Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2 2 2 2 2 2
0=a11 *n2 + 4*a13 *n1 + 4*a13 *n2 + b12 *kap*n2
Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
1 2 3 2 2 5 2 3 2
r13=(---*a11 *n1 *n2 *r4110 + 2*a13 *n1 *r4110 + 2*a13 *n1 *n2 *r4110
2
1 2 3 2 3 2 2
+ ---*b12 *kap*n1 *n2 *r4110)/(a13 *b12 *n2 )
2
r14=0
r15=0
r20=0
r21=0
r22=0
r23=0
r24=0
2*m3*n2*r4110
r27=---------------
2
b12
2*m3*n1*r4110
r28=---------------
2
b12
1 2 2 2 2 2 2 2 4
r29=( - ---*a11 *n1 *n2 *r4110 + a13 *kap*m3 *n2 *r4110 - 2*a13 *n1 *r4110
2
2 2 2 2 4 1 2 2 2
- 3*a13 *n1 *n2 *r4110 - a13 *n2 *r4110 - ---*b12 *kap*n1 *n2 *r4110)/(
2
2 2 2
a13 *b12 *n2 )
r210=0
r211=0
r212=0
r213=0
2
- n1 *r4110
r214=--------------
2
b12
r215=0
r217=0
r218=0
2*n1*n2*r4110
r219=---------------
2
b12
2
- n2 *r4110
r220=--------------
2
b12
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
- 2*a11*m3*r4110
r316=-------------------
2
b12
- 4*a13*m3*n1*r4110 - 2*b12*n1*n2*r4110
r317=------------------------------------------
2
b12 *n2
4*a13*m3*r4110 + 2*b12*n2*r4110
r318=---------------------------------
2
b12
2 2 2 8 2 3 8 2 2
r319=(---*a11 *n1*n2 *r4110 + ---*a13 *n1 *r4110 + ---*a13 *n1*n2 *r4110
3 3 3
2 2 2 2 2
+ ---*b12 *kap*n1*n2 *r4110)/(a13*b12 *n2 )
3
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
2*n1*r4110
r326=------------
b12
r327=0
r328=0
- 2*a11*n2*r4110
r329=-------------------
2
b12
r330=0
r331=0
r332=0
- 4*a13*n1*r4110
r333=-------------------
2
b12
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- 2*n2*r4110
r341=---------------
b12
r342=0
r343=0
- 2*a11*n1*r4110
r344=-------------------
2
b12
r345=0
r346=0
r347=0
2 2
- 4*a13*n1 *r4110 + 4*a13*n2 *r4110
r348=--------------------------------------
2
b12 *n2
r349=0
r350=0
r351=0
r352=0
4*a13*n1*r4110
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
r427= - r4110
r428=0
r429=0
r430= - r4110
r431=0
- 4*a13*r4110
r432=----------------
b12
- 4*a13*n1*r4110
r433=-------------------
b12*n2
2 2 2 2
- 4*a13 *n1 *r4110 - 4*a13 *n2 *r4110
r434=----------------------------------------
2 2
b12 *n2
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r446=2*r4110
r447=0
r448=0
r449=0
r450=0
4*a13*r4110
r451=-------------
b12
r452=0
2*a11*r4110
r453=-------------
b12
4*a11*a13*n1*r4110
r454=--------------------
2
b12 *n2
r455=0
r456=0
r457=r4110
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
r464=
2 2 2 2 2 2 2 2
- 2*a11 *n2 *r4110 - 4*a13 *n1 *r4110 - 8*a13 *n2 *r4110 - b12 *kap*n2 *r4110
--------------------------------------------------------------------------------
2 2
b12 *n2
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
r479=0
r480=0
r482=0
r483=2*r4110
r484=0
r485=0
4*a13*n1*r4110
r486=----------------
b12*n2
- 2*a11*r4110
r487=----------------
b12
r488=0
- 4*a11*a13*r4110
r489=--------------------
2
b12
r490=0
r491=0
r492=0
r493=0
r494=2*r4110
r495=0
r496=0
r497=0
r498=0
2
- 8*a13 *n1*r4110
r499=--------------------
2
b12 *n2
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4114=
2 2 2 2 2 2 2 2
- 2*a11 *n2 *r4110 - 8*a13 *n1 *r4110 - 4*a13 *n2 *r4110 - b12 *kap*n2 *r4110
--------------------------------------------------------------------------------
2 2
b12 *n2
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
n3=0
a33=0
a22=a11
- a13*n1
a23=-----------
n2
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:
n1, r4110, n2, a11, a13, m3, 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.
