Solution 4 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2
0=4*a13 + 4*a23 + b12 *kap
Expressions
The solution is given through the following expressions:
r10=0
r11=0
2
2*a13*m3*n1 *r494
r12=-------------------
4
b12 *kap
2
2*a13*m3 *n1*r494
r13=-------------------
4
b12
r14=0
r15=0
r20=0
r21=0
r22=0
r23=0
r24=0
2
4*a13*a23*m3*n1*r494 - 2*a13*b12*n1 *r494
r27=-------------------------------------------
4
b12 *kap
2 2
4*a13 *m3*n1*r494 - b12 *kap*m3*n1*r494
r28=-----------------------------------------
4
b12 *kap
2 2 1 2 2 2 1 2 2
- 2*a13 *n1 *r494 - ---*b12 *kap *m3 *r494 + ---*b12 *kap*n1 *r494
2 2
r29=---------------------------------------------------------------------
4
b12 *kap
2
2*a13*n1 *r494
r210=----------------
3
b12 *kap
r211=0
r212=0
r213=0
2 2 1 2 2
- 2*a13 *n1 *r494 + ---*b12 *kap*n1 *r494
2
r214=--------------------------------------------
4
b12 *kap
r215=0
r217=0
r218=0
r219=0
2 2
- 2*a13 *n1 *r494
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*a13*n1*r494
r312=------------------
2
b12 *kap
r314=0
- 2*a13*n1*r494
r315=------------------
2
b12 *kap
2 2
- 4*a13 *n1*r494 - 2*a23*b12*kap*m3*r494 + b12 *kap*n1*r494
r317=--------------------------------------------------------------
3
b12 *kap
4*a13*a23*n1*r494 - 2*a13*b12*kap*m3*r494
r318=-------------------------------------------
3
b12 *kap
2*a13*n1*r494
r319=---------------
2
b12
r320=0
2*a13*n1*r494
r321=---------------
2
b12 *kap
r322=0
r323=0
r324=0
r325=0
r327=0
r328=0
r329=0
r330=0
2 2
- 4*a13 *n1*r494 + b12 *kap*n1*r494
r331=--------------------------------------
3
b12 *kap
r332=0
3 2 2
- 4*a13 *n1*r494 - 4*a13*a23 *n1*r494 + a13*b12 *kap*n1*r494
r333=---------------------------------------------------------------
4
b12 *kap
r334=0
r335=0
r336=0
r337=0
2*a13*n1*r494
r338=---------------
2
b12 *kap
r339=0
r340=0
r342=0
r343=0
r344=0
r345=0
4*a13*a23*n1*r494
r346=-------------------
3
b12 *kap
2 2
- 4*a13 *n1*r494 + b12 *kap*n1*r494
r347=--------------------------------------
3
b12 *kap
- 2*a23*n1*r494
r348=------------------
2
b12
r349=0
r350=0
r351=0
4*a13*a23*n1*r494
r352=-------------------
3
b12 *kap
3 2 2
- 4*a13 *n1*r494 - 4*a13*a23 *n1*r494 - a13*b12 *kap*n1*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
r418=0
r421=0
r423=0
r424=0
1
r427=---*r494
2
r429=0
1
r430=---*r494
2
2*a13*r494
r432=------------
b12
- 2*a23*r494
r433=---------------
b12
2 2
2*a13 *r494 + 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
r447=0
r449=0
r450=0
r452=0
r453=0
r454=0
r455=0
r456=0
1
r457=---*r494
2
r458=0
r459=0
r460=0
2*a13*r494
r462=------------
b12
r463=0
2 2 1 2
4*a13 *r494 + 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
r482=0
r484=0
r485=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r495=0
- 2*a23*r494
r497=---------------
b12
2*a13*r494
r498=------------
b12
- 4*a13*a23*r494
r499=-------------------
2
b12
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
1
r4110=---*r494
2
r4112=0
- 2*a23*r494
r4113=---------------
b12
2 2 1 2
2*a13 *r494 + 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
n2=0
a33=0
a22=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, m3, a23, a13, n1, 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.
