Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
r10=0
m3*n2*n3*r453
r11=---------------
3
b12 *kap
m3*n1*n3*r453
r12=---------------
3
b12 *kap
2 3
kap*m3 *n3*r453 - n3 *r453
r13=----------------------------
3
b12 *kap
2
- n2*n3 *r453
r14=----------------
3
b12 *kap
2
- n1*n3 *r453
r15=----------------
3
b12 *kap
1 2 2 1 2 2
- ---*a22 *m3 *r453 - ---*b12 *kap*m3 *r453
2 2
r20=----------------------------------------------
3
a22*b12 *kap
r21=0
1 2 2 1 2 2
- ---*a22 *m3 *r453 - ---*b12 *kap*m3 *r453
2 2
r22=----------------------------------------------
3
a22*b12 *kap
r23=0
r24=0
- a22*n1*n3*r453 + b12*kap*m3*n2*r453
r27=----------------------------------------
2
a22*b12 *kap
- a22*m3*n1*r453 + b12*n2*n3*r453
r28=------------------------------------
3
b12 *kap
1 2 2 1 2 2 1 2 2
r29=( - ---*a22 *kap*m3 *r453 + ---*a22 *n1 *r453 - ---*b12 *kap*n2 *r453
2 2 2
1 2 2 3
- ---*b12 *kap*n3 *r453)/(a22*b12 *kap)
2
2 2
- a22 *m3*n2*r453 + a22*b12*n1*n3*r453 - b12 *kap*m3*n2*r453
r210=---------------------------------------------------------------
3
a22*b12 *kap
2*a22*m3*n3*r453
r211=------------------
3
b12 *kap
r212=0
- a22*n2*n3*r453
r213=-------------------
3
b12 *kap
1 2 2 1 2 2 1 2 2
r214=(---*a22 *n1 *r453 - ---*a22 *n2 *r453 - ---*a22 *n3 *r453
2 2 2
1 2 2 1 2 2 3
- ---*b12 *kap*n2 *r453 - ---*b12 *kap*n3 *r453)/(a22*b12 *kap)
2 2
- n2*n3*r453
r215=---------------
2
b12 *kap
2 2
a22 *m3*n3*r453 - b12 *kap*m3*n3*r453
r217=---------------------------------------
3
a22*b12 *kap
- a22*n1*n3*r453
r218=-------------------
3
b12 *kap
- a22*n1*n2*r453
r219=-------------------
3
b12 *kap
1 2 2 1 2 2 1 2 2 2
r220=( - ---*a22 *kap*m3 *r453 + ---*a22 *n3 *r453 - ---*b12 *kap *m3 *r453
2 2 2
1 2 2 1 2 2 3
- ---*b12 *kap*n2 *r453 + ---*b12 *kap*n3 *r453)/(a22*b12 *kap)
2 2
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
- n3*r453
r312=------------
b12*kap
2 2
- a22 *m3*r453 - b12 *kap*m3*r453
r314=------------------------------------
2
a22*b12 *kap
- n3*r453
r315=------------
b12*kap
a22*n1*r453
r317=-------------
2
b12 *kap
n2*r453
r318=---------
a22
r319=0
r320=0
n3*r453
r321=---------
b12*kap
r322=0
2 2
a22 *m3*r453 + b12 *kap*m3*r453
r323=---------------------------------
2
a22*b12 *kap
r324=0
r325=0
m3*r453
r327=---------
b12
2*a22*n3*r453
r328=---------------
2
b12 *kap
- n2*r453
r329=------------
b12
2 2
- a22 *m3*r453 - b12 *kap*m3*r453
r330=------------------------------------
3
b12 *kap
a22*n1*r453
r331=-------------
2
b12 *kap
2 2