{a23,r4110,a11,b12,a13}
Relevance for the application:
Modulo the following equation:
2 2 2 2 2 2 2 2
0=a11 *n2 + 4*a13 *n1 + 4*a13 *n2 + b12 *kap*n2
the system of equations related to the Hamiltonian HAM:
2 2
HAM=(u1 *a11*n2 + 2*u1*u3*a13*n2 + u1*v2*b12*n2 + u1*n1*n2 + u2 *a11*n2
2
- 2*u2*u3*a13*n1 - u2*v1*b12*n2 + u2*n2 + v3*m3*n2)/n2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 3 2 5 2 5 2 3 2 2
FI=u1 *u3 *( - 2*a11 *a13 *n2 - 8*a13 *n1 - 4*a13 *n2 - a13 *b12 *kap*n2 )
2 4 2 2 2 3 2 2 2 3 4
+ 4*u1 *u3*a13 *n1*n2 + u1 *v1 *a13 *b12 *n2 - u1 *a13 *n2
2 5 4 2 4 3
- 8*u1*u2*u3 *a13 *n1*n2 + u1*u2*u3*( - 4*a13 *n1 *n2 + 4*a13 *n2 )
3 2 2 3 3 3 4 2
+ 2*u1*u2*v1*v2*a13 *b12 *n2 + 2*u1*u2*a13 *n1*n2 - 4*u1*u3 *a11*a13 *n2
2 3 2 2 4
- 2*u1*u3 *v2*a11*a13 *b12*n2 + 4*u1*u3 *v3*a13 *b12*n1*n2
2 3 2 3 2 2
- 2*u1*u3 *a11*a13 *n1*n2 + 2*u1*u3*v1*v3*a13 *b12 *n2
3 3
- 2*u1*u3*v3*a13 *b12*n2
2 2 2 3 2 5 2 5 2 3 2 2
+ u2 *u3 *( - 2*a11 *a13 *n2 - 4*a13 *n1 - 8*a13 *n2 - a13 *b12 *kap*n2 )
2 4 2 2 2 3 2 2 2 3 2 2
- 4*u2 *u3*a13 *n1*n2 + u2 *v2 *a13 *b12 *n2 - u2 *a13 *n1 *n2
3 4 2 3 2
+ 4*u2*u3 *a11*a13 *n1*n2 + 2*u2*u3 *v1*a11*a13 *b12*n2
2 4 2 2 3 3
+ 4*u2*u3 *v3*a13 *b12*n2 - 2*u2*u3 *a11*a13 *n2
3 2 2 3 2
+ 2*u2*u3*v2*v3*a13 *b12 *n2 + 2*u2*u3*v3*a13 *b12*n1*n2
4 5 2 5 2 3 4
+ u3 *( - 4*a13 *n1 - 4*a13 *n2 ) - 4*u3 *v1*a13 *b12*n1*n2
3 4 2 3 2 2 2 2 8 4 3
- 4*u3 *v2*a13 *b12*n2 + u3 *(---*a11 *a13 *n1*n2 + ---*a13 *n1
3 3
8 4 2 2 2 2 2 2 2 3 2 2
+ ---*a13 *n1*n2 + ---*a13 *b12 *kap*n1*n2 ) - u3 *v1 *a13 *b12 *n2
3 3
2 4 2 3 3 2 2 3 2 2
+ u3 *v1*(4*a13 *m3*n2 + 2*a13 *b12*n2 ) - u3 *v2 *a13 *b12 *n2
2 4 3 2 2 3 2
+ u3 *v2*( - 4*a13 *m3*n1*n2 - 2*a13 *b12*n1*n2 ) - 2*u3 *v3*a11*a13 *m3*n2
2 1 2 2 2 3 2 2 3 4
+ u3 *( - ---*a11 *a13*n1 *n2 + a13 *kap*m3 *n2 - 2*a13 *n1
2
3 2 2 3 4 1 2 2 2
- 3*a13 *n1 *n2 - a13 *n2 - ---*a13*b12 *kap*n1 *n2 )
