Solution 1 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
1 2 2 1 2 2
---*a11 *m3 *r328 + ---*b12 *kap*m3 *r328
4 4
r20=-------------------------------------------
3
b12 *kap*n3
r21=0
1 2 2 1 2 2
---*a11 *m3 *r328 + ---*b12 *kap*m3 *r328
4 4
r22=-------------------------------------------
3
b12 *kap*n3
r23=0
r24=0
r27=0
r28=0
r210=0
m3*r328
r211=---------
b12
r212=0
r213=0
1 2 1 2
---*a11 *n3*r328 + ---*b12 *kap*n3*r328
4 4
r214=-----------------------------------------
3
b12 *kap
r215=0
1 2 3 2
---*a11 *m3*r328 + ---*b12 *kap*m3*r328
2 2
r217=-----------------------------------------
3
b12 *kap
r218=0
1 3 1 2
- ---*a11 *m3*r328 - ---*a11*b12 *kap*m3*r328
2 2
r219=------------------------------------------------
4
b12 *kap
1 2 2 1 2 2 1 2 2 2
r220=(---*a11 *kap*m3 *r328 - ---*a11 *n3 *r328 + ---*b12 *kap *m3 *r328
4 4 4
1 2 2 3
- ---*b12 *kap*n3 *r328)/(b12 *kap*n3)
4
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
1 2 1 2
---*a11 *m3*r328 + ---*b12 *kap*m3*r328
2 2
r314=-----------------------------------------
2
b12 *kap*n3
r315=r312
b12*r327 - m3*r453
r316=--------------------
b12
r317=0
r318=0
r320=0
r321=0
r322=0
1 2 1 2
- ---*a11 *m3*r328 - ---*b12 *kap*m3*r328
2 2
r323=--------------------------------------------
2
b12 *kap*n3
r324=0
r325=0
r326=0
r329=0
1 3 1 2
---*a11 *m3*r328 + ---*a11*b12 *kap*m3*r328
2 2
r330=---------------------------------------------
3
b12 *kap*n3
r331=0
r332=0
- a11*r328 + b12*kap*r312
r333=----------------------------
b12
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
1 2 3 2
- ---*a11 *r328 - ---*b12 *kap*r328
2 2
r342=--------------------------------------
2
b12 *kap
1 3 1 2 3
---*a11 *m3*r328 + ---*a11*b12 *kap*m3*r328 + b12 *kap*n3*r327
2 2
r343=----------------------------------------------------------------
3
b12 *kap*n3
r344=0
1 2 1 2
---*a11 *r328 + ---*b12 *kap*r328
2 2
r345=-----------------------------------
2
b12 *kap
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
- a11*r328 + 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
1 2 1 2
- ---*a11 *r328 - ---*b12 *kap*r328 - b12*kap*n3*r4110
4 4
r427=---------------------------------------------------------
b12*kap*n3
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
1 2 1 2
---*a11 *r328 + ---*b12 *kap*r328 + 2*b12*kap*n3*r4110
2 2
