Solution 12 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r10=0
1 2 1 2
- ---*c13 *m2*n1*r26 + c13*c23*m1*n1*r26 + ---*c23 *m2*n1*r26
2 2
r11=----------------------------------------------------------------
2
a33*c13*c23*m2 - a33*c23 *m1
1 2 1 2
---*c13 *m1*n1*r26 + c13*c23*m2*n1*r26 - ---*c23 *m1*n1*r26
2 2
r12=-------------------------------------------------------------
2
a33*c13*c23*m2 - a33*c23 *m1
r13=0
1 3 2 1 2 2
- ---*c13 *n1 *r26 - ---*c13*c23 *n1 *r26
2 2
r14=--------------------------------------------
2 3
a33*c13*c23 *m2 - a33*c23 *m1
1 2 2 1 2 2
---*c13 *n1 *r26 + ---*c23 *n1 *r26
2 2
r15=-------------------------------------
2
a33*c13*c23*m2 - a33*c23 *m1
1 2 1 2
---*c23*m1 *r26 + ---*c23*m2 *r26
2 2
r20=-----------------------------------
c13*m2*n1 - c23*m1*n1
2 2
c13 *n1*r26 + c23 *n1*r26
r21=---------------------------
a33*c13*m2 - a33*c23*m1
r22=0
3 2
c13 *n1*r26 + c13*c23 *n1*r26
r23=-------------------------------
2
a33*c13*c23*m2 - a33*c23 *m1
r24=0
r27=0
r28=0
r210=0
c13*m1*r26 + c23*m2*r26
r212=-------------------------
c13*m2 - c23*m1
r213=0
1 2 1 2
- ---*c13 *n1*r26 - ---*c23 *n1*r26
2 2
r214=--------------------------------------
2
c13*c23*m2 - c23 *m1
r215=0
- c13*m1*r26 - c23*m2*r26
r216=----------------------------
c13*m2 - c23*m1
r217=0
r218=0
r219=0
1 2 1 2
- ---*c13 *n1*r26 - ---*c23 *n1*r26
2 2
r220=--------------------------------------
2
c13*c23*m2 - c23 *m1
r30=0
r31=0
2
- 2*c13*c23*m1*r26 - 2*c23 *m2*r26
r32=-------------------------------------
c13*m2*n1 - c23*m1*n1
r33=0
r34=0
r35=0
r36=0
2
- 2*c13*c23*m1*r26 - 2*c23 *m2*r26
r37=-------------------------------------
c13*m2*n1 - c23*m1*n1
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
r321=0
r322=0
2 2
2*c13 *r26 + 2*c23 *r26
r323=-------------------------
c13*m2 - c23*m1
r324=0
r325=0
r326=0
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
r335=0
2 2
- 2*c13 *r26 - 2*c23 *r26
r336=----------------------------
c13*m2 - c23*m1
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
r40=0
r41=0
2 3
- 2*c13 *c23*r26 - 2*c23 *r26
r42=--------------------------------
c13*m2*n1 - c23*m1*n1
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
2 3
- 2*c13 *c23*r26 - 2*c23 *r26
r49=--------------------------------
c13*m2*n1 - c23*m1*n1
r410=0
r411=0
r412=0
r413=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
r427=0
r428=0
r429=0
r431=0
r432=0
r433=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
r446=0
r447=0
r448=0
r449=0
r450=0
r451=0
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
r488=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
r4107=0
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m3=0
n3=0
- c13*n1
n2=-----------
c23
c33=0
c22=0
c12=0
c11=0
b33=0
b32=0
b31=0
b23=0
b22=0
b21=0
b13=0
b12=0
b11=0
a23=0
a22=0
a13=0
a12=0
a11=0
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:
r26, m1, m2, n1, c23, c13, 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.
{c13*m2 - c23*m1,c23,c13,a33,n1,r26,c13 - i*c23,c13 + i*c23}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11,
a12,
a13,
a22,
a23,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
b33,
c11,
c12,
c22,
c33,
c13*n1 + c23*n2,
n3,
m3}$
The system of equations related to the Hamiltonian HAM:
- u2*c13*n1 2
HAM=u1*n1 + -------------- + u3 *a33 + 2*v1*v3*c13 + v1*m1 + 2*v2*v3*c23 + v2*m2
c23
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 3 2
