Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=4*a23 + b12 *kap
Expressions
The solution is given through the following expressions:
r10=0
2
- 2*a23*m3*n2 *r494
r11=----------------------
4
b12 *kap
r12=0
2
- 2*a23*m3 *n2*r494
r13=----------------------
4
b12
r14=0
r15=0
r20=0
r21=0
r22=0
r23=0
r24=0
2 2
- 4*a23 *m3*n2*r494 + b12 *kap*m3*n2*r494
r27=--------------------------------------------
4
b12 *kap
2
- 2*a23*n2 *r494
r28=-------------------
3
b12 *kap
2 2 1 2 2 2 1 2 2
2*a23 *n2 *r494 + ---*b12 *kap *m3 *r494 - ---*b12 *kap*n2 *r494
2 2
r29=------------------------------------------------------------------
4
b12 *kap
r210=0
r211=0
r212=0
r213=0
2 2
2*a23 *n2 *r494
r214=-----------------
4
b12 *kap
2
2*a23*n2 *r494
r215=----------------
3
b12 *kap
r217=0
r218=0
r219=0
2 2 1 2 2
2*a23 *n2 *r494 - ---*b12 *kap*n2 *r494
2
r220=-----------------------------------------
4
b12 *kap
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
2*a23*n2*r494
r310=---------------
2
b12 *kap
r311=0
2*a23*n2*r494
r312=---------------
2
b12 *kap
r313=0
r314=0
2*a23*n2*r494
r315=---------------
2
b12 *kap
r316=0
2*a23*m3*r494
r317=---------------
2
b12
2 2
- 4*a23 *n2*r494 + b12 *kap*n2*r494
r318=--------------------------------------
3
b12 *kap
3 1 2
10*a23 *n2*r494 + ---*a23*b12 *kap*n2*r494
2
r319=--------------------------------------------
4
b12 *kap
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
r326=0
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
3 2
4*a23 *n2*r494 + a23*b12 *kap*n2*r494
r333=---------------------------------------
4
b12 *kap
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
2 2
4*a23 *n2*r494 - b12 *kap*n2*r494
r341=-----------------------------------
3
b12 *kap
r342=0
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
3 2
4*a23 *n2*r494 - a23*b12 *kap*n2*r494
r353=---------------------------------------
4
b12 *kap
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
r427= - ---*r494
2
r428=0
r429=0
1
r430= - ---*r494
2
r431=0
r432=0
2*a23*r494
r433=------------
b12
2
- 2*a23 *r494
r434=----------------
2
b12
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r446=r494
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
r453=0
r454=0
r455=0
r456=0
1
r457=---*r494
2
r458=0
r459=0
r460=0
r461=0
r462=0
r463=0
2 1 2
- 2*a23 *r494 - ---*b12 *kap*r494
2
r464=------------------------------------
2
b12
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=r494
r484=0
r485=0
- 2*a23*r494
r486=---------------
b12
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=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
r4108=0
r4109=0
1
r4110=---*r494
2
r4111=0
r4112=0
r4113=0
2 1 2
- 4*a23 *r494 - ---*b12 *kap*r494
2
r4114=------------------------------------
2
b12
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
n1=0
a33=0
a22=0
a13=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:
r494, n2, m3, a23, 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.