a22 *n2*r453 + b12 *kap*n2*r453
r332=---------------------------------
2
a22*b12 *kap
2
- a22 *n3*r453
r333=-----------------
3
b12 *kap
2 2
- a22 *n2*r453 - b12 *kap*n2*r453
r334=------------------------------------
3
b12 *kap
r335=0
r336=0
r337=0
n3*r453
r338=---------
b12*kap
r339=0
r340=0
2 2
- a22 *n3*r453 + b12 *kap*n3*r453
r342=------------------------------------
2
a22*b12 *kap
m3*r453
r343=---------
b12
- n1*r453
r344=------------
b12
2 2
- a22 *n3*r453 - b12 *kap*n3*r453
r345=------------------------------------
2
a22*b12 *kap
- a22*n2*r453
r346=----------------
2
b12 *kap
a22*n1*r453
r347=-------------
2
b12 *kap
r348=0
2 2
- a22 *n1*r453 - b12 *kap*n1*r453
r349=------------------------------------
3
b12 *kap
r350=0
r351=0
n2*r453
r352=---------
a22
n3*r453
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
r418=0
r421=0
r423=0
r424=0
1
---*a22*r453
2
r427=--------------
b12*kap
r429=0
1
- ---*b12*r453
2
r430=-----------------
a22
r432=0
r433=0
r434=0
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
r454=0
r455=0
r456=0
1
---*a22*r453
2
r457=--------------
b12*kap
r458=0
r459=0
1 2 1 2
- ---*a22 *r453 - ---*b12 *kap*r453
2 2
r460=--------------------------------------
a22*b12*kap
r462=0
r463=0
1
- ---*a22*r453
2
r464=-----------------
b12
r465=0
r466=0
2 2
a22 *r453 + b12 *kap*r453
r467=---------------------------
2
b12 *kap
r468=0
1 3 1 2
- ---*a22 *r453 - ---*a22*b12 *kap*r453
2 2
r469=------------------------------------------
3
b12 *kap
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= - r453
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
a22*r453
r494=----------
b12*kap
r495=0
r497=0
r498=0
r499=0
r4100=0
2 2
- a22 *r453 - b12 *kap*r453
r4101=------------------------------
2
b12 *kap
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
1
- ---*b12*r453
2
r4110=-----------------
a22
r4112=0
r4113=0
1
---*b12*kap*r453
2
r4114=------------------
a22
r4115=0
r4116=0
r4117=0
r4118=0
1 2 1 2
---*a22 *r453 + ---*b12 *kap*r453
2 2
r4119=-----------------------------------
a22*b12
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
a33=0
a23=0
a13=0
2
- b12 *kap
a11=-------------
a22
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:
r453, n2, n1, n3, m3, a22, 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,
a22,
2 2
a22 + b12 *kap,
a11,
{n3,n2,n1,m3},
2 2
{r453,a22 *r453 + b12 *kap*r453}}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2
HAM=( - u1 *b12 *kap + u1*v2*a22*b12 + u1*a22*n1 + u2 *a22 - u2*v1*a22*b12
