Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r10=0
a22*m3*n2*n3*r494
r11=-------------------
4
b12 *kap
a22*m3*n1*n3*r494
r12=-------------------
4
b12 *kap
2 3
a22*kap*m3 *n3*r494 - a22*n3 *r494
r13=------------------------------------
4
b12 *kap
2
- a22*n2*n3 *r494
r14=--------------------
4
b12 *kap
2
- a22*n1*n3 *r494
r15=--------------------
4
b12 *kap
1 2 2 1 2 2
- ---*a22 *m3 *r494 - ---*b12 *kap*m3 *r494
2 2
r20=----------------------------------------------
4
b12 *kap
r21=0
1 2 2 1 2 2
- ---*a22 *m3 *r494 - ---*b12 *kap*m3 *r494
2 2
r22=----------------------------------------------
4
b12 *kap
r23=0
r24=0
- a22*n1*n3*r494 + b12*kap*m3*n2*r494
r27=----------------------------------------
3
b12 *kap
2
- a22 *m3*n1*r494 + a22*b12*n2*n3*r494
r28=-----------------------------------------
4
b12 *kap
1 2 2 1 2 2 1 2 2
r29=( - ---*a22 *kap*m3 *r494 + ---*a22 *n1 *r494 - ---*b12 *kap*n2 *r494
2 2 2
1 2 2 4
- ---*b12 *kap*n3 *r494)/(b12 *kap)
2
2 2
- a22 *m3*n2*r494 + a22*b12*n1*n3*r494 - b12 *kap*m3*n2*r494
r210=---------------------------------------------------------------
4
b12 *kap
2
2*a22 *m3*n3*r494
r211=-------------------
4
b12 *kap
r212=0
2
- a22 *n2*n3*r494
r213=--------------------
4
b12 *kap
1 2 2 1 2 2 1 2 2
r214=(---*a22 *n1 *r494 - ---*a22 *n2 *r494 - ---*a22 *n3 *r494
2 2 2
1 2 2 1 2 2 4
- ---*b12 *kap*n2 *r494 - ---*b12 *kap*n3 *r494)/(b12 *kap)
2 2
- a22*n2*n3*r494
r215=-------------------
3
b12 *kap
2 2
a22 *m3*n3*r494 - b12 *kap*m3*n3*r494
r217=---------------------------------------
4
b12 *kap
2
- a22 *n1*n3*r494
r218=--------------------
4
b12 *kap
2
- a22 *n1*n2*r494
r219=--------------------
4
b12 *kap
1 2 2 1 2 2 1 2 2 2
r220=( - ---*a22 *kap*m3 *r494 + ---*a22 *n3 *r494 - ---*b12 *kap *m3 *r494
2 2 2
1 2 2 1 2 2 4
- ---*b12 *kap*n2 *r494 + ---*b12 *kap*n3 *r494)/(b12 *kap)
2 2
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
- a22*n3*r494
r310=----------------
2
b12 *kap
r311=0
- a22*n3*r494
r312=----------------
2
b12 *kap
r313=0
2 2
- a22 *m3*r494 - b12 *kap*m3*r494
r314=------------------------------------
3
b12 *kap
- a22*n3*r494
r315=----------------
2
b12 *kap
- a22*m3*r494
r316=----------------
2
b12
2
a22 *n1*r494
r317=--------------
3
b12 *kap
n2*r494
r318=---------
b12
r319=0
r320=0
r321=0
r322=0
2 2
a22 *m3*r494 + b12 *kap*m3*r494
r323=---------------------------------
3
b12 *kap
r324=0
r325=0
2
- a22 *n1*r494
r326=-----------------
3
b12 *kap
r327=0
2
2*a22 *n3*r494