r446=--------------------------------------------------------
b12*kap*n3
r447=0
r448=0
r449=0
r450=0
r451=0
r452=r431
r454=0
r455=0
r456=0
1 2 1 2
---*a11 *r328 + ---*b12 *kap*r328 + b12*kap*n3*r4110
4 4
r457=------------------------------------------------------
b12*kap*n3
r458=0
r459=0
1 2 1 2
---*a11 *r328 + ---*b12 *kap*r328
4 4
r460=-----------------------------------
b12*kap*n3
r461=0
r462=0
r463=0
1 4 1 2 2 2
r464=( - ---*a11 *r328 - ---*a11 *b12 *kap*r328 - a11*b12 *kap*n3*r453
4 2
1 4 2 3 2 3
- ---*b12 *kap *r328 - b12 *kap *n3*r4110)/(b12 *kap*n3)
4
r465=0
r466=0
1 3 1 2
- ---*a11 *r328 - ---*a11*b12 *kap*r328
2 2
r467=------------------------------------------
2
b12 *kap*n3
r468=0
1 4 1 2 2
---*a11 *r328 + ---*a11 *b12 *kap*r328
4 4
r469=----------------------------------------
3
b12 *kap*n3
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
1 3 1 2 2
- ---*a11 *r328 - ---*a11*b12 *kap*r328 - b12 *kap*n3*r453
2 2
r487=-------------------------------------------------------------
2
b12 *kap*n3
r488=r431
r489=0
r490=0
r491=0
r492=0
r493=0
r494=2*r4110
r495=0
1 3 1 2
---*a11 *r328 + ---*a11*b12 *kap*r328
2 2
r496=---------------------------------------
2
b12 *kap*n3
r497=0
r498=0
r499=0
r4100=0
1 3 1 2
---*a11 *r328 + ---*a11*b12 *kap*r328
2 2
r4101=---------------------------------------
2
b12 *kap*n3
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
1 4 1 2 2 2
r4114=( - ---*a11 *r328 - ---*a11 *b12 *kap*r328 - a11*b12 *kap*n3*r453
4 4
3 2 3
- b12 *kap *n3*r4110)/(b12 *kap*n3)
r4115=0
r4116=0
r4117=0
r4118=0
1 4 1 2 2
---*a11 *r328 + ---*a11 *b12 *kap*r328
4 4
r4119=----------------------------------------
3
b12 *kap*n3
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=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, r328, r431, r434, r453, r4110,
m3, a11, n3, 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,n3,a11,a22}
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 + u3*n3 + v3*m3
2
has apart from the Hamiltonian and Casimirs the following 10 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
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 1 4 1 2 3
FI=u1 *u2 *(---*a11 *b12 + ---*a11 *b12 *kap)
4 4
2 2 1 4 1 2 3 2 3
+ u1 *u3 *( - ---*a11 *b12 - ---*a11 *b12 *kap) - u1 *u3*a11*b12 *kap*n3 +
4 4
2 1 2 2 1 2 2 1 3 2 2