+ u2*a22*n2 + u3*a22*n3 + v3*a22*m3)/a22
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 1 2 2 1 4 2 1 2 2 4 2
FI=u1 *u2 *(---*a22 *b12 *kap + ---*b12 *kap ) + ---*u1 *u3 *b12 *kap
2 2 2
2 2 1 2 2 4 2 3 2
+ u1 *u3*a22*b12 *kap*n3 - ---*u1 *v1 *b12 *kap + u1 *v1*b12 *kap*n2 + u1 *(
2
1 2 2 1 2 2 1 2 2 2 1 2 2
- ---*a22 *kap*m3 + ---*a22 *n3 - ---*b12 *kap *m3 - ---*b12 *kap*n2
2 2 2 2
1 2 2 2 3 3
+ ---*b12 *kap*n3 ) + u1*u2 *v2*( - a22 *b12 - a22*b12 *kap)
2
2 3 2 2 2
+ u1*u2 *( - a22 *n1 - a22*b12 *kap*n1) + u1*u2*v1*v2*a22 *b12
2 2
+ u1*u2*v1*a22 *b12*n1 - u1*u2*v2*a22 *b12*n2
2 3 2
+ u1*u2*v3*( - a22 *b12*n3 - b12 *kap*n3) - u1*u2*a22 *n1*n2
2 3 2 2 2
- u1*u3 *v2*a22*b12 *kap - u1*u3 *a22*b12 *kap*n1 + u1*u3*v1*a22*b12 *kap*m3
2 3 2
+ u1*u3*v2*( - a22 *b12*n3 + b12 *kap*n3) - u1*u3*a22 *n1*n3
2 2 2
+ u1*v1*v3*a22*b12 *n3 + u1*v1*(a22 *m3*n3 - b12 *kap*m3*n3)
2
- u1*v3*a22*b12*n2*n3 - u1*a22*n1*n3
4 1 4 1 2 2 3 3 3
+ u2 *( - ---*a22 - ---*a22 *b12 *kap) + u2 *v1*(a22 *b12 + a22*b12 *kap)
2 2
3 3 2 1 2 2 2 2
+ u2 *( - a22 *n2 - a22*b12 *kap*n2) - ---*u2 *u3 *a22 *b12 *kap
2
2 3 2 2 1 2 2 1 4
- u2 *u3*a22 *n3 + u2 *v1 *( - ---*a22 *b12 - ---*b12 *kap)
2 2
2 2 3 1 2 2 2 2
+ u2 *v1*(a22 *b12*n2 + b12 *kap*n2) + ---*u2 *v2 *a22 *b12
2
2 2 2 3 2 2
+ u2 *v2*a22 *b12*n1 + u2 *v3*( - a22 *m3 - a22*b12 *kap*m3) + u2 *(
1 2 2 1 2 2 1 2 2 1 2 2
---*a22 *n1 - ---*a22 *n2 - ---*a22 *n3 - ---*b12 *kap*n2
2 2 2 2
1 2 2 2 3 2 2
- ---*b12 *kap*n3 ) + u2*u3 *v1*a22*b12 *kap - u2*u3 *a22*b12 *kap*n2
2
2 2 2
+ 2*u2*u3*v1*a22 *b12*n3 + u2*u3*v2*a22*b12 *kap*m3 - u2*u3*a22 *n2*n3
2 3 2
+ u2*v1*v3*(a22 *b12*m3 + b12 *kap*m3) + u2*v2*v3*a22*b12 *n3
2
+ 2*u2*v2*a22 *m3*n3
2 2 2
+ u2*v3*( - a22 *m3*n2 + a22*b12*n1*n3 - b12 *kap*m3*n2) - u2*a22*n2*n3
1 2 2 4 2 3 1 2 2 2 2
- ---*u3 *v1 *b12 *kap + u3 *v1*b12 *kap*n2 + ---*u3 *v2 *a22 *b12
2 2
2 2 2
+ u3 *v2*a22 *b12*n1 + u3
1 2 2 1 2 2 1 2 2 1 2 2
*( - ---*a22 *kap*m3 + ---*a22 *n1 - ---*b12 *kap*n2 - ---*b12 *kap*n3 )
2 2 2 2
2 2 2 3
- u3*v1 *a22*b12 *n3 + u3*v1*v2*( - a22 *b12*m3 - b12 *kap*m3)
2 2 2
+ u3*v1*( - a22 *m3*n1 + a22*b12*n2*n3) - u3*v2 *a22*b12 *n3
2 2 3
+ u3*v2*( - a22*b12*n1*n3 + b12 *kap*m3*n2) + u3*(a22*kap*m3 *n3 - a22*n3 )
2 1 2 2 1 2 2
+ v1*a22*m3*n1*n3 + v2 *( - ---*a22 *m3 - ---*b12 *kap*m3 )
2 2
2 1 2 2 1 2 2
+ v2*a22*m3*n2*n3 + v3 *( - ---*a22 *m3 - ---*b12 *kap*m3 )
2 2
3 5 2 3 3 2 5 2
{HAM,FI} = u1 *u3*v1*b12 *kap + u1 *v1*a22*b12 *kap*n3 + u1 *u2*u3*v2*b12 *kap
2 2 2 4 2
