Solution 26 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
r13=0
r14=0
r15=0
- 8*a33*c12*r216 + b11*b21*r216
r20=----------------------------------
4*a33*b11
r21=0
r22=0
r23=0
- b11*r216
r24=-------------
4*a33
r26=0
r27=0
r28=0
r210=0
r212= - r216
r213=0
r214=0
r215=0
r217=0
r218=0
r219=0
r220=0
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=0
r311=0
r312=0
r313=0
r314=0
r315=0
r316=0
r317=0
r318=0
r319=0
r320=0
r323=0
r325=0
r326=0
r328=0
r329=0
r330=0
r332=0
r333=0
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
3 3 2 2 2
r40=( - 128*a33 *b11*c12*r420 + 128*a33 *c12 *r483 + 16*a33 *b11 *b21*r420
2 3 2 2
- 32*a33 *b11*b21*c12*r483 + 24*a33*b11 *c12*r453 + 2*a33*b11 *b21 *r483
4 3 2
- 3*b11 *b21*r453)/(64*a33 *b11 )
r41=0
3 2 2
r42=( - 64*a33 *c12*r420 + 8*a33 *b11*b21*r420 + 8*a33*b11 *c12*r453
3 3
- b11 *b21*r453)/(32*a33 *b11)
r43=0
r44=0
r45=0
2 2 2 2 2
r46=( - 16*a33 *b11 *r420 + 32*a33 *b11*c12*r483 + 256*a33 *c12 *r453
2 4
- 4*a33*b11 *b21*r483 - 64*a33*b11*b21*c12*r453 + 3*b11 *r453
2 2 3
+ 4*b11 *b21 *r453)/(64*a33 *b11)
r47=0
2 3
- 8*a33 *b11*r420 + b11 *r453
r48=--------------------------------
3
32*a33
3 2 2
r49=( - 64*a33 *c12*r420 + 8*a33 *b11*b21*r420 + 8*a33*b11 *c12*r453
3 3
- b11 *b21*r453)/(32*a33 *b11)
r410=0
2 2
a33*b11 *r483 + 16*a33*b11*c12*r453 - 2*b11 *b21*r453
r411=-------------------------------------------------------
3
32*a33
r412=0
2 3
- 8*a33 *b11*r420 + b11 *r453
r413=--------------------------------
3
32*a33
8*a33*c12*r453 - b11*b21*r453
r415=-------------------------------
2
8*a33
r416=0
r417=0
r418=0
r419=0
r421=0
- a33*b11*r483 - 8*a33*c12*r453 + b11*b21*r453
r422=-------------------------------------------------
2
4*a33
r423=0
r424=0
8*a33*c12*r453 - b11*b21*r453
r425=-------------------------------
4*a33*b11
r426=0
r427=0
r428=0
b11*r453
r429=----------
4*a33
r431=0
r432=0
r433=0
r435=0
2 2 3
- 16*a33 *b11*r420 + 32*a33 *c12*r483 - 4*a33*b11*b21*r483 + 3*b11 *r453
r439=---------------------------------------------------------------------------
2
16*a33 *b11
r442=0
2 2
- 8*a33 *r420 + b11 *r453
r444=----------------------------
2
8*a33
r445=0
b11*r453
r448=----------
2*a33
r450=0
r451=0
r454=0
r455=0
r458=0
r483
r460=------
2
r461=0
r463=0
r464=0
r465=0
r467=0
r468=0
r469=0
r470=0
2 2 3
16*a33 *b11*r420 - 32*a33 *c12*r483 + 4*a33*b11*b21*r483 - 3*b11 *r453
r471=------------------------------------------------------------------------
2
16*a33 *b11
r472=0
2 2
8*a33 *r420 - b11 *r453
r473=-------------------------
2
8*a33
8*a33*c12*r453 - b11*b21*r453
r474=-------------------------------
2
4*a33
r475=0
- a33*b11*r483 - 8*a33*c12*r453 + b11*b21*r453
r476=-------------------------------------------------
2
4*a33
r477=0
r478=2*r420
- a33*b11*r483 - 8*a33*c12*r453 + b11*b21*r453
r479=-------------------------------------------------
2
4*a33
r480=0
- b11*r453
r481=-------------
2*a33
r482=0
r484=0
r485=0
r486=0
r487= - r453
r488=0
r489=0
r490=0
r493=0
b11*r453
r495=----------
a33
r496=0
r498=0
r499=0
r4100=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r483
r4107=------
2
r4108=0
- b11*r453
r4109=-------------
a33
r4110=r483
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
r4125=0
m3=0
m2=0
m1=0
n3=0
n2=0
n1=0
3 2
32*a33*b21*c12 + b11 - 4*b11*b21
c33=------------------------------------
16*a33*b11
c23=0
c22=0
c13=0
c11=0
b32=0
b31=0
b23=0
b22=0
b13=0
b12= - b21
a23=0
a22=0
a13=0
a12=0
a11=0
b11
b33=-----
2
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:
b11, r216, r420, r483, r453, b21, c12, 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.
