Solution 4 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
2
- m3 *r4109
r24=--------------
2
b12
r27=0
r28=0
r210=0
5 2 3 2 2 6 2
a11 *m3 *r4109 + 2*a11 *b12 *kap*m3 *r4109 + b12 *kap *m3*r328
r211=----------------------------------------------------------------
7 2
b12 *kap
r212=0
r213=0
r214=0
r215=0
3 2 4
a11 *m3 *r4109 + b12 *kap*m3*r328
r217=-----------------------------------
5
b12 *kap
r218=0
6 2 4 2 2 1 3 4
r219=(a11 *m3 *r4109 + 2*a11 *b12 *kap*m3 *r4109 + ---*a11 *b12 *kap*m3*r328
2
1 6 2 8 2
+ ---*a11*b12 *kap *m3*r328)/(b12 *kap )
2
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
b12*r312 - 2*m3*r4109
r315=-----------------------
b12
b12*r327 - m3*r453
r316=--------------------
b12
r317=0
r318=0
r320=0
r321=0
r322=0
- 2*m3*r460
r323=--------------
b12
r324=0
r325=0
r326=0
r329=0
3
- 2*a11 *m3*r460
r330=-------------------
4
b12 *kap
r331=0
r332=0
6 3 4 7 3 6 3
a11 *m3*r4109 + a11 *b12 *kap*r328 + b12 *kap *r312 - b12 *kap *m3*r4109
r333=--------------------------------------------------------------------------
7 2
b12 *kap
r334=0
r335=0
r336=0
r337=0
2*m3*r4109
r338=------------
b12
r339=0
r340=0
r341=0
5 3 2 6 2
- a11 *m3*r4109 - 3*a11 *b12 *kap*m3*r4109 - b12 *kap *r328
r342=--------------------------------------------------------------
6 2
b12 *kap
3 4
- 2*a11 *m3*r460 + b12 *kap*r327
r343=-----------------------------------
4
b12 *kap
r344=0
3
2*a11 *m3*r4109
r345=-----------------
4
b12 *kap
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
1 3 2 3 2
r353=( - ---*a11 *r328 - a11 *kap*m3*r4109 - ---*a11*b12 *kap*r328
2 2
3 2 2 2 3
+ b12 *kap *r312 - 3*b12 *kap *m3*r4109)/(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
r427= - r460 - r4110
r428=0
r429=r4109
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= - 2*r4109
r449=0
r450=0
r451=0
5 5 2
- a11 *r4109 + b12 *kap *r431
r452=--------------------------------
5 2
b12 *kap
r454=0
r455=0
r456=0
r457=r460 + r4110
r458=0
r459= - r4109
2*a11*r4109
r461=-------------
b12
r462=0
r463=0
6 3 3 6 3 6 3
- a11 *r460 + a11 *b12 *kap*r453 - b12 *kap *r4110 - b12 *kap *r460
r464=----------------------------------------------------------------------
6 2
b12 *kap
r465=0
a11*r4109
r466=-----------
b12
3
2*a11 *r460
r467=-------------
3
b12 *kap
r468=0
6
a11 *r460
r469=-----------
6 2
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=2*r4110
r484=0
r485=0
r486=0
3 3
2*a11 *r460 - b12 *kap*r453
r487=-----------------------------
3
b12 *kap
3 3
- a11 *r4109 + b12 *kap*r431
r488=-------------------------------
3
b12 *kap
r489=0
r490=0
r491=0
r492=0
r493=0
r494=2*r4110
r495= - 2*r4109
3
- 2*a11 *r460
r496=----------------
3
b12 *kap
r497=0
r498=0
2
- a11 *r4109
r499=---------------
2
b12
r4100=0
3
- 2*a11 *r460
r4101=----------------
3
b12 *kap
3*a11*r4109
r4102=-------------
b12
r4103=0
r4104=kap*r4109
r4105=0
r4106=0
r4108=0
r4111=0
r4112=0
r4113=0
6 5 2 6 3
- a11 *r460 - a11*b12 *kap *r453 - b12 *kap *r4110
r4114=-----------------------------------------------------
6 2
b12 *kap
r4115=0
3
a11 *r4109
r4116=------------
3
b12 *kap
r4117=0
r4118=0
6
a11 *r460
r4119=-----------
6 2
b12 *kap
r4120=0
r4121=0
3
- a11 *r4109
r4122=---------------
3
b12 *kap
r4123=0
4
a11 *r4109
r4124=------------
4
b12 *kap
r4125=0
m2=0
m1=0
n3=0
n2=0
n1=0
1 3
- ---*a11
2
a33=-------------
2
b12 *kap
a23=0
3
- a11
a22=----------
2
b12 *kap
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, r328, r431, r434, r453, r460,
r4110, r4109, 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.
