 1
- ---*a22*n2*r488 - ---*b33*n2*r464
2 4
r331=--------------------------------------
2
a22
r332=0
r333=0
1
- ---*n2*r464
2
r334=----------------
a22
r335=0
r336=0
r337=0
r338=0
1
- ---*b33*n2*r488
4
r339=--------------------
2
a22
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
r346=0
1
- ---*n2*r488
2
r347=----------------
a22
r348=0
r349=0
r350=0
1
- ---*b33*n2*r464
4
r351=--------------------
2
a22
r352=0
r353=0
1
- ---*n2*r464
2
r354=----------------
a22
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
1 2
- ---*b33 *r464
4
r425=------------------
2
a22
r426=0
r427=0
r428=0
r429=0
r430=0
a22*r488 + b33*r464
r431=---------------------
a22
r432=0
r433=0
3
- 4*a22 *r26
r434=---------------
2
b33*n2
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
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=r464
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
m3=0
m2=0
m1=0
n3=0
n1=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= - a22
a23=0
a13=0
a12=0
a11=a22
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:
r14, r26, r488, r464, b33, n2, 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.
{b33,n2,r464,a11,a22}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11 - a22,
a12,
a13,
a23,
a22 + a33,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
c11,
c12,
c13,
c22,
c23,
a22*c33 + 1/4*b33**2,
n1,
n3,
m1,
m2,
m3}$
The system of equations related to the Hamiltonian HAM:
1 2 2
- ---*v3 *b33
2 2 2 4
HAM=u1 *a22 + u2 *a22 + u2*n2 - u3 *a22 + u3*v3*b33 + -----------------
a22
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
1 2 3 2 2 4 1 2 2
FI= - ---*u1 *u2*a22 *n2 + u1 *u3 *a22 - ---*u1 *v2*a22 *b33*n2
2 4
1 3 3 2 2 4 1 2 2 1 2 2 2
- ---*u2 *a22 *n2 + u2 *u3 *a22 - ---*u2 *v2*a22 *b33*n2 - ---*u2 *a22 *n2
2 4 4
1 2 3 1 2 1 2 2
+ ---*u2*u3 *a22 *n2 - ---*u2*u3*v3*a22 *b33*n2 + ---*u2*v3 *a22*b33 *n2
2 2 8
3 3 1 2 2 1 2 2 2 2
+ u3 *v3*a22 *b33 - ---*u3 *v2*a22 *b33*n2 - ---*u3 *v3 *a22 *b33
4 4
1 2 1 2 2 2 1 2 3
- ---*u3*v2*v3*a22*b33 *n2 - ----*v1 *b33 *n2 + ----*v2*v3 *b33 *n2
4 16 16
1 2 2 2
- ----*v3 *b33 *n2
16
which the program can not factorize further.
{HAM,FI} = 0
1 2 2 1
FI= - ---*u1*u2*v1*a22*n2 + u1*u3 *v1*a22 - ---*u1*v1*v2*b33*n2
2 4
1 2 2 2 1
- ---*u2 *v2*a22*n2 + u2*u3 *v2*a22 - ---*u2*u3*v3*a22*n2
2 2
1 2 3 2 1
- ---*u2*v2 *b33*n2 + u3 *v3*a22 - ---*u3*v2*v3*b33*n2
4 4
= a product of the elements of: {u1*v1 + u2*v2 + u3*v3,
1 2 2 1
- ---*u2*a22*n2 + u3 *a22 - ---*v2*b33*n2}
2 4
{HAM,FI} = 0
2 2 2 2 2 3 4 4
FI=u1*v1*a22*b33*n2 - u2 *a22 *n2 + 4*u2*u3 *a22 *n2 - 4*u3 *a22
2 2 2 1 2 2 2
+ 2*u3 *v2*a22 *b33*n2 + u3*v3*a22*b33*n2 + ---*v1 *b33 *n2
4
1 2 2 2
+ ---*v3 *b33 *n2
4
= a product of the elements of: {4,
1 2 1 2 2 2 2 3 4 4
---*u1*v1*a22*b33*n2 - ---*u2 *a22 *n2 + u2*u3 *a22 *n2 - u3 *a22
4 4
1 2 2 1 2 1 2 2 2
+ ---*u3 *v2*a22 *b33*n2 + ---*u3*v3*a22*b33*n2 + ----*v1 *b33 *n2
2 4 16
1 2 2 2
+ ----*v3 *b33 *n2 }
16
{HAM,FI} = 0
2 2 1
FI=u2*a22*n2 - 2*u3 *a22 + ---*v2*b33*n2
2
1 2 2 1
= a product of the elements of: {2,---*u2*a22*n2 - u3 *a22 + ---*v2*b33*n2}
2 4
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a22 + u2**2*a22 + u2*n2 - u3**2*a22 + u3*v3*b33 + ( - 1/4*v3**2*b33**2
)/a22$
FI= - 1/2*u1**2*u2*a22**3*n2 + u1**2*u3**2*a22**4 - 1/4*u1**2*v2*a22**2*b33*n2 -
1/2*u2**3*a22**3*n2 + u2**2*u3**2*a22**4 - 1/4*u2**2*v2*a22**2*b33*n2 - 1/4*u2
**2*a22**2*n2**2 + 1/2*u2*u3**2*a22**3*n2 - 1/2*u2*u3*v3*a22**2*b33*n2 + 1/8*u2*
v3**2*a22*b33**2*n2 + u3**3*v3*a22**3*b33 - 1/4*u3**2*v2*a22**2*b33*n2 - 1/4*u3
**2*v3**2*a22**2*b33**2 - 1/4*u3*v2*v3*a22*b33**2*n2 - 1/16*v1**2*b33**2*n2**2 +
1/16*v2*v3**2*b33**3*n2 - 1/16*v3**2*b33**2*n2**2$
FI= - 1/2*u1*u2*v1*a22*n2 + u1*u3**2*v1*a22**2 - 1/4*u1*v1*v2*b33*n2 - 1/2*u2**2
*v2*a22*n2 + u2*u3**2*v2*a22**2 - 1/2*u2*u3*v3*a22*n2 - 1/4*u2*v2**2*b33*n2 + u3
**3*v3*a22**2 - 1/4*u3*v2*v3*b33*n2$
FI=u1*v1*a22*b33*n2**2 - u2**2*a22**2*n2**2 + 4*u2*u3**2*a22**3*n2 - 4*u3**4*a22
**4 + 2*u3**2*v2*a22**2*b33*n2 + u3*v3*a22*b33*n2**2 + 1/4*v1**2*b33**2*n2**2 +
1/4*v3**2*b33**2*n2**2$
FI=u2*a22*n2 - 2*u3**2*a22**2 + 1/2*v2*b33*n2$