+ u1 *u2*v1*(a22 *b12 *kap*m3 + b12 *kap *m3)
2 3 2 2 5 2
+ u1 *u2*v2*a22*b12 *kap*n3 + u1 *u3 *v3*b12 *kap
2 4 2 3
- 2*u1 *u3*v1*v2*a22*b12 *kap - 2*u1 *u3*v1*a22*b12 *kap*n1
2 3
+ u1 *u3*v3*a22*b12 *kap*n3
2 2 2 4
+ u1 *v1*v2*( - a22 *b12 *n3 + b12 *kap*n3)
2 2 2
+ u1 *v1*( - a22 *b12*n1*n3 + a22*b12 *kap*m3*n2)
2 2 3 2 3
- u1*u2 *u3*v1*a22 *b12 *kap + u1*u2 *v1*a22*b12 *kap*n3
2 2 2 4 2
+ u1*u2 *v2*(a22 *b12 *kap*m3 + b12 *kap *m3)
2 4 3
+ 2*u1*u2*u3*v1 *a22*b12 *kap - 2*u1*u2*u3*v1*a22*b12 *kap*n2
2 4 3
- 2*u1*u2*u3*v2 *a22*b12 *kap - 2*u1*u2*u3*v2*a22*b12 *kap*n1
2 2 4 2
+ u1*u2*u3*v3*(a22 *b12 *kap*m3 + b12 *kap *m3)
2 2 2 4
+ u1*u2*v1 *(a22 *b12 *n3 - b12 *kap*n3)
2 3
+ u1*u2*v1*( - a22*b12 *kap*m3*n1 + b12 *kap*n2*n3)
2 2 2 4
+ u1*u2*v2 *( - a22 *b12 *n3 + b12 *kap*n3)
2 2
+ u1*u2*v2*( - a22 *b12*n1*n3 + a22*b12 *kap*m3*n2)
2 4 2 3
- 2*u1*u3 *v2*v3*a22*b12 *kap - 2*u1*u3 *v3*a22*b12 *kap*n1
3 5 2 4
- u1*u3*v1 *b12 *kap + 2*u1*u3*v1 *b12 *kap*n2
2 2 3 2 2
+ u1*u3*v1*v2 *a22 *b12 + 2*u1*u3*v1*v2*a22 *b12 *n1
2 2 3 2
+ u1*u3*v1*(a22 *b12*n1 - b12 *kap*n2 )
2 2 4
+ u1*u3*v2*v3*( - a22 *b12 *n3 + b12 *kap*n3)
2 2
+ u1*u3*v3*( - a22 *b12*n1*n3 + a22*b12 *kap*m3*n2)
3 3 2 2 2 4
- u1*v1 *a22*b12 *n3 + u1*v1 *v2*( - a22 *b12 *m3 - b12 *kap*m3)
2 2 2
+ u1*v1 *( - a22 *b12*m3*n1 + a22*b12 *n2*n3)
2 3
- u1*v1*v2 *a22*b12 *n3
2 3
+ u1*v1*v2*( - a22*b12 *n1*n3 + b12 *kap*m3*n2)
3 2 3 3 3
- u2 *u3*v2*a22 *b12 *kap + u2 *v2*a22*b12 *kap*n3
2 2 2 3 2 4
- u2 *u3 *v3*a22 *b12 *kap + 2*u2 *u3*v1*v2*a22*b12 *kap
2 3 2 3
- 2*u2 *u3*v2*a22*b12 *kap*n2 + u2 *u3*v3*a22*b12 *kap*n3
2 2 2 4
+ u2 *v1*v2*(a22 *b12 *n3 - b12 *kap*n3)
2 2 3
+ u2 *v2*( - a22*b12 *kap*m3*n1 + b12 *kap*n2*n3)
2 4 2 3
+ 2*u2*u3 *v1*v3*a22*b12 *kap - 2*u2*u3 *v3*a22*b12 *kap*n2
2 5 4
- u2*u3*v1 *v2*b12 *kap + 2*u2*u3*v1*v2*b12 *kap*n2
2 2 4 3 2 3
+ u2*u3*v1*v3*(a22 *b12 *n3 - b12 *kap*n3) + u2*u3*v2 *a22 *b12
2 2 2 2 2 3 2
+ 2*u2*u3*v2 *a22 *b12 *n1 + u2*u3*v2*(a22 *b12*n1 - b12 *kap*n2 )
2 3
+ u2*u3*v3*( - a22*b12 *kap*m3*n1 + b12 *kap*n2*n3)
2 3 2 2 2 4
- u2*v1 *v2*a22*b12 *n3 + u2*v1*v2 *( - a22 *b12 *m3 - b12 *kap*m3)
2 2 3 3
+ u2*v1*v2*( - a22 *b12*m3*n1 + a22*b12 *n2*n3) - u2*v2 *a22*b12 *n3
2 2 3 2 2 5
+ u2*v2 *( - a22*b12 *n1*n3 + b12 *kap*m3*n2) - u3 *v1 *v3*b12 *kap
2 4 2 2 2 3
+ 2*u3 *v1*v3*b12 *kap*n2 + u3 *v2 *v3*a22 *b12
2 2 2 2 2 2 3 2
+ 2*u3 *v2*v3*a22 *b12 *n1 + u3 *v3*(a22 *b12*n1 - b12 *kap*n2 )
2 3
- u3*v1 *v3*a22*b12 *n3
2 2 4
+ u3*v1*v2*v3*( - a22 *b12 *m3 - b12 *kap*m3)
2 2
