Solution 3 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r10=0
1 2
- ---*m2*n3 *r464
4
r11=--------------------
3
a22
1 2 1 2 3 1 2 3
---*a22*b32*m2*n3 *r464 - ---*b31 *n3 *r464 - ---*b32 *n3 *r464
4 8 8
r12=-----------------------------------------------------------------
4
a22 *b31
1 3
- ---*n3 *r464
4
r13=-----------------
3
a22
r14=0
r15=0
1 2 2 2 1 2
r20=(---*a22 *b32 *m2 *r464 - ---*a22*b31 *b32*m2*n3*r464
4 4
1 3 1 4 2 3 2 2 2
- ---*a22*b32 *m2*n3*r464 + ---*b31 *n3 *r464 + ----*b31 *b32 *n3 *r464
4 8 16
1 4 2 4 2
+ ----*b32 *n3 *r464)/(a22 *b31 )
16
r21=0
r23=0
1 2 1 2 1 2
- ---*a22*b32*m2 *r464 + ---*b31 *m2*n3*r464 + ---*b32 *m2*n3*r464
2 4 4
r24=---------------------------------------------------------------------
3
a22 *b31
1 2 2 2 1 2 2 2
r25=( - ---*a22 *b31 *m2 *r464 + ---*a22 *b32 *m2 *r464
4 4
1 2 1 3
- ---*a22*b31 *b32*m2*n3*r464 - ---*a22*b32 *m2*n3*r464
4 4
1 4 2 1 2 2 2 1 4 2 4
+ ----*b31 *n3 *r464 + ---*b31 *b32 *n3 *r464 + ----*b32 *n3 *r464)/(a22
16 8 16
2
*b31 )
r26=0
1 1 2
---*a22*m2*n3*r464 - ---*b32*n3 *r464
2 4
r27=---------------------------------------
3
a22
1 1 2 2
- ---*a22*b32*m2*n3*r464 + ---*b32 *n3 *r464
2 4
r28=-----------------------------------------------
3
a22 *b31
1 2
- ---*n3 *r464
4
r29=-----------------
2
a22
1
- ---*m2*n3*r464
2
r210=-------------------
2
a22
r212=0
r213=0
r214=0
1 1 2 2 1 2 2
---*a22*b32*m2*n3*r464 - ---*b31 *n3 *r464 - ---*b32 *n3 *r464
2 4 4
r215=----------------------------------------------------------------
3
a22 *b31
r216=0
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
1 2 1 2
---*b31 *n3*r464 + ---*b32 *n3*r464
4 4
r310=-------------------------------------
3
a22
r311=0
1
---*b32*m2*r464
2
r312=-----------------
2
a22
r313=0
1 2 1 2 1 2
r314=(---*a22*b31 *m2*r464 - ---*a22*b32 *m2*r464 + ---*b31 *b32*n3*r464
2 2 4
1 3 3
+ ---*b32 *n3*r464)/(a22 *b31)
4
1 1 2 1 2
- ---*a22*b32*m2*r464 + ---*b31 *n3*r464 + ---*b32 *n3*r464
2 4 4
r315=--------------------------------------------------------------
3
a22
r316=0
1
a22*m2*r464 + ---*b32*n3*r464
2
r317=-------------------------------
2
a22
2 1 2
- a22*b32*m2*r464 + b31 *n3*r464 + ---*b32 *n3*r464
2
r318=------------------------------------------------------
2
a22 *b31
n3*r464
r319=---------
a22
r320=0
r323=0
r325=0
1
- a22*m2*r464 - ---*b32*n3*r464
2
r326=----------------------------------
2
a22
r328=0
r329=0
r330=0
r332=0
n3*r464
r333=---------
a22
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
2 1 2
a22*b32*m2*r464 - b31 *n3*r464 - ---*b32 *n3*r464
2
r341=---------------------------------------------------
2
a22 *b31
r342=0
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
n3*r464
r353=---------
a22
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
1 2 1 2
---*b31 *r464 + ---*b32 *r464
4 4
r425=-------------------------------
2
a22
r426=0
1 2
---*b32 *r464
4
r427=---------------
2
a22
r428=0
1
---*b31*b32*r464
2
r429=------------------
2
a22
1 2
---*b31 *r464
4
r430=---------------
2
a22
r431=0
b32*r464
r432=----------
a22
b31*r464
r433=----------
a22
r434=r464
r435=0
r439=0
r442=0
r444=0
r445=0
r448=0
r450=0
- b32*r464
r451=-------------
a22
r453=0
r454=0
r455=0
r458=0
r460=0
r461=0
r463=0
r465=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
- b31*r464
r486=-------------
a22
r487=0
r488=0
r489=0
r490=0
r493=0
r495=0
r496=0
r498=0
r499=0
r4100=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
1 2 1 2
- a22*b32*m2 + ---*b31 *n3 + ---*b32 *n3
2 2
m1=-------------------------------------------
a22*b31
n2=0
n1=0
1 2 1 2
- ---*b31 - ---*b32
4 4
c33=------------------------
a22
c23=0
c22=0
c13=0
c12=0
c11=0
b33=0
b23=0
b22=0
b21=0
b13=0
b12=0
b11=0
a33=2*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:
r464, m2, n3, b32, 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.