r328=----------------
3
b12 *kap
- a22*n2*r494
r329=----------------
2
b12
3 2
- a22 *m3*r494 - a22*b12 *kap*m3*r494
r330=----------------------------------------
4
b12 *kap
r331=0
2 2
a22 *n2*r494 + b12 *kap*n2*r494
r332=---------------------------------
3
b12 *kap
3
- a22 *n3*r494
r333=-----------------
4
b12 *kap
3 2
- a22 *n2*r494 - a22*b12 *kap*n2*r494
r334=----------------------------------------
4
b12 *kap
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- n2*r494
r341=------------
b12
2 2
- a22 *n3*r494 + b12 *kap*n3*r494
r342=------------------------------------
3
b12 *kap
r343=0
- a22*n1*r494
r344=----------------
2
b12
2 2
- a22 *n3*r494 - b12 *kap*n3*r494
r345=------------------------------------
3
b12 *kap
2 2
- a22 *n2*r494 - b12 *kap*n2*r494
r346=------------------------------------
3
b12 *kap
r347=0
r348=0
3 2
- a22 *n1*r494 - a22*b12 *kap*n1*r494
r349=----------------------------------------
4
b12 *kap
r350=0
r351=0
r352=0
a22*n3*r494
r353=-------------
2
b12
r354=0
r355=0
r40=0
r41=0
r42=0
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
r49=0
r410=0
r411=0
r412=0
r413=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r423=0
r424=0
r426=0
1 2
---*a22 *r494
2
r427=---------------
2
b12 *kap
r428=0
r429=0
1
r430= - ---*r494
2
r431=0
r432=0
r433=0
r434=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
2
- a22 *r494
r446=--------------
2
b12 *kap
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
a22*r494
r453=----------
b12
r454=0
r455=0
r456=0
1 2
- ---*a22 *r494
2
r457=------------------
2
b12 *kap
r458=0
r459=0
1 2 1 2
- ---*a22 *r494 - ---*b12 *kap*r494
2 2
r460=--------------------------------------
2
b12 *kap
r461=0
r462=0
r463=0
1 2
- ---*a22 *r494
2
r464=------------------
2
b12
r465=0
r466=0
3 2
a22 *r494 + a22*b12 *kap*r494
r467=-------------------------------
3
b12 *kap
r468=0
1 4 1 2 2
- ---*a22 *r494 - ---*a22 *b12 *kap*r494
2 2
r469=-------------------------------------------
4
b12 *kap
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r479=0
r480=0
r482=0
r483=r494
r484=0
r485=0
r486=0
- a22*r494
r487=-------------
b12
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
3 2
- a22 *r494 - a22*b12 *kap*r494
r4101=----------------------------------
3
b12 *kap
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
1
r4110=---*r494
2
r4111=0
r4112=0
r4113=0
1
r4114=---*kap*r494
2
r4115=0
r4116=0
r4117=0
r4118=0
1 2 1 2
---*a22 *r494 + ---*b12 *kap*r494
2 2
