Solution 15 to problem over
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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
2 2 2 2 2 2 2
r20=( - a33 *c13 *m1 *r455 - a33 *c13 *m2 *r455 + 2*a33*b23*c13 *m2*n3*r455
2 2 2 2 2 2
- 2*a33*b23*c13*c23*m1*n3*r455 - b23 *c13 *n3 *r455 - b23 *c23 *n3 *r455)/
2 2 2 2 2 2
(a33 *b23 *c13 + a33 *b23 *c23 )
r21=0
r22=0
r23=0
r24=0
r26=0
r27=0
r28=0
r210=0
r212=0
r213=0
r214=0
r215=0
r216=0
r217=0
r218=0
r219=0
r220=0
r30=0
2 2
r31=(2*a33*c13 *m2*r455 - 4*a33*c13*c23*m1*r455 - 2*a33*c23 *m2*r455
2 2 2 2 2 2
- 2*b23*c13 *n3*r455 - 2*b23*c23 *n3*r455)/(a33 *c13 + a33 *c23 )
3 2
4*c13 *m1*r455 + 4*c13 *c23*m2*r455
r32=-------------------------------------
2 2 2 2
b23 *c13 + b23 *c23
r33=0
3 2 2
r34=( - 2*a33*c13 *m1*r455 - 4*a33*c13 *c23*m2*r455 + 2*a33*c13*c23 *m1*r455
2 3 2 3 2 2
+ 2*b23*c13 *c23*n3*r455 + 2*b23*c23 *n3*r455)/(a33 *c13 + a33 *c13*c23 )
r35=0
r36=0
3 2
4*c13 *m1*r455 + 4*c13 *c23*m2*r455
r37=-------------------------------------
2 2 2 2
b23 *c13 + b23 *c23
r38=0
r39=0
2 2
r310=(2*a33*c13 *m2*r455 - 2*a33*c13*c23*m1*r455 - 2*b23*c13 *n3*r455
2 2 2
- 2*b23*c23 *n3*r455)/(a33*b23*c13 + a33*b23*c23 )
r311=0
r312=0
r313=0
r314=0
r315=0
r316=0
r317=0
r318=0
r319=0
r320=0
2
2*c13 *m1*r455 + 2*c13*c23*m2*r455
r323=------------------------------------
2 2
b23*c13 + b23*c23
r325=0
r326=0
r328=0
r329=0
r330=0
r332=0
r333=0
r334=0
r335=0
2
- 2*c13 *m1*r455 - 2*c13*c23*m2*r455
r336=---------------------------------------
2 2
b23*c13 + b23*c23
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
2 2 2 2
b23 *c13 *r455 + b23 *c23 *r455
r40=---------------------------------
2 2
a33 *c13
- 4*c23*r455
r41=---------------
a33
2
4*c13 *r455
r42=-------------
2
b23
r43=0
r44=0
- 4*c13*r455
r45=---------------
a33
r46=0
r47=0
r48=0
2
4*c13 *r455
r49=-------------
2
b23
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
- 2*b23*r455
r435=---------------
a33
4*c13*r455
r439=------------
b23
r442=0
r444=0
r445=0
r448=0
r450=0
r451=0
r453=0
r454=0
r458=0
r460=0
r461=0
r463=0
r464=0
r465=0
r467=0
r468=0
r469=0
2*b23*c23*r455
r470=----------------
a33*c13
- 4*c13*r455
r471=---------------
b23
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
r493=0
r495=0
r496=0
r498=0
r499=0
r4100=0
r4102=0
r4103=0
r4104=0
r4105=r455
r4106=0
r4107=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
r4125=0
m3=0
n2=0
n1=0
2
- b23
c33=---------
a33
2 2 2 2
- b23 *c13 + b23 *c23
c22=--------------------------
2
a33*c13
2
b23 *c23
c12=----------
a33*c13
c11=0
b33=0
b32=0
b31=0
b22=0
b21=0
- b23*c23
b13=------------
c13
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:
r455, m1, m2, n3, b23, 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.