{a11,m3,b12}
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 3 2 3 3
HAM=(u1 *a11*b12 *kap + u1*v2*b12 *kap - u2 *a11 - u2*v1*b12 *kap
1 2 3 2 2
- ---*u3 *a11 + v3*b12 *kap*m3)/(b12 *kap)
2
has apart from the Hamiltonian and Casimirs the following 12 first integrals:
3 4 4 3 3 5 2 3 5
FI=u1 *u2*a11 *b12 *kap - u1 *v1*a11 *b12 *kap + u1 *u2*v2*a11 *b12 *kap
2 2 5 2 7 3 2 8 2
+ u1 *u3*( - a11 *b12 *kap *m3 - 3*b12 *kap *m3) + u1 *v1*v2*b12 *kap
3 8 3 2 7 2 2 2 6 2
+ u1*u2 *b12 *kap + 3*u1*u2 *v1*a11*b12 *kap - u1*u2*u3 *a11 *b12 *kap
2 8 2 3 4
- 2*u1*u2*v1 *b12 *kap + 2*u1*u2*v3*a11 *b12 *kap*m3
6 2 4 2 2 2 3 5
+ u1*u2*(a11 *m3 + 2*a11 *b12 *kap*m3 ) - u1*u3 *v1*a11 *b12 *kap
5 2 3 4 7 2
+ u1*u3*v2*( - a11 *b12 *m3 - 3*a11 *b12 *kap*m3) + 2*u1*v1*v3*b12 *kap *m3
3 3 2 3 7 2 2 7 2
+ u1*v1*a11 *b12 *kap*m3 + u2 *v2*a11*b12 *kap + 2*u2 *u3*v3*a11*b12 *kap
2 6 7 3 2 8 2
+ u2 *u3*(a11 *b12*m3 - b12 *kap *m3) - u2 *v1*v2*b12 *kap
2 5 3 8 2
- u2*u3 *v2*a11 *b12 - 2*u2*u3*v1*v3*b12 *kap
5 2 3 3 2 2 8 2
+ u2*v2*(a11 *b12*m3 + 2*a11 *b12 *kap*m3 ) + u3 *v1*v2*b12 *kap
2 7 2 6 2 2
- 2*u3*v1 *b12 *kap *m3 - v1*v2*b12 *kap *m3
which the program can not factorize further.
{HAM,FI} = 0
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 6 2 2 6 2 4 3 2 2 3 3
FI=u1 *u2 *a11 - u1 *u3 *a11 + u1 *b12 *kap *m3 - 2*u1*u2 *v2*a11 *b12 *kap
3 3 2 3 3
- 2*u1*u2*u3*v3*a11 *b12 *kap + 2*u1*u3 *v2*a11 *b12 *kap
3 2 4 6 3 3 3
- 2*u1*u3*v1*a11 *b12 *kap*m3 + u2 *a11 + 2*u2 *v1*a11 *b12 *kap
2 2 6 6 3 2 2 6 2 2 2 6 2
+ u2 *u3 *( - a11 - b12 *kap ) + u2 *v1 *b12 *kap + u2 *v2 *b12 *kap
2 3 2 6 2
- 2*u2 *v3*a11 *b12 *kap*m3 + 2*u2*u3*v2*v3*b12 *kap
5 2 2 2 6 2 5 2
- 2*u2*v1*v3*b12 *kap *m3 - u3 *v2 *b12 *kap + 2*u3*v1*v2*b12 *kap *m3
2 4 2 2 2 4 2 2
+ v2 *b12 *kap *m3 + v3 *b12 *kap *m3
which the program can not factorize further.