FI=u1 *( - a33*c13 *c23*n1 - a33*c23 *n1 )
2 2 4
+ u1*v2*v3*( - 4*a33*c13 *c23 *n1 - 4*a33*c23 *n1)
2 3
+ u1*v2*( - 2*a33*c13*c23 *m1*n1 - 2*a33*c23 *m2*n1)
2 3 3 3 2 2 2 3 2
+ u1*(c13 *c23*n1 + c23 *n1 ) + u2 *( - a33*c13 *c23*n1 - a33*c23 *n1 )
2 2 4
+ u2*v1*v3*(4*a33*c13 *c23 *n1 + 4*a33*c23 *n1)
2 3
+ u2*v1*(2*a33*c13*c23 *m1*n1 + 2*a33*c23 *m2*n1)
3 3 2 3
+ u2*( - c13 *n1 - c13*c23 *n1 )
2 3
+ u3*v3*(2*a33*c13*c23 *m2*n1 - 2*a33*c23 *m1*n1)
2 2 2 3 5
+ v1 *v3 *( - 4*a33*c13 *c23 - 4*a33*c23 )
2 3 4
+ v1 *v3*( - 4*a33*c13*c23 *m1 - 4*a33*c23 *m2)
3 2 3 2
+ v1*v3*(2*c13 *c23*n1 + 2*c13*c23 *n1 )
2 2 2 2 3 2
+ v1*(c13 *c23*m1*n1 + 2*c13*c23 *m2*n1 - c23 *m1*n1 )
2 2 2 3 5
+ v2 *v3 *( - 4*a33*c13 *c23 - 4*a33*c23 )
2 3 4
+ v2 *v3*( - 4*a33*c13*c23 *m1 - 4*a33*c23 *m2)
2 2 2 4 2
+ v2*v3*(2*c13 *c23 *n1 + 2*c23 *n1 )
2 2 2 2 3 2
+ v2*( - c13 *c23*m2*n1 + 2*c13*c23 *m1*n1 + c23 *m2*n1 )
2 3 2 3 2
+ v3 *(a33*c23 *m1 + a33*c23 *m2 )
= a product of the elements of: {4,
2 2 3 2
2 - a33*c13 *c23*n1 - a33*c23 *n1
u1 *------------------------------------
4
2 2 4
+ u1*v2*v3*( - a33*c13 *c23 *n1 - a33*c23 *n1)
2 3
- a33*c13*c23 *m1*n1 - a33*c23 *m2*n1
+ u1*v2*----------------------------------------
2
2 3 3 3 2 2 3 2
c13 *c23*n1 + c23 *n1 2 - a33*c13 *c23*n1 - a33*c23 *n1
+ u1*------------------------- + u2 *------------------------------------
4 4
2 2 4
+ u2*v1*v3*(a33*c13 *c23 *n1 + a33*c23 *n1)
2 3
a33*c13*c23 *m1*n1 + a33*c23 *m2*n1
+ u2*v1*-------------------------------------
2
3 3 2 3
- c13 *n1 - c13*c23 *n1
+ u2*----------------------------
4
2 3
a33*c13*c23 *m2*n1 - a33*c23 *m1*n1
+ u3*v3*-------------------------------------
2
2 2 2 3 5
+ v1 *v3 *( - a33*c13 *c23 - a33*c23 )
2 3 4
+ v1 *v3*( - a33*c13*c23 *m1 - a33*c23 *m2)
3 2 3 2
c13 *c23*n1 + c13*c23 *n1
+ v1*v3*-----------------------------
2
2 2 2 2 3 2
c13 *c23*m1*n1 + 2*c13*c23 *m2*n1 - c23 *m1*n1
+ v1*---------------------------------------------------
4
2 2 2 3 5
+ v2 *v3 *( - a33*c13 *c23 - a33*c23 )
2 3 4
+ v2 *v3*( - a33*c13*c23 *m1 - a33*c23 *m2)
2 2 2 4 2
c13 *c23 *n1 + c23 *n1
+ v2*v3*--------------------------
2
2 2 2 2 3 2
- c13 *c23*m2*n1 + 2*c13*c23 *m1*n1 + c23 *m2*n1
+ v2*------------------------------------------------------
4
3 2 3 2
2 a33*c23 *m1 + a33*c23 *m2
+ v3 *-----------------------------}
4
{HAM,FI} = 0
And again in machine readable form:
HAM=u1*n1 + ( - u2*c13*n1)/c23 + u3**2*a33 + 2*v1*v3*c13 + v1*m1 + 2*v2*v3*c23 +
v2*m2$
FI=u1**2*( - a33*c13**2*c23*n1**2 - a33*c23**3*n1**2) + u1*v2*v3*( - 4*a33*c13**
2*c23**2*n1 - 4*a33*c23**4*n1) + u1*v2*( - 2*a33*c13*c23**2*m1*n1 - 2*a33*c23**3
*m2*n1) + u1*(c13**2*c23*n1**3 + c23**3*n1**3) + u2**2*( - a33*c13**2*c23*n1**2
- a33*c23**3*n1**2) + u2*v1*v3*(4*a33*c13**2*c23**2*n1 + 4*a33*c23**4*n1) + u2*
v1*(2*a33*c13*c23**2*m1*n1 + 2*a33*c23**3*m2*n1) + u2*( - c13**3*n1**3 - c13*c23
**2*n1**3) + u3*v3*(2*a33*c13*c23**2*m2*n1 - 2*a33*c23**3*m1*n1) + v1**2*v3**2*(
- 4*a33*c13**2*c23**3 - 4*a33*c23**5) + v1**2*v3*( - 4*a33*c13*c23**3*m1 - 4*
a33*c23**4*m2) + v1*v3*(2*c13**3*c23*n1**2 + 2*c13*c23**3*n1**2) + v1*(c13**2*
c23*m1*n1**2 + 2*c13*c23**2*m2*n1**2 - c23**3*m1*n1**2) + v2**2*v3**2*( - 4*a33*
c13**2*c23**3 - 4*a33*c23**5) + v2**2*v3*( - 4*a33*c13*c23**3*m1 - 4*a33*c23**4*
m2) + v2*v3*(2*c13**2*c23**2*n1**2 + 2*c23**4*n1**2) + v2*( - c13**2*c23*m2*n1**
2 + 2*c13*c23**2*m1*n1**2 + c23**3*m2*n1**2) + v3**2*(a33*c23**3*m1**2 + a33*c23
**3*m2**2)$