r4119=-----------------------------------
2
b12
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
a33=0
a23=0
a13=0
2
- b12 *kap
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:
r494, n2, n1, n3, m3, a22, b12
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,
2 2
a22 + b12 *kap,
a22,
b12,
3 2 2 2
{a22 *r494 + a22*b12 *kap*r494,r494,a22 *r494 + b12 *kap*r494},
{m3,n1,n2,n3}}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2
HAM=( - u1 *b12 *kap + u1*v2*a22*b12 + u1*a22*n1 + u2 *a22 - u2*v1*a22*b12
+ u2*a22*n2 + u3*a22*n3 + v3*a22*m3)/a22
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 1 2 2 1 4 2 1 2 2 4 2
FI=u1 *u2 *(---*a22 *b12 *kap + ---*b12 *kap ) + ---*u1 *u3 *b12 *kap
2 2 2
2 2 1 2 2 4 2 1 2 2
+ u1 *u3*a22*b12 *kap*n3 + ---*u1 *v1 *b12 *kap + u1 *( - ---*a22 *kap*m3
2 2
1 2 2 1 2 2 2 1 2 2 1 2 2
+ ---*a22 *n3 - ---*b12 *kap *m3 - ---*b12 *kap*n2 + ---*b12 *kap*n3 )
2 2 2 2
2 3 3
+ u1*u2 *v2*( - a22 *b12 - a22*b12 *kap)
2 3 2 4
+ u1*u2 *( - a22 *n1 - a22*b12 *kap*n1) + u1*u2*v1*v2*b12 *kap
2 3
+ u1*u2*v2*( - a22 *b12*n2 - b12 *kap*n2)
2 3 2
+ u1*u2*v3*( - a22 *b12*n3 - b12 *kap*n3) - u1*u2*a22 *n1*n2
2 3 2 2 4
- u1*u3 *v2*a22*b12 *kap - u1*u3 *a22*b12 *kap*n1 + u1*u3*v1*v3*b12 *kap
2 3 3
+ u1*u3*v2*( - a22 *b12*n3 + b12 *kap*n3) - u1*u3*v3*b12 *kap*n2
2 2 2
- u1*u3*a22 *n1*n3 + u1*v1*(a22 *m3*n3 - b12 *kap*m3*n3)
2
- u1*v3*a22*b12*n2*n3 - u1*a22*n1*n3
4 1 4 1 2 2 3 3 3
+ u2 *( - ---*a22 - ---*a22 *b12 *kap) + u2 *v1*(a22 *b12 + a22*b12 *kap)
2 2
3 3 2 1 2 2 2 2
+ u2 *( - a22 *n2 - a22*b12 *kap*n2) - ---*u2 *u3 *a22 *b12 *kap
2
2 3 2 2 1 2 2 1 4
- u2 *u3*a22 *n3 + u2 *v1 *( - ---*a22 *b12 - ---*b12 *kap)
2 2
2 2 3 1 2 2 2 2
+ u2 *v1*(a22 *b12*n2 + b12 *kap*n2) - ---*u2 *v2 *a22 *b12
2
2 3 2 2 1 2 2 1 2 2
+ u2 *v3*( - a22 *m3 - a22*b12 *kap*m3) + u2 *(---*a22 *n1 - ---*a22 *n2
2 2
1 2 2 1 2 2 1 2 2
- ---*a22 *n3 - ---*b12 *kap*n2 - ---*b12 *kap*n3 )
2 2 2
2 3 2 2 2
+ u2*u3 *v1*a22*b12 *kap - u2*u3 *a22*b12 *kap*n2 + 2*u2*u3*v1*a22 *b12*n3
2 2 2 2
- u2*u3*v2*v3*a22 *b12 - u2*u3*v3*a22 *b12*n1 - u2*u3*a22 *n2*n3
2 3 2
+ u2*v1*v3*(a22 *b12*m3 + b12 *kap*m3) + 2*u2*v2*a22 *m3*n3
2 2 2
+ u2*v3*( - a22 *m3*n2 + a22*b12*n1*n3 - b12 *kap*m3*n2) - u2*a22*n2*n3
1 2 2 4 2 3 1 2 2 2 2
- ---*u3 *v1 *b12 *kap + u3 *v1*b12 *kap*n2 + ---*u3 *v2 *a22 *b12
2 2
2 2 2 2 2
+ u3 *v2*a22 *b12*n1 - u3 *v3*a22*b12 *kap*m3 + u3
1 2 2 1 2 2 1 2 2 1 2 2
*( - ---*a22 *kap*m3 + ---*a22 *n1 - ---*b12 *kap*n2 - ---*b12 *kap*n3 )
2 2 2 2