2
3 2 3 3
+ 2*u3*v1*a13 *m3*n1*n2 + 2*u3*v2*a13 *m3*n2
1 2 3 2 2 5 2 3 2 1 2 3 2
+ u3*(---*a11 *n1 *n2 + 2*a13 *n1 + 2*a13 *n1 *n2 + ---*b12 *kap*n1 *n2 )
2 2
1
= a product of the elements of: { - ---,
6
2 2 2 3 2 5 2 5 2 3 2 2
u1 *u3 *(12*a11 *a13 *n2 + 48*a13 *n1 + 24*a13 *n2 + 6*a13 *b12 *kap*n2 )
2 4 2 2 2 3 2 2 2 3 4
- 24*u1 *u3*a13 *n1*n2 - 6*u1 *v1 *a13 *b12 *n2 + 6*u1 *a13 *n2
2 5 4 2 4 3
+ 48*u1*u2*u3 *a13 *n1*n2 + u1*u2*u3*(24*a13 *n1 *n2 - 24*a13 *n2 )
3 2 2 3 3
- 12*u1*u2*v1*v2*a13 *b12 *n2 - 12*u1*u2*a13 *n1*n2
3 4 2 2 3 2
+ 24*u1*u3 *a11*a13 *n2 + 12*u1*u3 *v2*a11*a13 *b12*n2
2 4 2 3 2
- 24*u1*u3 *v3*a13 *b12*n1*n2 + 12*u1*u3 *a11*a13 *n1*n2
3 2 2 3 3 2 2
- 12*u1*u3*v1*v3*a13 *b12 *n2 + 12*u1*u3*v3*a13 *b12*n2 + u2 *u3
2 3 2 5 2 5 2 3 2 2
*(12*a11 *a13 *n2 + 24*a13 *n1 + 48*a13 *n2 + 6*a13 *b12 *kap*n2 )
2 4 2 2 2 3 2 2 2 3 2 2
+ 24*u2 *u3*a13 *n1*n2 - 6*u2 *v2 *a13 *b12 *n2 + 6*u2 *a13 *n1 *n2
3 4 2 3 2
- 24*u2*u3 *a11*a13 *n1*n2 - 12*u2*u3 *v1*a11*a13 *b12*n2
2 4 2 2 3 3
- 24*u2*u3 *v3*a13 *b12*n2 + 12*u2*u3 *a11*a13 *n2
3 2 2 3 2
- 12*u2*u3*v2*v3*a13 *b12 *n2 - 12*u2*u3*v3*a13 *b12*n1*n2
4 5 2 5 2 3 4
+ u3 *(24*a13 *n1 + 24*a13 *n2 ) + 24*u3 *v1*a13 *b12*n1*n2
3 4 2 3 2 2 2 4 3
+ 24*u3 *v2*a13 *b12*n2 + u3 *( - 4*a11 *a13 *n1*n2 - 16*a13 *n1
4 2 2 2 2 2 2 3 2 2
- 16*a13 *n1*n2 - 4*a13 *b12 *kap*n1*n2 ) + 6*u3 *v1 *a13 *b12 *n2
2 4 2 3 3 2 2 3 2 2
+ u3 *v1*( - 24*a13 *m3*n2 - 12*a13 *b12*n2 ) + 6*u3 *v2 *a13 *b12 *n2
2 4 3 2 2 3 2
+ u3 *v2*(24*a13 *m3*n1*n2 + 12*a13 *b12*n1*n2 ) + 12*u3 *v3*a11*a13 *m3*n2
2 2 2 2 3 2 2 3 4
+ u3 *(3*a11 *a13*n1 *n2 - 6*a13 *kap*m3 *n2 + 12*a13 *n1
3 2 2 3 4 2 2 2
+ 18*a13 *n1 *n2 + 6*a13 *n2 + 3*a13*b12 *kap*n1 *n2 )
3 2 3 3
- 12*u3*v1*a13 *m3*n1*n2 - 12*u3*v2*a13 *m3*n2 + u3
2 3 2 2 5 2 3 2 2 3 2
*( - 3*a11 *n1 *n2 - 12*a13 *n1 - 12*a13 *n1 *n2 - 3*b12 *kap*n1 *n2 )}