{HAM,FI} = 0
2 2 2 2 3 2 2 3
FI= - u1 *u3 *a11*b12 *kap - u1*u3 *v2*b12 *kap + u2 *u3 *a11
2 3 2 2
+ u2*u3 *v1*b12 *kap - u3 *v3*b12 *kap*m3
= a product of the elements of: {u3,
u3,
2 2 3 2 3 3
- u1 *a11*b12 *kap - u1*v2*b12 *kap + u2 *a11 + u2*v1*b12 *kap
2
- v3*b12 *kap*m3}
{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 1 3 3 3
FI=u1 *u3*( - ---*a11 *b12 - ---*a11*b12 *kap)
2 2
1 3 1 2 4
+ u1*u2*(---*a11 *m3 + ---*a11*b12 *kap*m3) - u1*u3*v2*b12 *kap
2 2
3 2 3 4
+ u1*v1*b12 *kap*m3 + u2 *u3*a11 *b12 + u2*u3*v1*b12 *kap
3
+ u2*v2*b12 *kap*m3
1
= a product of the elements of: { - ---,
2
2 3 3 3 2
u1 *u3*(a11 *b12 + 3*a11*b12 *kap) + u1*u2*( - a11 *m3 - a11*b12 *kap*m3)
4 3 2 3
+ 2*u1*u3*v2*b12 *kap - 2*u1*v1*b12 *kap*m3 - 2*u2 *u3*a11 *b12
4 3
- 2*u2*u3*v1*b12 *kap - 2*u2*v2*b12 *kap*m3}
{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*b12**2*kap + u1*v2*b12**3*kap - u2**2*a11**3 - u2*v1*b12**3*kap -
1/2*u3**2*a11**3 + v3*b12**2*kap*m3)/(b12**2*kap)$
FI=u1**3*u2*a11**4*b12**4*kap - u1**3*v1*a11**3*b12**5*kap + u1**2*u2*v2*a11**3*
b12**5*kap + u1**2*u3*( - a11**2*b12**5*kap**2*m3 - 3*b12**7*kap**3*m3) + u1**2*
v1*v2*b12**8*kap**2 + u1*u2**3*b12**8*kap**3 + 3*u1*u2**2*v1*a11*b12**7*kap**2 -
u1*u2*u3**2*a11**2*b12**6*kap**2 - 2*u1*u2*v1**2*b12**8*kap**2 + 2*u1*u2*v3*a11
**3*b12**4*kap*m3 + u1*u2*(a11**6*m3**2 + 2*a11**4*b12**2*kap*m3**2) - u1*u3**2*
v1*a11**3*b12**5*kap + u1*u3*v2*( - a11**5*b12**2*m3 - 3*a11**3*b12**4*kap*m3) +
2*u1*v1*v3*b12**7*kap**2*m3 + u1*v1*a11**3*b12**3*kap*m3**2 + u2**3*v2*a11*b12
**7*kap**2 + 2*u2**2*u3*v3*a11*b12**7*kap**2 + u2**2*u3*(a11**6*b12*m3 - b12**7*
kap**3*m3) - u2**2*v1*v2*b12**8*kap**2 - u2*u3**2*v2*a11**5*b12**3 - 2*u2*u3*v1*
v3*b12**8*kap**2 + u2*v2*(a11**5*b12*m3**2 + 2*a11**3*b12**3*kap*m3**2) + u3**2*
v1*v2*b12**8*kap**2 - 2*u3*v1**2*b12**7*kap**2*m3 - v1*v2*b12**6*kap**2*m3**2$
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*u2**2*a11**6 - u1**2*u3**2*a11**6 + u1**2*b12**4*kap**3*m3**2 - 2*u1*u2
**2*v2*a11**3*b12**3*kap - 2*u1*u2*u3*v3*a11**3*b12**3*kap + 2*u1*u3**2*v2*a11**
3*b12**3*kap - 2*u1*u3*v1*a11**3*b12**2*kap*m3 + u2**4*a11**6 + 2*u2**3*v1*a11**
3*b12**3*kap + u2**2*u3**2*( - a11**6 - b12**6*kap**3) + u2**2*v1**2*b12**6*kap
**2 + u2**2*v2**2*b12**6*kap**2 - 2*u2**2*v3*a11**3*b12**2*kap*m3 + 2*u2*u3*v2*
v3*b12**6*kap**2 - 2*u2*v1*v3*b12**5*kap**2*m3 - u3**2*v2**2*b12**6*kap**2 + 2*
u3*v1*v2*b12**5*kap**2*m3 + v2**2*b12**4*kap**2*m3**2 + v3**2*b12**4*kap**2*m3**
2$
FI= - u1**2*u3**2*a11*b12**2*kap - u1*u3**2*v2*b12**3*kap + u2**2*u3**2*a11**3 +
u2*u3**2*v1*b12**3*kap - u3**2*v3*b12**2*kap*m3$
FI=u3**4$
FI=u1*u3**2*v1 + u2*u3**2*v2 + u3**3*v3$
FI=u1**2*u3*( - 1/2*a11**3*b12 - 3/2*a11*b12**3*kap) + u1*u2*(1/2*a11**3*m3 + 1/
2*a11*b12**2*kap*m3) - u1*u3*v2*b12**4*kap + u1*v1*b12**3*kap*m3 + u2**2*u3*a11
**3*b12 + u2*u3*v1*b12**4*kap + u2*v2*b12**3*kap*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$