u1 *(---*a11 *b12*kap*m3 - ---*a11 *b12*n3 + ---*b12 *kap *m3
4 4 4
1 3 2 2 1 3 2 1 4
- ---*b12 *kap*n3 ) + u1*u2 *v2*(---*a11 *b12 + ---*a11*b12 *kap)
4 2 2
1 3 2 1 4
+ u1*u2*u3*v3*(---*a11 *b12 + ---*a11*b12 *kap)
2 2
1 2 2 1 4
+ u1*u2*v3*(---*a11 *b12 *n3 + ---*b12 *kap*n3)
2 2
1 3 1 2
+ u1*u2*( - ---*a11 *m3*n3 - ---*a11*b12 *kap*m3*n3)
2 2
2 1 3 2 1 4
+ u1*u3 *v2*( - ---*a11 *b12 - ---*a11*b12 *kap)
2 2
1 3 1 3
+ u1*u3*v1*(---*a11 *b12*m3 + ---*a11*b12 *kap*m3)
2 2
1 2 2 3 4
+ u1*u3*v2*( - ---*a11 *b12 *n3 - ---*b12 *kap*n3)
2 2
1 2 3 3
+ u1*v1*(---*a11 *b12*m3*n3 + ---*b12 *kap*m3*n3)
2 2
4 1 4 1 2 3
+ u2 *(---*a11 *b12 + ---*a11 *b12 *kap)
4 4
3 1 3 2 1 4
+ u2 *v1*( - ---*a11 *b12 - ---*a11*b12 *kap)
2 2
2 2 1 4 1 2 3 1 5 2
+ u2 *u3 *( - ---*a11 *b12 - ---*a11 *b12 *kap - ---*b12 *kap )
4 2 4
2 3 2 2 1 2 3 1 5
- u2 *u3*a11*b12 *kap*n3 + u2 *v1 *(---*a11 *b12 + ---*b12 *kap)
4 4
2 2 1 2 3 1 5
+ u2 *v2 *(---*a11 *b12 + ---*b12 *kap)
4 4
2 1 3 1 3
+ u2 *v3*(---*a11 *b12*m3 + ---*a11*b12 *kap*m3)
2 2
2 1 2 2 1 3 2 4
+ u2 *(---*a11 *b12*n3 + ---*b12 *kap*n3 ) + u2*u3*v1*b12 *kap*n3
4 4
1 2 3 1 5
+ u2*u3*v2*v3*(---*a11 *b12 + ---*b12 *kap)
2 2
1 2 2 1 4 3
+ u2*v1*v3*( - ---*a11 *b12 *m3 - ---*b12 *kap*m3) + u2*v2*b12 *kap*m3*n3
2 2
2 2 1 2 3 1 5
+ u3 *v2 *( - ---*a11 *b12 - ---*b12 *kap)
4 4
1 2 2 1 4
+ u3*v1*v2*(---*a11 *b12 *m3 + ---*b12 *kap*m3)
2 2
2 1 2 2 1 3 2
+ v2 *(---*a11 *b12*m3 + ---*b12 *kap*m3 )
4 4
2 1 2 2 1 3 2
+ v3 *(---*a11 *b12*m3 + ---*b12 *kap*m3 )
4 4
1
= a product of the elements of: {---,
4
2 2 4 2 3 2 2 4 2 3
u1 *u2 *(a11 *b12 + a11 *b12 *kap) + u1 *u3 *( - a11 *b12 - a11 *b12 *kap)
2 3
- 4*u1 *u3*a11*b12 *kap*n3
2 2 2 2 2 3 2 2 3 2
+ u1 *(a11 *b12*kap*m3 - a11 *b12*n3 + b12 *kap *m3 - b12 *kap*n3 )
2 3 2 4
+ u1*u2 *v2*(2*a11 *b12 + 2*a11*b12 *kap)
3 2 4
+ u1*u2*u3*v3*(2*a11 *b12 + 2*a11*b12 *kap)
2 2 4
+ u1*u2*v3*(2*a11 *b12 *n3 + 2*b12 *kap*n3)
3 2
+ u1*u2*( - 2*a11 *m3*n3 - 2*a11*b12 *kap*m3*n3)
2 3 2 4
+ u1*u3 *v2*( - 2*a11 *b12 - 2*a11*b12 *kap)
3 3
+ u1*u3*v1*(2*a11 *b12*m3 + 2*a11*b12 *kap*m3)
2 2 4
+ u1*u3*v2*( - 2*a11 *b12 *n3 - 6*b12 *kap*n3)
2 3
+ u1*v1*(2*a11 *b12*m3*n3 + 6*b12 *kap*m3*n3)
4 4 2 3 3 3 2 4
+ u2 *(a11 *b12 + a11 *b12 *kap) + u2 *v1*( - 2*a11 *b12 - 2*a11*b12 *kap)
2 2 4 2 3 5 2
+ u2 *u3 *( - a11 *b12 - 2*a11 *b12 *kap - b12 *kap )