{b33,a33}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{ - b11 + 2*b33,
a11,
a12,
a13,
a22,
a23,
b12 + b21,
b13,
b22,
b23,
b31,
b32,
c11,
c13,
c22,
c23,
16*a33*b11*c33 - 32*a33*b21*c12 - b11**3 + 4*b11*b21**2,
n1,
n2,
n3,
m1,
m2,
m3}$
The system of equations related to the Hamiltonian HAM:
2 u3*v3*b11
HAM=u1*v1*b11 - u1*v2*b21 + u2*v1*b21 + u3 *a33 + ----------- + 2*v1*v2*c12
2
3 2
2 32*a33*b21*c12 + b11 - 4*b11*b21
+ v3 *------------------------------------
16*a33*b11
has apart from the Hamiltonian and Casimirs the following 4 first integrals:
2 2 2 2 2 2 2 3
FI= - 64*u1 *v1*v2*a33 *b11 + 64*u1*u2*v1 *a33 *b11 - 64*u1*u3 *v2*a33 *b11
2 2 3 2 2
- 32*u1*u3*v2*v3*a33 *b11 + u1*v1 *( - 128*a33 *b11*c12 + 16*a33*b11 *b21)
2 2 2
+ u1*v1*v2 *( - 128*a33 *b11*c12 + 16*a33*b11 *b21)
2 2 2 3 3
+ u1*v1*v3 *(128*a33 *b11*c12 - 16*a33*b11 *b21) - 8*u1*v2 *a33*b11
2 3 2 3 2 2
- 12*u1*v2*v3 *a33*b11 + 64*u2*u3 *v1*a33 *b11 + 32*u2*u3*v1*v3*a33 *b11
3 3 2 3 2 2 2
+ 8*u2*v1 *a33*b11 + 12*u2*v1*v3 *a33*b11 + 16*u3 *v1*v2*a33 *b11
2 2 3 2
+ u3 *v3 *(128*a33 *c12 - 16*a33 *b11*b21)
2 2 2
+ u3*v1 *v3*( - 128*a33 *b11*c12 + 16*a33*b11 *b21)
3 2 2 3 4
+ u3*v3 *(64*a33 *b11*c12 - 8*a33*b11 *b21) + 2*v1 *v2*b11
2 2 2 3
+ v1 *v2 *(32*a33*b11 *c12 - 4*b11 *b21)
2 2 2 3 3 4
+ v1 *v3 *(16*a33*b11 *c12 - 2*b11 *b21) + 2*v1*v2 *b11
2 2 2 4 2 2
+ v1*v2*v3 *(256*a33 *c12 - 64*a33*b11*b21*c12 + 3*b11 + 4*b11 *b21 )
2 2 2 3
+ v2 *v3 *(16*a33*b11 *c12 - 2*b11 *b21)
4 2 3
+ v3 *(24*a33*b11 *c12 - 3*b11 *b21)
= a product of the elements of: {64,
2 2 2 2 2 2 2 3
- u1 *v1*v2*a33 *b11 + u1*u2*v1 *a33 *b11 - u1*u3 *v2*a33 *b11
2 2 2 2
- u1*u3*v2*v3*a33 *b11 3 - 8*a33 *b11*c12 + a33*b11 *b21
+ -------------------------- + u1*v1 *----------------------------------
2 4
2 2
2 - 8*a33 *b11*c12 + a33*b11 *b21
+ u1*v1*v2 *----------------------------------
4
2 2 3 3
2 8*a33 *b11*c12 - a33*b11 *b21 - u1*v2 *a33*b11
+ u1*v1*v3 *------------------------------- + --------------------
4 8
2 3 2 2
- 3*u1*v2*v3 *a33*b11 2 3 u2*u3*v1*v3*a33 *b11
+ ------------------------- + u2*u3 *v1*a33 *b11 + -----------------------