2 2 2 3
- u3*v1 *a22*b12 *n3 + u3*v1*v2*( - a22 *b12*m3 - b12 *kap*m3)
2 2 2
+ u3*v1*( - a22 *m3*n1 + a22*b12*n2*n3) - u3*v2 *a22*b12 *n3
2 2 2
+ u3*v2*( - a22*b12*n1*n3 + b12 *kap*m3*n2) - u3*v3 *a22*b12 *n3
2 3
+ u3*(a22*kap*m3 *n3 - a22*n3 ) + v1*a22*m3*n1*n3
2 1 2 2 1 2 2
+ v2 *( - ---*a22 *m3 - ---*b12 *kap*m3 ) + v2*a22*m3*n2*n3
2 2
2 1 2 2 1 2 2
+ v3 *( - ---*a22 *m3 - ---*b12 *kap*m3 )
2 2
1
= a product of the elements of: { - ---,
2
2 2 2 2 4 2 2 2 4 2
u1 *u2 *( - a22 *b12 *kap - b12 *kap ) - u1 *u3 *b12 *kap
2 2 2 2 4 2
- 2*u1 *u3*a22*b12 *kap*n3 - u1 *v1 *b12 *kap + u1
2 2 2 2 2 2 2 2 2 2 2
*(a22 *kap*m3 - a22 *n3 + b12 *kap *m3 + b12 *kap*n2 - b12 *kap*n3 )
2 3 3
+ u1*u2 *v2*(2*a22 *b12 + 2*a22*b12 *kap)
2 3 2 4
+ u1*u2 *(2*a22 *n1 + 2*a22*b12 *kap*n1) - 2*u1*u2*v1*v2*b12 *kap
2 3
+ u1*u2*v2*(2*a22 *b12*n2 + 2*b12 *kap*n2)
2 3 2
+ u1*u2*v3*(2*a22 *b12*n3 + 2*b12 *kap*n3) + 2*u1*u2*a22 *n1*n2
2 3 2 2
+ 2*u1*u3 *v2*a22*b12 *kap + 2*u1*u3 *a22*b12 *kap*n1
4 2 3
- 2*u1*u3*v1*v3*b12 *kap + u1*u3*v2*(2*a22 *b12*n3 - 2*b12 *kap*n3)
3 2
+ 2*u1*u3*v3*b12 *kap*n2 + 2*u1*u3*a22 *n1*n3
2 2
+ u1*v1*( - 2*a22 *m3*n3 + 2*b12 *kap*m3*n3) + 2*u1*v3*a22*b12*n2*n3
2 4 4 2 2
+ 2*u1*a22*n1*n3 + u2 *(a22 + a22 *b12 *kap)
3 3 3
+ u2 *v1*( - 2*a22 *b12 - 2*a22*b12 *kap)
3 3 2 2 2 2 2
+ u2 *(2*a22 *n2 + 2*a22*b12 *kap*n2) + u2 *u3 *a22 *b12 *kap
2 3 2 2 2 2 4
+ 2*u2 *u3*a22 *n3 + u2 *v1 *(a22 *b12 + b12 *kap)
2 2 3 2 2 2 2
+ u2 *v1*( - 2*a22 *b12*n2 - 2*b12 *kap*n2) + u2 *v2 *a22 *b12
2 3 2
+ u2 *v3*(2*a22 *m3 + 2*a22*b12 *kap*m3)
2 2 2 2 2 2 2 2 2 2 2
+ u2 *( - a22 *n1 + a22 *n2 + a22 *n3 + b12 *kap*n2 + b12 *kap*n3 )
2 3 2 2
- 2*u2*u3 *v1*a22*b12 *kap + 2*u2*u3 *a22*b12 *kap*n2
2 2 2 2
- 4*u2*u3*v1*a22 *b12*n3 + 2*u2*u3*v2*v3*a22 *b12 + 2*u2*u3*v3*a22 *b12*n1
2 2 3
+ 2*u2*u3*a22 *n2*n3 + u2*v1*v3*( - 2*a22 *b12*m3 - 2*b12 *kap*m3)
2
- 4*u2*v2*a22 *m3*n3
2 2
+ u2*v3*(2*a22 *m3*n2 - 2*a22*b12*n1*n3 + 2*b12 *kap*m3*n2)
2 2 2 4 2 3
+ 2*u2*a22*n2*n3 + u3 *v1 *b12 *kap - 2*u3 *v1*b12 *kap*n2
2 2 2 2 2 2 2 2
- u3 *v2 *a22 *b12 - 2*u3 *v2*a22 *b12*n1 + 2*u3 *v3*a22*b12 *kap*m3
2 2 2 2 2 2 2 2 2
+ u3 *(a22 *kap*m3 - a22 *n1 + b12 *kap*n2 + b12 *kap*n3 )
2 2 2 3
+ 2*u3*v1 *a22*b12 *n3 + u3*v1*v2*(2*a22 *b12*m3 + 2*b12 *kap*m3)
2 2 2
+ u3*v1*(2*a22 *m3*n1 - 2*a22*b12*n2*n3) + 2*u3*v2 *a22*b12 *n3
2 2 2
+ u3*v2*(2*a22*b12*n1*n3 - 2*b12 *kap*m3*n2) + 2*u3*v3 *a22*b12 *n3
2 3
+ u3*( - 2*a22*kap*m3 *n3 + 2*a22*n3 ) - 2*v1*a22*m3*n1*n3
2 2 2 2 2