2 3 2 2 2 3 5
- 4*u2 *u3*a11*b12 *kap*n3 + u2 *v1 *(a11 *b12 + b12 *kap)
2 2 2 3 5
+ u2 *v2 *(a11 *b12 + b12 *kap)
2 3 3
+ u2 *v3*(2*a11 *b12*m3 + 2*a11*b12 *kap*m3)
2 2 2 3 2 4
+ u2 *(a11 *b12*n3 + b12 *kap*n3 ) + 4*u2*u3*v1*b12 *kap*n3
2 3 5
+ u2*u3*v2*v3*(2*a11 *b12 + 2*b12 *kap)
2 2 4 3
+ u2*v1*v3*( - 2*a11 *b12 *m3 - 2*b12 *kap*m3) + 4*u2*v2*b12 *kap*m3*n3
2 2 2 3 5
+ u3 *v2 *( - a11 *b12 - b12 *kap)
2 2 4
+ u3*v1*v2*(2*a11 *b12 *m3 + 2*b12 *kap*m3)
2 2 2 3 2 2 2 2 3 2
+ v2 *(a11 *b12*m3 + b12 *kap*m3 ) + v3 *(a11 *b12*m3 + 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 + u1*v2*b12 + u2**2*a11 - u2*v1*b12 + 1/2*u3**2*a11 + u3*n3 + 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=u3**4$
FI=u1*u3**2*v1 + u2*u3**2*v2 + u3**3*v3$
FI=u1**2*u2**2*(1/4*a11**4*b12 + 1/4*a11**2*b12**3*kap) + u1**2*u3**2*( - 1/4*
a11**4*b12 - 1/4*a11**2*b12**3*kap) - u1**2*u3*a11*b12**3*kap*n3 + u1**2*(1/4*
a11**2*b12*kap*m3**2 - 1/4*a11**2*b12*n3**2 + 1/4*b12**3*kap**2*m3**2 - 1/4*b12
**3*kap*n3**2) + u1*u2**2*v2*(1/2*a11**3*b12**2 + 1/2*a11*b12**4*kap) + u1*u2*u3
*v3*(1/2*a11**3*b12**2 + 1/2*a11*b12**4*kap) + u1*u2*v3*(1/2*a11**2*b12**2*n3 +
1/2*b12**4*kap*n3) + u1*u2*( - 1/2*a11**3*m3*n3 - 1/2*a11*b12**2*kap*m3*n3) + u1
*u3**2*v2*( - 1/2*a11**3*b12**2 - 1/2*a11*b12**4*kap) + u1*u3*v1*(1/2*a11**3*b12
*m3 + 1/2*a11*b12**3*kap*m3) + u1*u3*v2*( - 1/2*a11**2*b12**2*n3 - 3/2*b12**4*
kap*n3) + u1*v1*(1/2*a11**2*b12*m3*n3 + 3/2*b12**3*kap*m3*n3) + u2**4*(1/4*a11**
4*b12 + 1/4*a11**2*b12**3*kap) + u2**3*v1*( - 1/2*a11**3*b12**2 - 1/2*a11*b12**4
*kap) + u2**2*u3**2*( - 1/4*a11**4*b12 - 1/2*a11**2*b12**3*kap - 1/4*b12**5*kap
**2) - u2**2*u3*a11*b12**3*kap*n3 + u2**2*v1**2*(1/4*a11**2*b12**3 + 1/4*b12**5*
kap) + u2**2*v2**2*(1/4*a11**2*b12**3 + 1/4*b12**5*kap) + u2**2*v3*(1/2*a11**3*
b12*m3 + 1/2*a11*b12**3*kap*m3) + u2**2*(1/4*a11**2*b12*n3**2 + 1/4*b12**3*kap*
n3**2) + u2*u3*v1*b12**4*kap*n3 + u2*u3*v2*v3*(1/2*a11**2*b12**3 + 1/2*b12**5*
kap) + u2*v1*v3*( - 1/2*a11**2*b12**2*m3 - 1/2*b12**4*kap*m3) + u2*v2*b12**3*kap
*m3*n3 + u3**2*v2**2*( - 1/4*a11**2*b12**3 - 1/4*b12**5*kap) + u3*v1*v2*(1/2*a11
**2*b12**2*m3 + 1/2*b12**4*kap*m3) + v2**2*(1/4*a11**2*b12*m3**2 + 1/4*b12**3*
kap*m3**2) + v3**2*(1/4*a11**2*b12*m3**2 + 1/4*b12**3*kap*m3**2)$
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$