16 2
3 3 2 3 2 2 2
u2*v1 *a33*b11 3*u2*v1*v3 *a33*b11 u3 *v1*v2*a33 *b11
+ ----------------- + ---------------------- + ---------------------
8 16 4
3 2
2 2 8*a33 *c12 - a33 *b11*b21
+ u3 *v3 *---------------------------
4
2 2
2 - 8*a33 *b11*c12 + a33*b11 *b21
+ u3*v1 *v3*----------------------------------
4
2 2 3 4
3 8*a33 *b11*c12 - a33*b11 *b21 v1 *v2*b11
+ u3*v3 *------------------------------- + -------------
8 32
2 3 2 3
2 2 8*a33*b11 *c12 - b11 *b21 2 2 8*a33*b11 *c12 - b11 *b21
+ v1 *v2 *--------------------------- + v1 *v3 *---------------------------
16 32
3 4
v1*v2 *b11
+ -------------
32
2 2 4 2 2
2 256*a33 *c12 - 64*a33*b11*b21*c12 + 3*b11 + 4*b11 *b21
+ v1*v2*v3 *-----------------------------------------------------------
64
2 3 2 3
2 2 8*a33*b11 *c12 - b11 *b21 4 24*a33*b11 *c12 - 3*b11 *b21
+ v2 *v3 *--------------------------- + v3 *------------------------------}
32 64
{HAM,FI} = {512,
- u1*v1 - u2*v2 - u3*v3,
b11,
a33,
2 2 2
- u3*v1 *a33*b11 8*a33 *c12 - a33*b11*b21
-------------------- + u3*v1*v2*--------------------------
32 8
2 2
u3*v2 *a33*b11
+ -----------------
32
3 2
2 - 64*a33*b21*c12 + b11 + 8*b11*b21
+ v1 *v3*---------------------------------------
128
2 2 3
4*a33*b11*c12 - b11 *b21 v2 *v3*b11
+ v1*v2*v3*-------------------------- + -------------}
16 128
2 2 2 2 2 2 2 2 2 2
FI=32*u1 *v1 *a33 *b11 + 16*u1 *v2 *a33 *b11 + 32*u1*u3*v1*v3*a33 *b11
3 3 2 3
- 8*u1*v1 *a33*b11 - 8*u1*v1*v2 *a33*b11
2 2 2 2 2 2 2
+ u1*v2*v3 *( - 64*a33 *b11*c12 + 8*a33*b11 *b21) + 16*u2 *v1 *a33 *b11
2 2 2 2 3
+ u2*v1*v3 *(64*a33 *b11*c12 - 8*a33*b11 *b21) - 8*u3*v1 *v3*a33*b11
2 2 4 2 2 3
+ v1 *v2 *b11 + v1*v2*v3 *(16*a33*b11 *c12 - 2*b11 *b21)
4 2 2 2 2
+ v3 *(64*a33 *c12 - 16*a33*b11*b21*c12 + b11 *b21 )
= a product of the elements of: {32,
2 2 2 2
2 2 2 2 u1 *v2 *a33 *b11 2 2
u1 *v1 *a33 *b11 + ------------------- + u1*u3*v1*v3*a33 *b11
2
3 3 2 3
- u1*v1 *a33*b11 - u1*v1*v2 *a33*b11
+ -------------------- + -----------------------
4 4
2 2 2 2 2 2
2 - 8*a33 *b11*c12 + a33*b11 *b21 u2 *v1 *a33 *b11
