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
1 2
---*m3 *r448
2
r24=--------------
2
b12
r27=0
r28=0
r210=0
1 3 2 4
- ---*a11 *m3 *r448 + b12 *kap*m3*r328
2
r211=-----------------------------------------
5
b12 *kap
r212=0
r213=0
r214=0
r215=0
1 3 2 4
- ---*a11 *m3 *r448 + b12 *kap*m3*r328
2
r217=-----------------------------------------
5
b12 *kap
r218=0
1 2
- ---*kap*m3 *r448
2
r219=---------------------
2
b12
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
r311=0
r312=r310
r313=0
2*m3*r460
r314=-----------
b12
b12*r310 + m3*r448
r315=--------------------
b12
b12*r343 + m3*r487
r316=--------------------
b12
r317=0
r318=0
r320=0
r321=0
r322=0
- 2*m3*r460
r323=--------------
b12
r324=0
r325=0
r326=0
2
- 2*a11*m3*r460 + b12 *r343
r327=------------------------------
2
b12
r329=0
2*a11*m3*r460
r330=---------------
2
b12
r331=0
r332=0
1 4 3 2 2 4
r333=( - ---*a11 *m3*r448 - ---*a11 *b12 *kap*m3*r448 - a11*b12 *kap*r328
2 2
5 2 5
+ b12 *kap *r310)/(b12 *kap)
r334=0
r335=0
r336=0
r337=0
- m3*r448
r338=------------
b12
r339=0
r340=0
r341=0
2
- a11*m3*r448 - b12 *r328
r342=----------------------------
2
b12
r344=0
a11*m3*r448
r345=-------------
2
b12
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
1 4 1 2 2 4
r353=(---*a11 *m3*r448 + ---*a11 *b12 *kap*m3*r448 - a11*b12 *kap*r328
2 8
5 2 5 4 2 5
+ b12 *kap *r310 + ---*b12 *kap *m3*r448)/(b12 *kap)
8
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 - r460
2
r428=0
1
r429= - ---*r448
2
1
r430= - ---*r494
2
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=r494 + 2*r460
r447=0
r449=0
r450=0
r451=0
1 3 3
- ---*a11 *r448 + b12 *kap*r431
2
r452=----------------------------------
3
b12 *kap
3 3
2*a11 *r460 - b12 *kap*r487
r453=-----------------------------
3
b12 *kap
r454=0
r455=0
r456=0
1
r457=---*r494 + r460
2
r458=0
1
r459=---*r448
2
3
a11 *r448
r461=-----------
3
b12 *kap
r462=0
r463=0
6 5 2 6 3 1 6 3
a11 *r460 + a11*b12 *kap *r487 - b12 *kap *r460 - ---*b12 *kap *r494
2
r464=----------------------------------------------------------------------
6 2
b12 *kap
r465=0
1 3
---*a11 *r448
2
r466=---------------
3
b12 *kap
3
2*a11 *r460
r467=-------------
3
b12 *kap
r468=0
4
- a11 *r460
r469=--------------
4
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=r494
r484=0
r485=0
r486=0
1 3 3
---*a11 *r448 + b12 *kap*r431
2
r488=-------------------------------
3
b12 *kap
r489=0
r490=0
r491=0
r492=0
r493=0
r495=r448
3
- 2*a11 *r460
r496=----------------
3
b12 *kap
r497=0
r498=0
6 1 4 2
a11 *r448 + ---*a11 *b12 *kap*r448
2
r499=------------------------------------
6 2
b12 *kap
r4100=0
3
- 2*a11 *r460
r4101=----------------
3
b12 *kap
3 3
---*a11 *r448
2
r4102=---------------
3
b12 *kap
r4103=0
1 4
- ---*a11 *r448
2
r4104=------------------
4
b12 *kap
r4105=0
r4106=0
r4108=0
1
r4109= - ---*r448
2
1
r4110=---*r494
2
r4111=0
r4112=0
r4113=0
r4114
6 4 2 5 2 1 6 3
- a11 *r460 - 2*a11 *b12 *kap*r460 + a11*b12 *kap *r487 - ---*b12 *kap *r494
2
=-------------------------------------------------------------------------------
6 2
b12 *kap
r4115=0
1 3
- ---*a11 *r448
2
r4116=------------------
3
b12 *kap
r4117=0
r4118=0
4
- a11 *r460
r4119=--------------
4
b12 *kap
r4120=0
r4121=0
1 3
---*a11 *r448
2
r4122=---------------
3
b12 *kap
r4123=0
1
r4124= - ---*kap*r448
2
r4125=0
m2=0
m1=0
n3=0
n2=0
n1=0
1 3
- ---*a11
2
a33=-------------
2
b12 *kap
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:
r434, r319, r29, r13, r310, r431, r460, r328, r343, r487,
r494, r448, 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 2
HAM=(2*u1 *a11*b12 *kap + 2*u1*v2*b12 *kap + 2*u2 *a11*b12 *kap
3 2 3 2 2
- 2*u2*v1*b12 *kap - u3 *a11 + 2*v3*b12 *kap*m3)/(2*b12 *kap)
has apart from the Hamiltonian and Casimirs the following 12 first integrals:
3 6 3 3 3 3 2 3 3
FI= - 4*u1 *u2*b12 *kap + 4*u1 *v1*a11 *b12 *kap - 4*u1 *u2*v2*a11 *b12 *kap
2 4 2 3 2 5 3
+ u1 *u3*(4*a11 *b12*kap*m3 + a11 *b12 *kap *m3 + 5*b12 *kap *m3)
2 6 2 3 4 2
- 4*u1 *v1*v2*b12 *kap - 4*u1*u2 *a11 *b12 *kap
2 3 3 2 6 4 2
+ 12*u1*u2 *v1*a11 *b12 *kap + u1*u2*u3 *(8*a11 + 4*a11 *b12 *kap)
2 6 2 4 2
+ 8*u1*u2*v1 *b12 *kap + 8*u1*u2*v3*a11*b12 *kap *m3
4 3 2 2 3 3
- 4*u1*u2*b12 *kap *m3 + 4*u1*u3 *v1*a11 *b12 *kap
4 2 5 2
- 8*u1*u3*v2*a11*b12 *kap *m3 - 8*u1*v1*v3*b12 *kap *m3
3 2 3 3 3
- 4*u1*v1*a11 *b12*kap*m3 + 4*u2 *v2*a11 *b12 *kap
2 3 3
+ 8*u2 *u3*v3*a11 *b12 *kap
2 4 2 3 2
+ u2 *u3*( - 4*a11 *b12*kap*m3 - 12*a11 *b12 *kap *m3)
2 6 2 2 3 3
+ 4*u2 *v1*v2*b12 *kap - 4*u2*u3 *v2*a11 *b12 *kap
6 2 3 2 2 6 2
+ 8*u2*u3*v1*v3*b12 *kap - 4*u2*v2*a11 *b12*kap*m3 - 4*u3 *v1*v2*b12 *kap
2 5 2 4 2 2
+ 8*u3*v1 *b12 *kap *m3 + 4*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 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
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
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 2 4 2 2 2 6 4 2
FI= - u1 *u2 *a11 *b12 *kap + u1 *u3 *( - a11 - 2*a11 *b12 *kap)
2 4 3 2 2 3 3
+ u1 *b12 *kap *m3 - 2*u1*u2 *v2*a11 *b12 *kap
3 3 4 4 2 3 3 3
- 2*u1*u2*u3*v3*a11 *b12 *kap - u2 *a11 *b12 *kap + 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 4 2 2 3 3
+ 2*u2 *v3*a11*b12 *kap *m3 + 2*u2*u3 *v1*a11 *b12 *kap
6 2 4 2
+ 2*u2*u3*v2*v3*b12 *kap - 2*u2*u3*v2*a11*b12 *kap *m3
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 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 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
FI=u3
which the program can not factorize further.
{HAM,FI} = 0
2
FI=u3
= a product of the elements of: {u3,u3}
{HAM,FI} = 0
3
FI=u3
= a product of the elements of: {u3,u3,u3}
{HAM,FI} = 0
4
FI=u3
= a product of the elements of: {u3,u3,u3,u3}
{HAM,FI} = 0
And again in machine readable form:
HAM=(2*u1**2*a11*b12**2*kap + 2*u1*v2*b12**3*kap + 2*u2**2*a11*b12**2*kap - 2*u2
*v1*b12**3*kap - u3**2*a11**3 + 2*v3*b12**2*kap*m3)/(2*b12**2*kap)$
FI= - 4*u1**3*u2*b12**6*kap**3 + 4*u1**3*v1*a11**3*b12**3*kap - 4*u1**2*u2*v2*
a11**3*b12**3*kap + u1**2*u3*(4*a11**4*b12*kap*m3 + a11**2*b12**3*kap**2*m3 + 5*
b12**5*kap**3*m3) - 4*u1**2*v1*v2*b12**6*kap**2 - 4*u1*u2**3*a11**4*b12**2*kap +
12*u1*u2**2*v1*a11**3*b12**3*kap + u1*u2*u3**2*(8*a11**6 + 4*a11**4*b12**2*kap)
+ 8*u1*u2*v1**2*b12**6*kap**2 + 8*u1*u2*v3*a11*b12**4*kap**2*m3 - 4*u1*u2*b12**
4*kap**3*m3**2 + 4*u1*u3**2*v1*a11**3*b12**3*kap - 8*u1*u3*v2*a11*b12**4*kap**2*
m3 - 8*u1*v1*v3*b12**5*kap**2*m3 - 4*u1*v1*a11**3*b12*kap*m3**2 + 4*u2**3*v2*a11
**3*b12**3*kap + 8*u2**2*u3*v3*a11**3*b12**3*kap + u2**2*u3*( - 4*a11**4*b12*kap
*m3 - 12*a11**2*b12**3*kap**2*m3) + 4*u2**2*v1*v2*b12**6*kap**2 - 4*u2*u3**2*v2*
a11**3*b12**3*kap + 8*u2*u3*v1*v3*b12**6*kap**2 - 4*u2*v2*a11**3*b12*kap*m3**2 -
4*u3**2*v1*v2*b12**6*kap**2 + 8*u3*v1**2*b12**5*kap**2*m3 + 4*v1*v2*b12**4*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*u3**2*a11 + u1*u3**2*v2*b12 + u2**2*u3**2*a11 - u2*u3**2*v1*b12 + u3**2
*v3*m3$
FI=u1*u3*v1 + u2*u3*v2 + u3**2*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**2*u2**2*a11**4*b12**2*kap + u1**2*u3**2*( - a11**6 - 2*a11**4*b12**2*
kap) + 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 - u2**4*a11**4*b12**2*kap + 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*b12**4*kap**2*m3 + 2*u2*u3**2*v1*a11**3*b12**3*
kap + 2*u2*u3*v2*v3*b12**6*kap**2 - 2*u2*u3*v2*a11*b12**4*kap**2*m3 - 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*u3**2*v1 + u2*u3**2*v2 + u3**3*v3$
FI=u1**2*u3*kap + u2**2*u3*kap + u3*v1**2 + u3*v2**2 + u3*v3**2$
FI=u3$
FI=u3**2$
FI=u3**3$
FI=u3**4$