{b23,r455,a33}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11,
a12,
a13,
a22,
a23,
b11,
b12,
b13*c13 + b23*c23,
b21,
b22,
b31,
b32,
b33,
c11,
a33*c12*c13 - b23**2*c23,
a33*c13**2*c22 + b23**2*c13**2 - b23**2*c23**2,
a33*c33 + b23**2,
n1,
n2,
m3}$
The system of equations related to the Hamiltonian HAM:
2
- u1*v3*b23*c23 2 2*v1*v2*b23 *c23
HAM=------------------ + u2*v3*b23 + u3 *a33 + u3*n3 + ------------------
c13 a33*c13
2 2 2 2
2 - b23 *c13 + b23 *c23
+ 2*v1*v3*c13 + v1*m1 + v2 *-------------------------- + 2*v2*v3*c23
2
a33*c13
2 2
- v3 *b23
+ v2*m2 + -------------
a33
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 4 2 2 2 2
FI=u1 *v3 *(a33 *b23 *c13 + a33 *b23 *c13 *c23 )
2 2 5 2 3 2
+ u1*v2*v3 *( - 4*a33 *b23*c13 - 4*a33 *b23*c13 *c23 )
2 4 2 3
+ u1*v2*v3*( - 2*a33 *b23*c13 *m1 - 2*a33 *b23*c13 *c23*m2)
3 3 3 3 3
+ u1*v3 *(2*a33*b23 *c13 *c23 + 2*a33*b23 *c13*c23 )
2 2 2 2 4 2 2 2 2
+ u2 *v3 *(a33 *b23 *c13 + a33 *b23 *c13 *c23 )
2 2 5 2 3 2
+ u2*v1*v3 *(4*a33 *b23*c13 + 4*a33 *b23*c13 *c23 )
2 4 2 3
+ u2*v1*v3*(2*a33 *b23*c13 *m1 + 2*a33 *b23*c13 *c23*m2)
3 3 4 3 2 2 2
+ u2*v3 *( - 2*a33*b23 *c13 - 2*a33*b23 *c13 *c23 ) + u3*v3 *(
2 4 2 3 2 4
2*a33 *b23*c13 *m2 - 2*a33 *b23*c13 *c23*m1 - 2*a33*b23 *c13 *n3
2 2 2 2 2 2 6 2 4 2
- 2*a33*b23 *c13 *c23 *n3) + v1 *v3 *(4*a33 *c13 + 4*a33 *c13 *c23 )
2 2 5 2 4
+ v1 *v3*(4*a33 *c13 *m1 + 4*a33 *c13 *c23*m2)
3 2 5 2 3 2 2
+ v1*v3 *( - 4*a33*b23 *c13 - 4*a33*b23 *c13 *c23 ) + v1*v3 *(
2 4 2 3 2 2 2
- 2*a33*b23 *c13 *m1 - 4*a33*b23 *c13 *c23*m2 + 2*a33*b23 *c13 *c23 *m1
3 3 3 3
+ 2*b23 *c13 *c23*n3 + 2*b23 *c13*c23 *n3)
2 2 2 6 2 4 2
+ v2 *v3 *(4*a33 *c13 + 4*a33 *c13 *c23 )
2 2 5 2 4
+ v2 *v3*(4*a33 *c13 *m1 + 4*a33 *c13 *c23*m2)
3 2 4 2 2 3 2
+ v2*v3 *( - 4*a33*b23 *c13 *c23 - 4*a33*b23 *c13 *c23 ) + v2*v3 *(
2 4 2 3 2 2 2
2*a33*b23 *c13 *m2 - 4*a33*b23 *c13 *c23*m1 - 2*a33*b23 *c13 *c23 *m2
3 4 3 2 2
- 2*b23 *c13 *n3 - 2*b23 *c13 *c23 *n3)