+ u3*v1*v3*( - a22 *b12*m3*n1 + a22*b12 *n2*n3)
2 3
- u3*v2 *v3*a22*b12 *n3
2 3
+ u3*v2*v3*( - a22*b12 *n1*n3 + b12 *kap*m3*n2)
And again in machine readable form:
HAM=( - u1**2*b12**2*kap + u1*v2*a22*b12 + u1*a22*n1 + u2**2*a22**2 - u2*v1*a22*
b12 + u2*a22*n2 + u3*a22*n3 + v3*a22*m3)/a22$
FI=u1**2*u2**2*(1/2*a22**2*b12**2*kap + 1/2*b12**4*kap**2) + 1/2*u1**2*u3**2*b12
**4*kap**2 + u1**2*u3*a22*b12**2*kap*n3 - 1/2*u1**2*v1**2*b12**4*kap + u1**2*v1*
b12**3*kap*n2 + u1**2*( - 1/2*a22**2*kap*m3**2 + 1/2*a22**2*n3**2 - 1/2*b12**2*
kap**2*m3**2 - 1/2*b12**2*kap*n2**2 + 1/2*b12**2*kap*n3**2) + u1*u2**2*v2*( -
a22**3*b12 - a22*b12**3*kap) + u1*u2**2*( - a22**3*n1 - a22*b12**2*kap*n1) + u1*
u2*v1*v2*a22**2*b12**2 + u1*u2*v1*a22**2*b12*n1 - u1*u2*v2*a22**2*b12*n2 + u1*u2
*v3*( - a22**2*b12*n3 - b12**3*kap*n3) - u1*u2*a22**2*n1*n2 - u1*u3**2*v2*a22*
b12**3*kap - u1*u3**2*a22*b12**2*kap*n1 + u1*u3*v1*a22*b12**2*kap*m3 + u1*u3*v2*
( - a22**2*b12*n3 + b12**3*kap*n3) - u1*u3*a22**2*n1*n3 + u1*v1*v3*a22*b12**2*n3
+ u1*v1*(a22**2*m3*n3 - b12**2*kap*m3*n3) - u1*v3*a22*b12*n2*n3 - u1*a22*n1*n3
**2 + u2**4*( - 1/2*a22**4 - 1/2*a22**2*b12**2*kap) + u2**3*v1*(a22**3*b12 + a22
*b12**3*kap) + u2**3*( - a22**3*n2 - a22*b12**2*kap*n2) - 1/2*u2**2*u3**2*a22**2
*b12**2*kap - u2**2*u3*a22**3*n3 + u2**2*v1**2*( - 1/2*a22**2*b12**2 - 1/2*b12**
4*kap) + u2**2*v1*(a22**2*b12*n2 + b12**3*kap*n2) + 1/2*u2**2*v2**2*a22**2*b12**
2 + u2**2*v2*a22**2*b12*n1 + u2**2*v3*( - a22**3*m3 - a22*b12**2*kap*m3) + u2**2
*(1/2*a22**2*n1**2 - 1/2*a22**2*n2**2 - 1/2*a22**2*n3**2 - 1/2*b12**2*kap*n2**2
- 1/2*b12**2*kap*n3**2) + u2*u3**2*v1*a22*b12**3*kap - u2*u3**2*a22*b12**2*kap*
n2 + 2*u2*u3*v1*a22**2*b12*n3 + u2*u3*v2*a22*b12**2*kap*m3 - u2*u3*a22**2*n2*n3
+ u2*v1*v3*(a22**2*b12*m3 + b12**3*kap*m3) + u2*v2*v3*a22*b12**2*n3 + 2*u2*v2*
a22**2*m3*n3 + u2*v3*( - a22**2*m3*n2 + a22*b12*n1*n3 - b12**2*kap*m3*n2) - u2*
a22*n2*n3**2 - 1/2*u3**2*v1**2*b12**4*kap + u3**2*v1*b12**3*kap*n2 + 1/2*u3**2*
v2**2*a22**2*b12**2 + u3**2*v2*a22**2*b12*n1 + u3**2*( - 1/2*a22**2*kap*m3**2 +
1/2*a22**2*n1**2 - 1/2*b12**2*kap*n2**2 - 1/2*b12**2*kap*n3**2) - u3*v1**2*a22*
b12**2*n3 + u3*v1*v2*( - a22**2*b12*m3 - b12**3*kap*m3) + u3*v1*( - a22**2*m3*n1
+ a22*b12*n2*n3) - u3*v2**2*a22*b12**2*n3 + u3*v2*( - a22*b12*n1*n3 + b12**2*
kap*m3*n2) + u3*(a22*kap*m3**2*n3 - a22*n3**3) + v1*a22*m3*n1*n3 + v2**2*( - 1/2
*a22**2*m3**2 - 1/2*b12**2*kap*m3**2) + v2*a22*m3*n2*n3 + v3**2*( - 1/2*a22**2*
m3**2 - 1/2*b12**2*kap*m3**2)$