Solution 3 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
0=4*a13 *n1 + 4*a13 *n3 + b12 *kap*n1
Expressions
The solution is given through the following expressions:
r10=0
r11=0
- m3*n1*r310
r12=---------------
2
b12
2 2
- kap*m3 *r310 + n3 *r310
r13=----------------------------
2
b12
r14=0
n1*n3*r310
r15=------------
2
b12
r20=0
r21=0
r22=0
r23=0
r24=0
n1*r310
r27=---------
b12
- 4*a13*m3*r310
r28=------------------
2
b12
2 2 2
- a13*kap*m3 *r310 + 2*a13*n1 *r310 + a13*n3 *r310
r29=-----------------------------------------------------
2
b12 *n1
- n1*r310
r210=------------
b12
4*a13*m3*n3*r310
r211=------------------
2
b12 *n1
r212=0
r213=0
2*a13*n1*r310
r214=---------------
2
b12
r215=0
4*a13*m3*n3*r310
r217=------------------
2
b12 *n1
r218=0
r219=0
a13*n1*r310
r220=-------------
2
b12
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r311=0
r312=r310
r313=0
r314=0
r315=r310
2
- 4*a13 *m3*n3*r310
r316=----------------------
2 2
b12 *n1
4*a13*r310
r317=------------
b12
2
- 4*a13 *m3*r310
r318=-------------------
2
b12 *n1
2
4*a13 *r310
r319=-------------
2
b12
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
- 4*a13*r310
r326=---------------
b12
r327=0
4*a13*n3*r310
r328=---------------
b12*n1
r329=0
r330=0
r331=0
r332=0
2 2 2 2 2 2
8*a13 *n1 *r310 + 8*a13 *n3 *r310 + b12 *kap*n1 *r310
r333=-------------------------------------------------------
2 2
b12 *n1
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
- 4*a13*n3*r310
r342=------------------
b12*n1
r343=0
2
- 8*a13 *n3*r310
r344=-------------------
2
b12 *n1
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
2 2 2 2 2 2
4*a13 *n1 *r310 + 8*a13 *n3 *r310 + b12 *kap*n1 *r310
r353=-------------------------------------------------------
2 2
b12 *n1
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
a13*r310
r427=----------
n1
r428=0
r429=0
a13*r310
r430=----------
n1
r431=0
2
4*a13 *r310
r432=-------------
b12*n1
r433=0
3
4*a13 *r310
r434=-------------
2
b12 *n1
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
- 2*a13*r310
r446=---------------
n1
r447=0
r448=0
r449=0
r450=0
2
- 4*a13 *r310
r451=----------------
b12*n1
r452=0
2
4*a13 *n3*r310
r453=----------------
2
b12*n1
r454=0
r455=0
r456=0
- a13*r310
r457=-------------
n1
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
3 2 3 2 2 2
8*a13 *n1 *r310 + 8*a13 *n3 *r310 + a13*b12 *kap*n1 *r310
r464=-----------------------------------------------------------
2 3
b12 *n1
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
- 2*a13*r310
r483=---------------
n1
r484=0
r485=0
r486=0
2
- 4*a13 *n3*r310
r487=-------------------
2
b12*n1
r488=0
3
- 8*a13 *n3*r310
r489=-------------------
2 2
b12 *n1
r490=0
r491=0
r492=0
r493=0
- 2*a13*r310
r494=---------------
n1
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
- a13*r310
r4110=-------------
n1
r4111=0
r4112=0
r4113=0
3 2 3 2 2 2
4*a13 *n1 *r310 + 8*a13 *n3 *r310 + a13*b12 *kap*n1 *r310
r4114=-----------------------------------------------------------
2 3
b12 *n1
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
n2=0
a33=0
a23=0
- 2*a13*n3
a22=-------------
n1
- 2*a13*n3
a11=-------------
n1
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:
r310, m3, n1, n3, a13, 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.