4 4 4 4 2 2 4 4 2 2 4 2
+ v3 *(b23 *c13 + 2*b23 *c13 *c23 + b23 *c23 ) + v3 *( - a33 *c13 *m1
2 4 2 4 3
- a33 *c13 *m2 + 2*a33*b23*c13 *m2*n3 - 2*a33*b23*c13 *c23*m1*n3
2 4 2 2 2 2 2
- b23 *c13 *n3 - b23 *c13 *c23 *n3 )
= a product of the elements of: {v3,
2 2 2 4 2 2 2 2
u1 *v3*(a33 *b23 *c13 + a33 *b23 *c13 *c23 )
2 5 2 3 2
+ u1*v2*v3*( - 4*a33 *b23*c13 - 4*a33 *b23*c13 *c23 )
2 4 2 3
+ u1*v2*( - 2*a33 *b23*c13 *m1 - 2*a33 *b23*c13 *c23*m2)
2 3 3 3 3
+ u1*v3 *(2*a33*b23 *c13 *c23 + 2*a33*b23 *c13*c23 )
2 2 2 4 2 2 2 2
+ u2 *v3*(a33 *b23 *c13 + a33 *b23 *c13 *c23 )
2 5 2 3 2
+ u2*v1*v3*(4*a33 *b23*c13 + 4*a33 *b23*c13 *c23 )
2 4 2 3
+ u2*v1*(2*a33 *b23*c13 *m1 + 2*a33 *b23*c13 *c23*m2)
2 3 4 3 2 2
+ u2*v3 *( - 2*a33*b23 *c13 - 2*a33*b23 *c13 *c23 ) + u3*v3*(
2 4 2 3 2 4
2*a33 *b23*c13 *m2 - 2*a33 *b23*c13 *c23*m1 - 2*a33*b23 *c13 *n3
2 2 2 2 2 6 2 4 2
- 2*a33*b23 *c13 *c23 *n3) + v1 *v3*(4*a33 *c13 + 4*a33 *c13 *c23 )
2 2 5 2 4
+ v1 *(4*a33 *c13 *m1 + 4*a33 *c13 *c23*m2)
2 2 5 2 3 2
+ v1*v3 *( - 4*a33*b23 *c13 - 4*a33*b23 *c13 *c23 ) + v1*v3*(
2 4 2 3 2 2 2
- 2*a33*b23 *c13 *m1 - 4*a33*b23 *c13 *c23*m2 + 2*a33*b23 *c13 *c23 *m1
3 3 3 3
+ 2*b23 *c13 *c23*n3 + 2*b23 *c13*c23 *n3)
2 2 6 2 4 2
+ v2 *v3*(4*a33 *c13 + 4*a33 *c13 *c23 )
2 2 5 2 4
+ v2 *(4*a33 *c13 *m1 + 4*a33 *c13 *c23*m2)
2 2 4 2 2 3
+ v2*v3 *( - 4*a33*b23 *c13 *c23 - 4*a33*b23 *c13 *c23 ) + v2*v3*(
2 4 2 3 2 2 2
2*a33*b23 *c13 *m2 - 4*a33*b23 *c13 *c23*m1 - 2*a33*b23 *c13 *c23 *m2
3 4 3 2 2
- 2*b23 *c13 *n3 - 2*b23 *c13 *c23 *n3)
3 4 4 4 2 2 4 4 2 4 2
+ v3 *(b23 *c13 + 2*b23 *c13 *c23 + b23 *c23 ) + v3*( - a33 *c13 *m1
2 4 2 4 3
- a33 *c13 *m2 + 2*a33*b23*c13 *m2*n3 - 2*a33*b23*c13 *c23*m1*n3
2 4 2 2 2 2 2
- b23 *c13 *n3 - b23 *c13 *c23 *n3 )}
{HAM,FI} = {2,
b23,
b23,
a33,
a33,
v3,
u1*v1 + u2*v2 + u3*v3,
c13,
3 2 2 3
u1*v3*(b23*c13 + b23*c13*c23 ) + u2*v3*(b23*c13 *c23 + b23*c23 )
3 3 2 2
