Solution 3 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=a11 + b12 *kap
Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
r14=0
r15=0
2
m3 *r460
r20=----------
2
b12
r21=0
2
m3 *r460
r22=----------
2
b12
r23=0
r24=0
r27=0
r28=0
r210=0
- m3*r342
r211=------------
b12
r212=0
r213=0
r214=0
r215=0
- m3*r342
r217=------------
b12
r218=0
r219=0
2
kap*m3 *r460
r220=--------------
2
b12
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=r312
r311=0
r313=0
2*m3*r460
r314=-----------
b12
r315=r312
2
2*a11*m3*r460 + b12 *r327 + b12*m3*r487
r316=-----------------------------------------
2
b12
r317=0
r318=0
r320=0
r321=0
r322=0
- 2*m3*r460
r323=--------------
b12
r324=0
r325=0
r326=0
r328= - r342
r329=0
2*a11*m3*r460
r330=---------------
2
b12
r331=0
r332=0
a11*r342 + b12*kap*r312
r333=-------------------------
b12
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
2
2*a11*m3*r460 + b12 *r327
r343=---------------------------
2
b12
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
a11*r342 + b12*kap*r312
r353=-------------------------
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= - r460 - r4110
r428=0
r429=0
r430= - r4110
r432=0
r433=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r446=2*r460 + 2*r4110
r447=0
r448=0
r449=0
r450=0
r451=0
r452=r431
- 2*a11*r460 - b12*r487
r453=--------------------------
b12
r454=0
r455=0
r456=0
r457=r460 + r4110
r458=0
r459=0
r461=0
r462=0
r463=0
2 2 2
a11 *r460 + a11*b12*r487 - b12 *kap*r4110 - b12 *kap*r460
r464=-----------------------------------------------------------
2
b12
r465=0
r466=0
- 2*a11*r460
r467=---------------
b12
r468=0
2
a11 *r460
r469=-----------
2
b12
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
r486=0
r488=r431
r489=0
r490=0
r491=0
r492=0
r493=0
r494=2*r4110
r495=0
2*a11*r460
r496=------------
b12
r497=0
r498=0
r499=0
r4100=0
2*a11*r460
r4101=------------
b12
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
2 2
a11 *r460 + a11*b12*r487 - b12 *kap*r4110
r4114=-------------------------------------------
2
b12
r4115=0
r4116=0
r4117=0
r4118=0
2
a11 *r460
r4119=-----------
2
b12
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
n3=0
n2=0
n1=0
1
a33=---*a11
2
a23=0
a22=a11
a13=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:
r13, r29, r312, r319, r327, r342, r431, r434, r460, r487,
r4110, a11, 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.
{a22,a11,b12,m3}
Relevance for the application:
Modulo the following equation:
2 2
0=a11 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2 1 2
HAM=u1 *a11 + u1*v2*b12 + u2 *a11 - u2*v1*b12 + ---*u3 *a11 + v3*m3
2
has apart from the Hamiltonian and Casimirs the following 11 first integrals:
2 2 2 2 2 2
FI= - u1 *u3 *kap + u1 *v1 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2 *u3 *kap
2 2 2 2 2 2
+ u2 *v2 + 2*u2*u3*v2*v3 - u3 *v1 - u3 *v2
which the program can not factorize further.
{HAM,FI} = 0
2 2 2 2 2 2 2
FI=u1 *u3 *a11 + u1*u3 *v2*b12 + u2 *u3 *a11 - u2*u3 *v1*b12 + u3 *v3*m3
= a product of the elements of: {u3,
u3,
2 2
u1 *a11 + u1*v2*b12 + u2 *a11 - u2*v1*b12 + v3*m3}
{HAM,FI} = 0
2 2 2 2 2 2 2 2 2
FI=u1 *u2 *a11 + u1 *u3 *a11 + u1 *kap*m3 + 2*u1*u2 *v2*a11*b12
4 2 3
+ 2*u1*u2*u3*v3*a11*b12 + 2*u1*u3*v1*a11*m3 + u2 *a11 - 2*u2 *v1*a11*b12
2 2 2 2 2 2 2 2 2 2 2
+ u2 *u3 *(a11 - b12 *kap) + u2 *v1 *b12 + u2 *v2 *b12 + 2*u2 *v3*a11*m3
2 2
- 2*u2*u3 *v1*a11*b12 + 2*u2*u3*v2*v3*b12 - 2*u2*v1*v3*b12*m3
2 2 2 2 2 2 2 2
- u3 *v2 *b12 + 2*u3 *v3*a11*m3 + 2*u3*v1*v2*b12*m3 + v2 *m3 + v3 *m3
which the program can not factorize further.