3 2
{n1,a13,b12,n3,{8*a13 *n3*r310 + 2*a13*b12 *kap*n3*r310,r310}}
Relevance for the application:
Modulo the following equation:
2 2 2 2 2 2
0=4*a13 *n1 + 4*a13 *n3 + b12 *kap*n1
the system of equations related to the Hamiltonian HAM:
2 2 2
HAM=( - 2*u1 *a13*n3 + 2*u1*u3*a13*n1 + u1*v2*b12*n1 + u1*n1 - 2*u2 *a13*n3
- u2*v1*b12*n1 + u3*n1*n3 + v3*m3*n1)/n1
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 3 2 3 2 2 2
FI=u1 *u3 *(4*a13 *n1 + 8*a13 *n3 + a13*b12 *kap*n1 )
2 2 3 2 2 2 3 2 2 2 2
+ u1 *u3*(4*a13 *n1 + 8*a13 *n1*n3 + b12 *kap*n1 ) - u1 *v1 *a13*b12 *n1
2 4 2 2 3 3
+ u1 *a13*n1 - 2*u1*u2*v1*v2*a13*b12 *n1 - 8*u1*u3 *a13 *n1*n3
2 2 2 2 2
- 4*u1*u3 *v2*a13 *b12*n1*n3 - 8*u1*u3 *a13 *n1 *n3
2 2 2
- 2*u1*u3*v1*v3*a13*b12 *n1 - 4*u1*u3*v2*a13*b12*n1 *n3
2 4
+ 4*u1*v1*a13*m3*n1 *n3 + u1*n1 *n3
2 2 3 2 3 2 2 2
+ u2 *u3 *(8*a13 *n1 + 8*a13 *n3 + a13*b12 *kap*n1 )
2 2 3 2 2 2 3 2 2 2 2
+ u2 *u3*(8*a13 *n1 + 8*a13 *n1*n3 + b12 *kap*n1 ) - u2 *v2 *a13*b12 *n1
2 4 2 2 2 2 2
+ 2*u2 *a13*n1 + 4*u2*u3 *v1*a13 *b12*n1*n3 - 4*u2*u3 *v3*a13 *b12*n1
2 2 2
+ 4*u2*u3*v1*a13*b12*n1 *n3 - 2*u2*u3*v2*v3*a13*b12 *n1
3 2 4
- 4*u2*u3*v3*a13*b12*n1 + 4*u2*v2*a13*m3*n1 *n3 - u2*v3*b12*n1
4 3 2 3 2 2 3 2 3
+ 4*u3 *a13 *n1 + 4*u3 *v2*a13 *b12*n1 + 4*u3 *a13 *n1
2 2 2 2 2 2 2 2 2 2 2
+ u3 *v1 *a13*b12 *n1 - 4*u3 *v1*a13 *m3*n1 + u3 *v2 *a13*b12 *n1
2 3 2 2
+ 4*u3 *v2*a13*b12*n1 - 4*u3 *v3*a13 *m3*n1*n3
2 2 2 4 2 2 2 2 3
+ u3 *( - a13*kap*m3 *n1 + 2*a13*n1 + a13*n1 *n3 ) + u3*v1 *b12 *n1
3 2 2 3 4 2 2 3
- 4*u3*v1*a13*m3*n1 + u3*v2 *b12 *n1 + u3*v2*b12*n1 + u3*v3 *b12 *n1
2 3 3 2 4
+ u3*( - kap*m3 *n1 + n1 *n3 ) - v1*m3*n1
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=( - 2*u1**2*a13*n3 + 2*u1*u3*a13*n1 + u1*v2*b12*n1 + u1*n1**2 - 2*u2**2*a13*
n3 - u2*v1*b12*n1 + u3*n1*n3 + v3*m3*n1)/n1$
FI=u1**2*u3**2*(4*a13**3*n1**2 + 8*a13**3*n3**2 + a13*b12**2*kap*n1**2) + u1**2*
u3*(4*a13**2*n1**3 + 8*a13**2*n1*n3**2 + b12**2*kap*n1**3) - u1**2*v1**2*a13*b12
**2*n1**2 + u1**2*a13*n1**4 - 2*u1*u2*v1*v2*a13*b12**2*n1**2 - 8*u1*u3**3*a13**3
*n1*n3 - 4*u1*u3**2*v2*a13**2*b12*n1*n3 - 8*u1*u3**2*a13**2*n1**2*n3 - 2*u1*u3*
v1*v3*a13*b12**2*n1**2 - 4*u1*u3*v2*a13*b12*n1**2*n3 + 4*u1*v1*a13*m3*n1**2*n3 +
u1*n1**4*n3 + u2**2*u3**2*(8*a13**3*n1**2 + 8*a13**3*n3**2 + a13*b12**2*kap*n1
**2) + u2**2*u3*(8*a13**2*n1**3 + 8*a13**2*n1*n3**2 + b12**2*kap*n1**3) - u2**2*
v2**2*a13*b12**2*n1**2 + 2*u2**2*a13*n1**4 + 4*u2*u3**2*v1*a13**2*b12*n1*n3 - 4*
u2*u3**2*v3*a13**2*b12*n1**2 + 4*u2*u3*v1*a13*b12*n1**2*n3 - 2*u2*u3*v2*v3*a13*
b12**2*n1**2 - 4*u2*u3*v3*a13*b12*n1**3 + 4*u2*v2*a13*m3*n1**2*n3 - u2*v3*b12*n1
**4 + 4*u3**4*a13**3*n1**2 + 4*u3**3*v2*a13**2*b12*n1**2 + 4*u3**3*a13**2*n1**3
+ u3**2*v1**2*a13*b12**2*n1**2 - 4*u3**2*v1*a13**2*m3*n1**2 + u3**2*v2**2*a13*
b12**2*n1**2 + 4*u3**2*v2*a13*b12*n1**3 - 4*u3**2*v3*a13**2*m3*n1*n3 + u3**2*( -
a13*kap*m3**2*n1**2 + 2*a13*n1**4 + a13*n1**2*n3**2) + u3*v1**2*b12**2*n1**3 -
4*u3*v1*a13*m3*n1**3 + u3*v2**2*b12**2*n1**3 + u3*v2*b12*n1**4 + u3*v3**2*b12**2
*n1**3 + u3*( - kap*m3**2*n1**3 + n1**3*n3**2) - v1*m3*n1**4$