2 2
{b12,n1,a13,a23,{r494,4*a13 *r494 + b12 *kap*r494}}
Relevance for the application:
Modulo the following equation:
2 2 2
0=4*a13 + 4*a23 + b12 *kap
the system of equations related to the Hamiltonian HAM:
HAM=2*u1*u3*a13 + u1*v2*b12 + u1*n1 + 2*u2*u3*a23 - u2*v1*b12 + v3*m3
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 2 2 1 4 2
FI=u1 *u3 *(2*a13 *b12 *kap + 4*a23 *b12 *kap + ---*b12 *kap )
2
2 3
- 2*u1 *u3*v1*a23*b12 *kap
2 3 2 2
+ u1 *u3*( - 4*a13 *n1 - 4*a13*a23 *n1 - a13*b12 *kap*n1)
1 2 2 4 2 2 2 2
+ ---*u1 *v1 *b12 *kap + 4*u1 *v1*a13*a23*b12*n1 - 2*u1 *a13 *n1
2
2 2 3
- 4*u1*u2*u3 *a13*a23*b12 *kap + 2*u1*u2*u3*v1*a13*b12 *kap
3 2
- 2*u1*u2*u3*v2*a23*b12 *kap - 2*u1*u2*u3*a23*b12 *kap*n1
4 2 3
+ u1*u2*v1*v2*b12 *kap + u1*u2*v1*( - 4*a13 *b12*n1 + b12 *kap*n1)
2
+ 4*u1*u2*v2*a13*a23*b12*n1 + 2*u1*v1*v3*a13*b12 *n1
2 2 2 2 2 2 1 4 2
+ u2 *u3 *(4*a13 *b12 *kap + 2*a23 *b12 *kap + ---*b12 *kap )
2
2 3
+ 2*u2 *u3*v2*a13*b12 *kap
2 3 2 2
+ u2 *u3*( - 4*a13 *n1 - 4*a13*a23 *n1 + a13*b12 *kap*n1)
1 2 2 4 2 2 3
+ ---*u2 *v2 *b12 *kap + u2 *v2*( - 4*a13 *b12*n1 + b12 *kap*n1)
2
2 2 2 1 2 2 2
+ u2 *( - 2*a13 *n1 + ---*b12 *kap*n1 ) + 2*u2*v2*v3*a13*b12 *n1
2
2 4 2 2 2 2
+ 2*u2*v3*a13*b12*n1 + u3 *(2*a13 *b12 *kap + 2*a23 *b12 *kap)
3 3 3 3 3 2
- 2*u3 *v1*a23*b12 *kap + 2*u3 *v2*a13*b12 *kap + 2*u3 *a13*b12 *kap*n1
1 2 2 4 2 2
+ ---*u3 *v1 *b12 *kap + u3 *v1*(4*a13*a23*b12*n1 - 2*a13*b12 *kap*m3)
2
1 2 2 4
+ ---*u3 *v2 *b12 *kap
2
2 2 2 3
+ u3 *v2*( - 4*a13 *b12*n1 - 2*a23*b12 *kap*m3 + b12 *kap*n1)
2 2 2 1 2 2 2 1 2 2
+ u3 *( - 2*a13 *n1 - ---*b12 *kap *m3 + ---*b12 *kap*n1 )
2 2
2 2 2 2
- 2*u3*v1 *a13*b12 *n1 + u3*v1*(4*a13 *m3*n1 - b12 *kap*m3*n1)
2 2 2
- 2*u3*v2 *a13*b12 *n1 + u3*v2*(4*a13*a23*m3*n1 - 2*a13*b12*n1 )
2 2
+ 2*u3*a13*kap*m3 *n1 + 2*v1*a13*m3*n1
3 3 2 2
{HAM,FI} = 8*u1 *v1*a13 *b12*n1 + 16*u1 *u2*v1*a13 *a23*b12*n1