{HAM,FI} = 0
4
FI=u3
= a product of the elements of: {u3,u3,u3,u3}
{HAM,FI} = 0
2 2 3
FI=u1*u3 *v1 + u2*u3 *v2 + u3 *v3
= a product of the elements of: {u3,
u3,
u1*v1 + u2*v2 + u3*v3}
{HAM,FI} = 0
2 2
FI=u1 *u3*a11 + u1*u3*v2*b12 - u1*v1*m3 + u2 *u3*a11 - u2*u3*v1*b12 - u2*v2*m3
which the program can not factorize further.
{HAM,FI} = 0
2
FI=u1*u3*v1 + u2*u3*v2 + u3 *v3
= a product of the elements of: {u3,u1*v1 + u2*v2 + u3*v3}
{HAM,FI} = 0
3
FI=u3
= a product of the elements of: {u3,u3,u3}
{HAM,FI} = 0
2 2 2 2 2
FI=u1 *u3*kap + u2 *u3*kap + u3*v1 + u3*v2 + u3*v3
2 2 2 2 2
= a product of the elements of: {u3,u1 *kap + u2 *kap + v1 + v2 + v3 }
{HAM,FI} = 0
2
FI=u3
= a product of the elements of: {u3,u3}
{HAM,FI} = 0
FI=u3
which the program can not factorize further.
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u1*v2*b12 + u2**2*a11 - u2*v1*b12 + 1/2*u3**2*a11 + v3*m3$
FI= - u1**2*u3**2*kap + u1**2*v1**2 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2**2*u3**
2*kap + u2**2*v2**2 + 2*u2*u3*v2*v3 - u3**2*v1**2 - u3**2*v2**2$
FI=u1**2*u3**2*a11 + u1*u3**2*v2*b12 + u2**2*u3**2*a11 - u2*u3**2*v1*b12 + u3**2
*v3*m3$
FI=u1**2*u2**2*a11**2 + u1**2*u3**2*a11**2 + u1**2*kap*m3**2 + 2*u1*u2**2*v2*a11
*b12 + 2*u1*u2*u3*v3*a11*b12 + 2*u1*u3*v1*a11*m3 + u2**4*a11**2 - 2*u2**3*v1*a11
*b12 + u2**2*u3**2*(a11**2 - b12**2*kap) + u2**2*v1**2*b12**2 + u2**2*v2**2*b12
**2 + 2*u2**2*v3*a11*m3 - 2*u2*u3**2*v1*a11*b12 + 2*u2*u3*v2*v3*b12**2 - 2*u2*v1
*v3*b12*m3 - u3**2*v2**2*b12**2 + 2*u3**2*v3*a11*m3 + 2*u3*v1*v2*b12*m3 + v2**2*
m3**2 + v3**2*m3**2$
FI=u3**4$
FI=u1*u3**2*v1 + u2*u3**2*v2 + u3**3*v3$
FI=u1**2*u3*a11 + u1*u3*v2*b12 - u1*v1*m3 + u2**2*u3*a11 - u2*u3*v1*b12 - u2*v2*
m3$
FI=u1*u3*v1 + u2*u3*v2 + u3**2*v3$
FI=u3**3$
FI=u1**2*u3*kap + u2**2*u3*kap + u3*v1**2 + u3*v2**2 + u3*v3**2$
FI=u3**2$
FI=u3$