+ v1*v3*(2*c13 *c23 + 2*c13*c23 ) + v1*(c13 *c23*m1 + c13*c23 *m2)
4 2 2 3 2
+ v2*v3*( - 2*c13 - 2*c13 *c23 ) + v2*( - c13 *m1 - c13 *c23*m2)}
And again in machine readable form:
HAM=( - u1*v3*b23*c23)/c13 + u2*v3*b23 + u3**2*a33 + u3*n3 + (2*v1*v2*b23**2*c23
)/(a33*c13) + 2*v1*v3*c13 + v1*m1 + v2**2*( - b23**2*c13**2 + b23**2*c23**2)/(
a33*c13**2) + 2*v2*v3*c23 + v2*m2 + ( - v3**2*b23**2)/a33$
FI=u1**2*v3**2*(a33**2*b23**2*c13**4 + a33**2*b23**2*c13**2*c23**2) + u1*v2*v3**
2*( - 4*a33**2*b23*c13**5 - 4*a33**2*b23*c13**3*c23**2) + u1*v2*v3*( - 2*a33**2*
b23*c13**4*m1 - 2*a33**2*b23*c13**3*c23*m2) + u1*v3**3*(2*a33*b23**3*c13**3*c23
+ 2*a33*b23**3*c13*c23**3) + u2**2*v3**2*(a33**2*b23**2*c13**4 + a33**2*b23**2*
c13**2*c23**2) + u2*v1*v3**2*(4*a33**2*b23*c13**5 + 4*a33**2*b23*c13**3*c23**2)
+ u2*v1*v3*(2*a33**2*b23*c13**4*m1 + 2*a33**2*b23*c13**3*c23*m2) + u2*v3**3*( -
2*a33*b23**3*c13**4 - 2*a33*b23**3*c13**2*c23**2) + u3*v3**2*(2*a33**2*b23*c13**
4*m2 - 2*a33**2*b23*c13**3*c23*m1 - 2*a33*b23**2*c13**4*n3 - 2*a33*b23**2*c13**2
*c23**2*n3) + v1**2*v3**2*(4*a33**2*c13**6 + 4*a33**2*c13**4*c23**2) + v1**2*v3*
(4*a33**2*c13**5*m1 + 4*a33**2*c13**4*c23*m2) + v1*v3**3*( - 4*a33*b23**2*c13**5
- 4*a33*b23**2*c13**3*c23**2) + v1*v3**2*( - 2*a33*b23**2*c13**4*m1 - 4*a33*b23
**2*c13**3*c23*m2 + 2*a33*b23**2*c13**2*c23**2*m1 + 2*b23**3*c13**3*c23*n3 + 2*
b23**3*c13*c23**3*n3) + v2**2*v3**2*(4*a33**2*c13**6 + 4*a33**2*c13**4*c23**2) +
v2**2*v3*(4*a33**2*c13**5*m1 + 4*a33**2*c13**4*c23*m2) + v2*v3**3*( - 4*a33*b23
**2*c13**4*c23 - 4*a33*b23**2*c13**2*c23**3) + v2*v3**2*(2*a33*b23**2*c13**4*m2
- 4*a33*b23**2*c13**3*c23*m1 - 2*a33*b23**2*c13**2*c23**2*m2 - 2*b23**3*c13**4*
n3 - 2*b23**3*c13**2*c23**2*n3) + v3**4*(b23**4*c13**4 + 2*b23**4*c13**2*c23**2
+ b23**4*c23**4) + v3**2*( - a33**2*c13**4*m1**2 - a33**2*c13**4*m2**2 + 2*a33*
b23*c13**4*m2*n3 - 2*a33*b23*c13**3*c23*m1*n3 - b23**2*c13**4*n3**2 - b23**2*c13
**2*c23**2*n3**2)$