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a11*n2 + 2*u1*u3*a13*n2 + u1*v2*b12*n2 + u1*n1*n2 + u2**2*a11*n2 - 2*
u2*u3*a13*n1 - u2*v1*b12*n2 + u2*n2**2 + v3*m3*n2)/n2$
FI=u1**2*u3**2*( - 2*a11**2*a13**3*n2**2 - 8*a13**5*n1**2 - 4*a13**5*n2**2 - a13
**3*b12**2*kap*n2**2) + 4*u1**2*u3*a13**4*n1*n2**2 + u1**2*v1**2*a13**3*b12**2*
n2**2 - u1**2*a13**3*n2**4 - 8*u1*u2*u3**2*a13**5*n1*n2 + u1*u2*u3*( - 4*a13**4*
n1**2*n2 + 4*a13**4*n2**3) + 2*u1*u2*v1*v2*a13**3*b12**2*n2**2 + 2*u1*u2*a13**3*
n1*n2**3 - 4*u1*u3**3*a11*a13**4*n2**2 - 2*u1*u3**2*v2*a11*a13**3*b12*n2**2 + 4*
u1*u3**2*v3*a13**4*b12*n1*n2 - 2*u1*u3**2*a11*a13**3*n1*n2**2 + 2*u1*u3*v1*v3*
a13**3*b12**2*n2**2 - 2*u1*u3*v3*a13**3*b12*n2**3 + u2**2*u3**2*( - 2*a11**2*a13
**3*n2**2 - 4*a13**5*n1**2 - 8*a13**5*n2**2 - a13**3*b12**2*kap*n2**2) - 4*u2**2
*u3*a13**4*n1*n2**2 + u2**2*v2**2*a13**3*b12**2*n2**2 - u2**2*a13**3*n1**2*n2**2
+ 4*u2*u3**3*a11*a13**4*n1*n2 + 2*u2*u3**2*v1*a11*a13**3*b12*n2**2 + 4*u2*u3**2
*v3*a13**4*b12*n2**2 - 2*u2*u3**2*a11*a13**3*n2**3 + 2*u2*u3*v2*v3*a13**3*b12**2
*n2**2 + 2*u2*u3*v3*a13**3*b12*n1*n2**2 + u3**4*( - 4*a13**5*n1**2 - 4*a13**5*n2
**2) - 4*u3**3*v1*a13**4*b12*n1*n2 - 4*u3**3*v2*a13**4*b12*n2**2 + u3**3*(2/3*
a11**2*a13**2*n1*n2**2 + 8/3*a13**4*n1**3 + 8/3*a13**4*n1*n2**2 + 2/3*a13**2*b12
**2*kap*n1*n2**2) - u3**2*v1**2*a13**3*b12**2*n2**2 + u3**2*v1*(4*a13**4*m3*n2**
2 + 2*a13**3*b12*n2**3) - u3**2*v2**2*a13**3*b12**2*n2**2 + u3**2*v2*( - 4*a13**
4*m3*n1*n2 - 2*a13**3*b12*n1*n2**2) - 2*u3**2*v3*a11*a13**3*m3*n2**2 + u3**2*( -
1/2*a11**2*a13*n1**2*n2**2 + a13**3*kap*m3**2*n2**2 - 2*a13**3*n1**4 - 3*a13**3
*n1**2*n2**2 - a13**3*n2**4 - 1/2*a13*b12**2*kap*n1**2*n2**2) + 2*u3*v1*a13**3*
m3*n1*n2**2 + 2*u3*v2*a13**3*m3*n2**3 + u3*(1/2*a11**2*n1**3*n2**2 + 2*a13**2*n1
**5 + 2*a13**2*n1**3*n2**2 + 1/2*b12**2*kap*n1**3*n2**2)$