2 3 2 3
+ 8*u1 *u2*v2*a13 *b12*n1 + 8*u1 *u3*v3*a13 *b12*n1
2 2
- 4*u1 *v1*v3*a13*a23*b12 *n1
2 3 3
+ u1*u2 *v1*( - 8*a13 *b12*n1 - 2*a13*b12 *kap*n1)
2 2 2
+ 16*u1*u2 *v2*a13 *a23*b12*n1 + 16*u1*u2*u3*v3*a13 *a23*b12*n1
2 2 2
+ 4*u1*u2*v1*v3*a13 *b12 *n1 - 4*u1*u2*v2*v3*a13*a23*b12 *n1
3 5 2 2 2 4
- u1*u3 *v1*b12 *kap - 4*u1*u3 *v1 *a23*b12 *kap
2 4 2 3
+ 4*u1*u3 *v1*v2*a13*b12 *kap + 6*u1*u3 *v1*a13*b12 *kap*n1
3 5
+ u1*u3*v1 *b12 *kap
2 2 3
+ u1*u3*v1 *(8*a13*a23*b12 *n1 - 2*a13*b12 *kap*m3)
2 5
+ u1*u3*v1*v2 *b12 *kap + u1*u3*v1*v2
2 2 3 4
*( - 8*a13 *b12 *n1 - 2*a23*b12 *kap*m3 + 2*b12 *kap*n1)
2 2 3 2
+ u1*u3*v1*( - 4*a13 *b12*n1 + b12 *kap*n1 )
2 2 3 3
- 4*u1*u3*v3 *a13*a23*b12 *n1 - 2*u1*v1 *a13*b12 *n1
2 2 3
+ u1*v1 *(4*a13 *b12*m3*n1 - b12 *kap*m3*n1)
2 3
- 2*u1*v1*v2 *a13*b12 *n1
2 2
+ u1*v1*v2*(4*a13*a23*b12*m3*n1 - 2*a13*b12 *n1 )
3 3 3
+ u2 *v2*( - 8*a13 *b12*n1 - 2*a13*b12 *kap*n1)
2 3 3
+ u2 *u3*v3*( - 8*a13 *b12*n1 - 2*a13*b12 *kap*n1)
2 2 2 3 5 2
+ 4*u2 *v2*v3*a13 *b12 *n1 - u2*u3 *v2*b12 *kap
2 4 2 2 4
- 4*u2*u3 *v1*v2*a23*b12 *kap + 4*u2*u3 *v2 *a13*b12 *kap
2 3 2 5
+ 6*u2*u3 *v2*a13*b12 *kap*n1 + u2*u3*v1 *v2*b12 *kap
2 3
+ u2*u3*v1*v2*(8*a13*a23*b12 *n1 - 2*a13*b12 *kap*m3)
3 5
+ u2*u3*v2 *b12 *kap
2 2 2 3 4
+ u2*u3*v2 *( - 8*a13 *b12 *n1 - 2*a23*b12 *kap*m3 + 2*b12 *kap*n1)
2 2 3 2
+ u2*u3*v2*( - 4*a13 *b12*n1 + b12 *kap*n1 )
2 2 2 2 3
+ 4*u2*u3*v3 *a13 *b12 *n1 - 2*u2*v1 *v2*a13*b12 *n1
2 3
+ u2*v1*v2*(4*a13 *b12*m3*n1 - b12 *kap*m3*n1)
3 3
- 2*u2*v2 *a13*b12 *n1
2 2 2 4 5 2
+ u2*v2 *(4*a13*a23*b12*m3*n1 - 2*a13*b12 *n1 ) - u3 *v3*b12 *kap
3 4 3 4
- 4*u3 *v1*v3*a23*b12 *kap + 4*u3 *v2*v3*a13*b12 *kap
3 3 2 2 5
+ 6*u3 *v3*a13*b12 *kap*n1 + u3 *v1 *v3*b12 *kap
2 2 3
+ u3 *v1*v3*(8*a13*a23*b12 *n1 - 2*a13*b12 *kap*m3)
2 2 5
+ u3 *v2 *v3*b12 *kap
2 2 2 3 4