{b12,a23,r494,{n2,m3}}
Relevance for the application:
Modulo the following equation:
2 2
0=4*a23 + b12 *kap
the system of equations related to the Hamiltonian HAM:
HAM=u1*v2*b12 + 2*u2*u3*a23 - u2*v1*b12 + u2*n2 + v3*m3
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 4 2
FI=u1 *u3 *( - 8*a23 *b12 *kap - b12 *kap )
2 3 2 2 2 4
+ u1 *u3*(8*a23 *n2 - 2*a23*b12 *kap*n2) + u1 *v1 *b12 *kap
2 2 2 2 2 4
+ u1 *(4*a23 *n2 - b12 *kap*n2 ) + 2*u1*u2*v1*v2*b12 *kap
2 3 4
- 4*u1*u3 *v3*a23*b12 *kap + 2*u1*u3*v1*v3*b12 *kap
2 3 2
+ u1*u3*v3*(8*a23 *b12*n2 - 2*b12 *kap*n2) + 4*u1*v3*a23*b12*n2
2 2 2 2 4 2
+ u2 *u3 *( - 4*a23 *b12 *kap - b12 *kap )
2 3 2 2 2 4 2 2 2
+ u2 *u3*(8*a23 *n2 + 2*a23*b12 *kap*n2) + u2 *v2 *b12 *kap + 4*u2 *a23 *n2
4 4 2 2 3 3
+ 2*u2*u3*v2*v3*b12 *kap - 4*u3 *a23 *b12 *kap + 4*u3 *v1*a23*b12 *kap
3 3 2 2 2 4
+ u3 *(20*a23 *n2 + a23*b12 *kap*n2) - u3 *v1 *b12 *kap
2 2 3 2 2 4
+ u3 *v1*( - 8*a23 *b12*n2 + 2*b12 *kap*n2) - u3 *v2 *b12 *kap
2 2 2 2 2 2 2 2 2 2
+ 4*u3 *v2*a23*b12 *kap*m3 + u3 *(4*a23 *n2 + b12 *kap *m3 - b12 *kap*n2 )
2 2 2 2 2
+ 4*u3*v1 *a23*b12 *n2 - 4*u3*v1*a23*b12*n2 + 4*u3*v2 *a23*b12 *n2
2 2 2 2
+ u3*v2*( - 8*a23 *m3*n2 + 2*b12 *kap*m3*n2) + 4*u3*v3 *a23*b12 *n2
2 2
- 4*u3*a23*kap*m3 *n2 - 4*v2*a23*m3*n2
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1*v2*b12 + 2*u2*u3*a23 - u2*v1*b12 + u2*n2 + v3*m3$
FI=u1**2*u3**2*( - 8*a23**2*b12**2*kap - b12**4*kap**2) + u1**2*u3*(8*a23**3*n2
- 2*a23*b12**2*kap*n2) + u1**2*v1**2*b12**4*kap + u1**2*(4*a23**2*n2**2 - b12**2
*kap*n2**2) + 2*u1*u2*v1*v2*b12**4*kap - 4*u1*u3**2*v3*a23*b12**3*kap + 2*u1*u3*
v1*v3*b12**4*kap + u1*u3*v3*(8*a23**2*b12*n2 - 2*b12**3*kap*n2) + 4*u1*v3*a23*
b12*n2**2 + u2**2*u3**2*( - 4*a23**2*b12**2*kap - b12**4*kap**2) + u2**2*u3*(8*
a23**3*n2 + 2*a23*b12**2*kap*n2) + u2**2*v2**2*b12**4*kap + 4*u2**2*a23**2*n2**2
+ 2*u2*u3*v2*v3*b12**4*kap - 4*u3**4*a23**2*b12**2*kap + 4*u3**3*v1*a23*b12**3*
kap + u3**3*(20*a23**3*n2 + a23*b12**2*kap*n2) - u3**2*v1**2*b12**4*kap + u3**2*
v1*( - 8*a23**2*b12*n2 + 2*b12**3*kap*n2) - u3**2*v2**2*b12**4*kap + 4*u3**2*v2*
a23*b12**2*kap*m3 + u3**2*(4*a23**2*n2**2 + b12**2*kap**2*m3**2 - b12**2*kap*n2
**2) + 4*u3*v1**2*a23*b12**2*n2 - 4*u3*v1*a23*b12*n2**2 + 4*u3*v2**2*a23*b12**2*
n2 + u3*v2*( - 8*a23**2*m3*n2 + 2*b12**2*kap*m3*n2) + 4*u3*v3**2*a23*b12**2*n2 -
4*u3*a23*kap*m3**2*n2 - 4*v2*a23*m3*n2**2$