{a11,a22,r464}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11 - a22,
a12,
a13,
a23,
- 2*a22 + a33,
b11,
b12,
b13,
b21,
b22,
b23,
b33,
c11,
c12,
c13,
c22,
c23,
a22*c33 + 1/4*b31**2 + 1/4*b32**2,
n1,
n2,
a22*b31*m1 + a22*b32*m2 - 1/2*b31**2*n3 - 1/2*b32**2*n3,
m3}$
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a22 + u2 *a22 + 2*u3 *a22 + u3*v1*b31 + u3*v2*b32 + u3*n3
1 2 1 2
- a22*b32*m2 + ---*b31 *n3 + ---*b32 *n3
2 2
+ v1*------------------------------------------- + v2*m2
a22*b31
1 2 1 2
- ---*b31 - ---*b32
2 4 4
+ v3 *------------------------
a22
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 4 2 2 3 2 2 3 3
FI=u1 *u3 *a22 *b31 + u1 *u3*a22 *b31 *n3 - u1*u3 *v3*a22 *b31
3 2 3 1 2 2
+ u1*u3*v3*(a22 *b31*b32*m2 - a22 *b31 *n3 - ---*a22 *b31*b32 *n3)
2
1 2 1 3 2 1 2 2
+ u1*v3*(---*a22 *b31*b32*m2*n3 - ---*a22*b31 *n3 - ---*a22*b31*b32 *n3 )
2 4 4
2 2 4 2 2 3 2 2 3 2
+ u2 *u3 *a22 *b31 + u2 *u3*a22 *b31 *n3 - u2*u3 *v3*a22 *b31 *b32
3 2 1 2 2
+ u2*u3*v3*( - a22 *b31 *m2 - ---*a22 *b31 *b32*n3)
2
1 2 2 4 4 2 3 3 3
- ---*u2*v3*a22 *b31 *m2*n3 + u3 *a22 *b31 + u3 *v1*a22 *b31
2
3 3 2 3 3 2 1 2 2 2 4
+ u3 *v2*a22 *b31 *b32 + u3 *a22 *b31 *n3 + ---*u3 *v1 *a22 *b31
4
1 2 2 3
+ ---*u3 *v1*v2*a22 *b31 *b32
2
2 3 2 3 1 2 2
+ u3 *v1*( - a22 *b31*b32*m2 + a22 *b31 *n3 + ---*a22 *b31*b32 *n3)
2
1 2 2 2 2 2 2 3 2 1 2 2
+ ---*u3 *v2 *a22 *b31 *b32 + u3 *v2*(a22 *b31 *m2 + ---*a22 *b31 *b32*n3)
4 2
2 2 1 2 4 1 2 2 2 1 2 2 2 2
+ u3 *v3 *(---*a22 *b31 + ---*a22 *b31 *b32 ) - ---*u3 *a22 *b31 *n3
4 4 4
2 1 2 2 1 4 1 2 2
+ u3*v1 *( - ---*a22 *b31 *b32*m2 + ---*a22*b31 *n3 + ---*a22*b31 *b32 *n3)
2 4 4
1 2 3 1 2 2 1 3
+ u3*v1*v2*(---*a22 *b31 *m2 - ---*a22 *b31*b32 *m2 + ---*a22*b31 *b32*n3
2 2 4
1 3
+ ---*a22*b31*b32 *n3)
4
1 2 1 2 2
+ u3*v1*( - ---*a22 *b31*b32*m2*n3 + ---*a22*b31*b32 *n3 )
2 4
1 2 2 2
+ ---*u3*v2 *a22 *b31 *b32*m2
2
1 2 2 1 2 2
+ u3*v2*(---*a22 *b31 *m2*n3 - ---*a22*b31 *b32*n3 )
2 4
2 1 4 1 2 2 1 2 3
+ u3*v3 *(---*a22*b31 *n3 + ---*a22*b31 *b32 *n3) - ---*u3*a22*b31 *n3 +
4 4 4
2 1 2 2 2 1 2 2 2 1 2
v1 *( - ---*a22 *b31 *m2 + ---*a22 *b32 *m2 - ---*a22*b31 *b32*m2*n3
4 4 4
1 3 1 4 2 1 2 2 2
- ---*a22*b32 *m2*n3 + ----*b31 *n3 + ---*b31 *b32 *n3
4 16 8
1 4 2
+ ----*b32 *n3 ) + v1*v2
16
1 2 2 1 3 1 2
*( - ---*a22 *b31*b32*m2 + ---*a22*b31 *m2*n3 + ---*a22*b31*b32 *m2*n3)
2 4 4
1 2 1 3 3 1 2 3
+ v1*(---*a22*b31*b32*m2*n3 - ---*b31 *n3 - ---*b31*b32 *n3 )
4 8 8
1 2 2 2 1 2 2 2 1 2
- ---*v2*a22*b31 *m2*n3 + v3 *(---*a22 *b32 *m2 - ---*a22*b31 *b32*m2*n3
4 4 4
1 3 1 4 2 3 2 2 2 1 4 2
- ---*a22*b32 *m2*n3 + ---*b31 *n3 + ----*b31 *b32 *n3 + ----*b32 *n3 )
4 8 16 16
which the program can not factorize further.