+ u3 *v2*v3*( - 8*a13 *b12 *n1 - 2*a23*b12 *kap*m3 + 2*b12 *kap*n1)
2 2 2 3 2
+ u3 *v3*( - 4*a13 *b12*n1 + b12 *kap*n1 )
2 3
- 2*u3*v1 *v3*a13*b12 *n1
2 3
+ u3*v1*v3*(4*a13 *b12*m3*n1 - b12 *kap*m3*n1)
2 3
- 2*u3*v2 *v3*a13*b12 *n1
2 2
+ u3*v2*v3*(4*a13*a23*b12*m3*n1 - 2*a13*b12 *n1 )
And again in machine readable form:
HAM=2*u1*u3*a13 + u1*v2*b12 + u1*n1 + 2*u2*u3*a23 - u2*v1*b12 + v3*m3$
FI=u1**2*u3**2*(2*a13**2*b12**2*kap + 4*a23**2*b12**2*kap + 1/2*b12**4*kap**2) -
2*u1**2*u3*v1*a23*b12**3*kap + u1**2*u3*( - 4*a13**3*n1 - 4*a13*a23**2*n1 - a13
*b12**2*kap*n1) + 1/2*u1**2*v1**2*b12**4*kap + 4*u1**2*v1*a13*a23*b12*n1 - 2*u1
**2*a13**2*n1**2 - 4*u1*u2*u3**2*a13*a23*b12**2*kap + 2*u1*u2*u3*v1*a13*b12**3*
kap - 2*u1*u2*u3*v2*a23*b12**3*kap - 2*u1*u2*u3*a23*b12**2*kap*n1 + u1*u2*v1*v2*
b12**4*kap + u1*u2*v1*( - 4*a13**2*b12*n1 + b12**3*kap*n1) + 4*u1*u2*v2*a13*a23*
b12*n1 + 2*u1*v1*v3*a13*b12**2*n1 + u2**2*u3**2*(4*a13**2*b12**2*kap + 2*a23**2*
b12**2*kap + 1/2*b12**4*kap**2) + 2*u2**2*u3*v2*a13*b12**3*kap + u2**2*u3*( - 4*
a13**3*n1 - 4*a13*a23**2*n1 + a13*b12**2*kap*n1) + 1/2*u2**2*v2**2*b12**4*kap +
u2**2*v2*( - 4*a13**2*b12*n1 + b12**3*kap*n1) + u2**2*( - 2*a13**2*n1**2 + 1/2*
b12**2*kap*n1**2) + 2*u2*v2*v3*a13*b12**2*n1 + 2*u2*v3*a13*b12*n1**2 + u3**4*(2*
a13**2*b12**2*kap + 2*a23**2*b12**2*kap) - 2*u3**3*v1*a23*b12**3*kap + 2*u3**3*
v2*a13*b12**3*kap + 2*u3**3*a13*b12**2*kap*n1 + 1/2*u3**2*v1**2*b12**4*kap + u3
**2*v1*(4*a13*a23*b12*n1 - 2*a13*b12**2*kap*m3) + 1/2*u3**2*v2**2*b12**4*kap +
u3**2*v2*( - 4*a13**2*b12*n1 - 2*a23*b12**2*kap*m3 + b12**3*kap*n1) + u3**2*( -
2*a13**2*n1**2 - 1/2*b12**2*kap**2*m3**2 + 1/2*b12**2*kap*n1**2) - 2*u3*v1**2*
a13*b12**2*n1 + u3*v1*(4*a13**2*m3*n1 - b12**2*kap*m3*n1) - 2*u3*v2**2*a13*b12**
2*n1 + u3*v2*(4*a13*a23*m3*n1 - 2*a13*b12*n1**2) + 2*u3*a13*kap*m3**2*n1 + 2*v1*
a13*m3*n1**2$