+ u1*v2*v3 *---------------------------------- + -------------------
4 2
2 2 2 3
2 8*a33 *b11*c12 - a33*b11 *b21 - u3*v1 *v3*a33*b11
+ u2*v1*v3 *------------------------------- + -----------------------
4 4
2 2 4 2 3
v1 *v2 *b11 2 8*a33*b11 *c12 - b11 *b21
+ -------------- + v1*v2*v3 *---------------------------
32 16
2 2 2 2
4 64*a33 *c12 - 16*a33*b11*b21*c12 + b11 *b21
+ v3 *-----------------------------------------------}
32
{HAM,FI} = {64,
u1*v1 + u2*v2 + u3*v3,
b11,
a33,
2
2 u1*v2*v3*a33*b11 2
u1*u3*v2*a33 *b11 + ------------------- + u2*u3*v1*a33 *b11
4
2 2
u2*v1*v3*a33*b11 - u3*v1 *a33*b11*b21
+ ------------------- + -----------------------
4 2
2 2
- u3*v1*v2*a33*b11 2 - 4*a33*b11*c12 + b11 *b21
+ ---------------------- + v1 *v3*-----------------------------
4 4
3 2
32*a33*b21*c12 - b11 - 4*b11*b21
+ v1*v2*v3*------------------------------------}
16
2 3 2
FI=8*u1*v1 *v2*a33*b11 + 4*u1*v2 *a33*b11 + 4*u1*v2*v3 *a33*b11
3 2
- 4*u2*v1 *a33*b11 - 4*u2*v1*v3 *a33*b11 + 4*u3*v1*v2*v3*a33*b11
3 2 2 2 3 2
- v1 *v2*b11 + v1 *v3 *( - 8*a33*c12 + b11*b21) - v1*v2 *b11
2 2 2 2
- v1*v2*v3 *b11 + v2 *v3 *( - 8*a33*c12 + b11*b21)
4
+ v3 *( - 8*a33*c12 + b11*b21)
= a product of the elements of: {8,
3 2
2 u1*v2 *a33*b11 u1*v2*v3 *a33*b11
u1*v1 *v2*a33*b11 + ---------------- + -------------------
2 2
3 2
- u2*v1 *a33*b11 - u2*v1*v3 *a33*b11 u3*v1*v2*v3*a33*b11
+ ------------------- + ---------------------- + ---------------------
2 2 2
3 2 3 2
- v1 *v2*b11 2 2 - 8*a33*c12 + b11*b21 - v1*v2 *b11
+ ---------------- + v1 *v3 *------------------------ + ----------------
8 8 8
2 2
- v1*v2*v3 *b11 2 2 - 8*a33*c12 + b11*b21
+ ------------------- + v2 *v3 *------------------------
8 8
4 - 8*a33*c12 + b11*b21
+ v3 *------------------------}
8
{HAM,FI} = {8,
- u1*v1 - u2*v2 - u3*v3,
b11,
a33,
2
2 2 - v1 *v3*b11
u3*v1 *a33 - u3*v2 *a33 + --------------- + v1*v2*v3*b21
4
2
- v2 *v3*b11
+ ---------------}
4
2 2
FI=4*u1*v2*a33*b11 - 4*u2*v1*a33*b11 - v1*v2*b11 + v3 *( - 8*a33*c12 + b11*b21)
= a product of the elements of: {4,
2
- v1*v2*b11
u1*v2*a33*b11 - u2*v1*a33*b11 + ---------------