{HAM,FI} = {2,
- a22,
a22,
u1*v1 + u2*v2 + u3*v3,
b31,
1
u3*a22 + ---*n3,
2
2
- u1*u3*a22*b31*b32 - u1*a22*b31*m2 + u2*u3*a22*b31
1 2 1 2
+ u2*( - a22*b32*m2 + ---*b31 *n3 + ---*b32 *n3)}
2 2
And again in machine readable form:
HAM=u1**2*a22 + u2**2*a22 + 2*u3**2*a22 + u3*v1*b31 + u3*v2*b32 + u3*n3 + v1*( -
a22*b32*m2 + 1/2*b31**2*n3 + 1/2*b32**2*n3)/(a22*b31) + v2*m2 + v3**2*( - 1/4*
b31**2 - 1/4*b32**2)/a22$
FI=u1**2*u3**2*a22**4*b31**2 + u1**2*u3*a22**3*b31**2*n3 - u1*u3**2*v3*a22**3*
b31**3 + u1*u3*v3*(a22**3*b31*b32*m2 - a22**2*b31**3*n3 - 1/2*a22**2*b31*b32**2*
n3) + u1*v3*(1/2*a22**2*b31*b32*m2*n3 - 1/4*a22*b31**3*n3**2 - 1/4*a22*b31*b32**
2*n3**2) + u2**2*u3**2*a22**4*b31**2 + u2**2*u3*a22**3*b31**2*n3 - u2*u3**2*v3*
a22**3*b31**2*b32 + u2*u3*v3*( - a22**3*b31**2*m2 - 1/2*a22**2*b31**2*b32*n3) -
1/2*u2*v3*a22**2*b31**2*m2*n3 + u3**4*a22**4*b31**2 + u3**3*v1*a22**3*b31**3 +
u3**3*v2*a22**3*b31**2*b32 + u3**3*a22**3*b31**2*n3 + 1/4*u3**2*v1**2*a22**2*b31
**4 + 1/2*u3**2*v1*v2*a22**2*b31**3*b32 + u3**2*v1*( - a22**3*b31*b32*m2 + a22**
2*b31**3*n3 + 1/2*a22**2*b31*b32**2*n3) + 1/4*u3**2*v2**2*a22**2*b31**2*b32**2 +
u3**2*v2*(a22**3*b31**2*m2 + 1/2*a22**2*b31**2*b32*n3) + u3**2*v3**2*(1/4*a22**
2*b31**4 + 1/4*a22**2*b31**2*b32**2) - 1/4*u3**2*a22**2*b31**2*n3**2 + u3*v1**2*
( - 1/2*a22**2*b31**2*b32*m2 + 1/4*a22*b31**4*n3 + 1/4*a22*b31**2*b32**2*n3) +
u3*v1*v2*(1/2*a22**2*b31**3*m2 - 1/2*a22**2*b31*b32**2*m2 + 1/4*a22*b31**3*b32*
n3 + 1/4*a22*b31*b32**3*n3) + u3*v1*( - 1/2*a22**2*b31*b32*m2*n3 + 1/4*a22*b31*
b32**2*n3**2) + 1/2*u3*v2**2*a22**2*b31**2*b32*m2 + u3*v2*(1/2*a22**2*b31**2*m2*
n3 - 1/4*a22*b31**2*b32*n3**2) + u3*v3**2*(1/4*a22*b31**4*n3 + 1/4*a22*b31**2*
b32**2*n3) - 1/4*u3*a22*b31**2*n3**3 + v1**2*( - 1/4*a22**2*b31**2*m2**2 + 1/4*
a22**2*b32**2*m2**2 - 1/4*a22*b31**2*b32*m2*n3 - 1/4*a22*b32**3*m2*n3 + 1/16*b31
**4*n3**2 + 1/8*b31**2*b32**2*n3**2 + 1/16*b32**4*n3**2) + v1*v2*( - 1/2*a22**2*
b31*b32*m2**2 + 1/4*a22*b31**3*m2*n3 + 1/4*a22*b31*b32**2*m2*n3) + v1*(1/4*a22*
b31*b32*m2*n3**2 - 1/8*b31**3*n3**3 - 1/8*b31*b32**2*n3**3) - 1/4*v2*a22*b31**2*
m2*n3**2 + v3**2*(1/4*a22**2*b32**2*m2**2 - 1/4*a22*b31**2*b32*m2*n3 - 1/4*a22*
b32**3*m2*n3 + 1/8*b31**4*n3**2 + 3/16*b31**2*b32**2*n3**2 + 1/16*b32**4*n3**2)$