+ v2 *(a22 *m3 + b12 *kap*m3 ) - 2*v2*a22*m3*n2*n3
2 2 2 2 2
+ v3 *(a22 *m3 + b12 *kap*m3 )}
{HAM,FI} = 0
And again in machine readable form:
HAM=( - u1**2*b12**2*kap + u1*v2*a22*b12 + u1*a22*n1 + u2**2*a22**2 - u2*v1*a22*
b12 + u2*a22*n2 + u3*a22*n3 + v3*a22*m3)/a22$
FI=u1**2*u2**2*(1/2*a22**2*b12**2*kap + 1/2*b12**4*kap**2) + 1/2*u1**2*u3**2*b12
**4*kap**2 + u1**2*u3*a22*b12**2*kap*n3 + 1/2*u1**2*v1**2*b12**4*kap + u1**2*( -
1/2*a22**2*kap*m3**2 + 1/2*a22**2*n3**2 - 1/2*b12**2*kap**2*m3**2 - 1/2*b12**2*
kap*n2**2 + 1/2*b12**2*kap*n3**2) + u1*u2**2*v2*( - a22**3*b12 - a22*b12**3*kap)
+ u1*u2**2*( - a22**3*n1 - a22*b12**2*kap*n1) + u1*u2*v1*v2*b12**4*kap + u1*u2*
v2*( - a22**2*b12*n2 - b12**3*kap*n2) + u1*u2*v3*( - a22**2*b12*n3 - b12**3*kap*
n3) - u1*u2*a22**2*n1*n2 - u1*u3**2*v2*a22*b12**3*kap - u1*u3**2*a22*b12**2*kap*
n1 + u1*u3*v1*v3*b12**4*kap + u1*u3*v2*( - a22**2*b12*n3 + b12**3*kap*n3) - u1*
u3*v3*b12**3*kap*n2 - u1*u3*a22**2*n1*n3 + u1*v1*(a22**2*m3*n3 - b12**2*kap*m3*
n3) - u1*v3*a22*b12*n2*n3 - u1*a22*n1*n3**2 + u2**4*( - 1/2*a22**4 - 1/2*a22**2*
b12**2*kap) + u2**3*v1*(a22**3*b12 + a22*b12**3*kap) + u2**3*( - a22**3*n2 - a22
*b12**2*kap*n2) - 1/2*u2**2*u3**2*a22**2*b12**2*kap - u2**2*u3*a22**3*n3 + u2**2
*v1**2*( - 1/2*a22**2*b12**2 - 1/2*b12**4*kap) + u2**2*v1*(a22**2*b12*n2 + b12**
3*kap*n2) - 1/2*u2**2*v2**2*a22**2*b12**2 + u2**2*v3*( - a22**3*m3 - a22*b12**2*
kap*m3) + u2**2*(1/2*a22**2*n1**2 - 1/2*a22**2*n2**2 - 1/2*a22**2*n3**2 - 1/2*
b12**2*kap*n2**2 - 1/2*b12**2*kap*n3**2) + u2*u3**2*v1*a22*b12**3*kap - u2*u3**2
*a22*b12**2*kap*n2 + 2*u2*u3*v1*a22**2*b12*n3 - u2*u3*v2*v3*a22**2*b12**2 - u2*
u3*v3*a22**2*b12*n1 - u2*u3*a22**2*n2*n3 + u2*v1*v3*(a22**2*b12*m3 + b12**3*kap*
m3) + 2*u2*v2*a22**2*m3*n3 + u2*v3*( - a22**2*m3*n2 + a22*b12*n1*n3 - b12**2*kap
*m3*n2) - u2*a22*n2*n3**2 - 1/2*u3**2*v1**2*b12**4*kap + u3**2*v1*b12**3*kap*n2
+ 1/2*u3**2*v2**2*a22**2*b12**2 + u3**2*v2*a22**2*b12*n1 - u3**2*v3*a22*b12**2*
kap*m3 + u3**2*( - 1/2*a22**2*kap*m3**2 + 1/2*a22**2*n1**2 - 1/2*b12**2*kap*n2**
2 - 1/2*b12**2*kap*n3**2) - u3*v1**2*a22*b12**2*n3 + u3*v1*v2*( - a22**2*b12*m3
- b12**3*kap*m3) + u3*v1*( - a22**2*m3*n1 + a22*b12*n2*n3) - u3*v2**2*a22*b12**2
*n3 + u3*v2*( - a22*b12*n1*n3 + b12**2*kap*m3*n2) - u3*v3**2*a22*b12**2*n3 + u3*
(a22*kap*m3**2*n3 - a22*n3**3) + v1*a22*m3*n1*n3 + v2**2*( - 1/2*a22**2*m3**2 -
1/2*b12**2*kap*m3**2) + v2*a22*m3*n2*n3 + v3**2*( - 1/2*a22**2*m3**2 - 1/2*b12**
2*kap*m3**2)$