4
2 - 8*a33*c12 + b11*b21
+ v3 *------------------------}
4
{HAM,FI} = 0
And again in machine readable form:
HAM=u1*v1*b11 - u1*v2*b21 + u2*v1*b21 + u3**2*a33 + (u3*v3*b11)/2 + 2*v1*v2*c12
+ v3**2*(32*a33*b21*c12 + b11**3 - 4*b11*b21**2)/(16*a33*b11)$
FI= - 64*u1**2*v1*v2*a33**2*b11**2 + 64*u1*u2*v1**2*a33**2*b11**2 - 64*u1*u3**2*
v2*a33**3*b11 - 32*u1*u3*v2*v3*a33**2*b11**2 + u1*v1**3*( - 128*a33**2*b11*c12 +
16*a33*b11**2*b21) + u1*v1*v2**2*( - 128*a33**2*b11*c12 + 16*a33*b11**2*b21) +
u1*v1*v3**2*(128*a33**2*b11*c12 - 16*a33*b11**2*b21) - 8*u1*v2**3*a33*b11**3 -
12*u1*v2*v3**2*a33*b11**3 + 64*u2*u3**2*v1*a33**3*b11 + 32*u2*u3*v1*v3*a33**2*
b11**2 + 8*u2*v1**3*a33*b11**3 + 12*u2*v1*v3**2*a33*b11**3 + 16*u3**2*v1*v2*a33
**2*b11**2 + u3**2*v3**2*(128*a33**3*c12 - 16*a33**2*b11*b21) + u3*v1**2*v3*( -
128*a33**2*b11*c12 + 16*a33*b11**2*b21) + u3*v3**3*(64*a33**2*b11*c12 - 8*a33*
b11**2*b21) + 2*v1**3*v2*b11**4 + v1**2*v2**2*(32*a33*b11**2*c12 - 4*b11**3*b21)
+ v1**2*v3**2*(16*a33*b11**2*c12 - 2*b11**3*b21) + 2*v1*v2**3*b11**4 + v1*v2*v3
**2*(256*a33**2*c12**2 - 64*a33*b11*b21*c12 + 3*b11**4 + 4*b11**2*b21**2) + v2**
2*v3**2*(16*a33*b11**2*c12 - 2*b11**3*b21) + v3**4*(24*a33*b11**2*c12 - 3*b11**3
*b21)$
FI=32*u1**2*v1**2*a33**2*b11**2 + 16*u1**2*v2**2*a33**2*b11**2 + 32*u1*u3*v1*v3*
a33**2*b11**2 - 8*u1*v1**3*a33*b11**3 - 8*u1*v1*v2**2*a33*b11**3 + u1*v2*v3**2*(
- 64*a33**2*b11*c12 + 8*a33*b11**2*b21) + 16*u2**2*v1**2*a33**2*b11**2 + u2*v1*
v3**2*(64*a33**2*b11*c12 - 8*a33*b11**2*b21) - 8*u3*v1**2*v3*a33*b11**3 + v1**2*
v2**2*b11**4 + v1*v2*v3**2*(16*a33*b11**2*c12 - 2*b11**3*b21) + v3**4*(64*a33**2
*c12**2 - 16*a33*b11*b21*c12 + b11**2*b21**2)$
FI=8*u1*v1**2*v2*a33*b11 + 4*u1*v2**3*a33*b11 + 4*u1*v2*v3**2*a33*b11 - 4*u2*v1
**3*a33*b11 - 4*u2*v1*v3**2*a33*b11 + 4*u3*v1*v2*v3*a33*b11 - v1**3*v2*b11**2 +
v1**2*v3**2*( - 8*a33*c12 + b11*b21) - v1*v2**3*b11**2 - v1*v2*v3**2*b11**2 + v2
**2*v3**2*( - 8*a33*c12 + b11*b21) + v3**4*( - 8*a33*c12 + b11*b21)$
FI=4*u1*v2*a33*b11 - 4*u2*v1*a33*b11 - v1*v2*b11**2 + v3**2*( - 8*a33*c12 + b11*
b21)$