Solution 2 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
r10=0
1 2 1 2 3 1 2 3
---*a11*b31*m1*n3 *r4114 - ---*b31 *n3 *r4114 - ---*b32 *n3 *r4114
4 8 8
r11=--------------------------------------------------------------------
4
a11 *b32
1 2
- ---*m1*n3 *r4114
4
r12=---------------------
3
a11
1 3
- ---*n3 *r4114
4
r13=------------------
3
a11
r14=0
r15=0
1 2 2 2 1 3
r20=(---*a11 *b31 *m1 *r4114 - ---*a11*b31 *m1*n3*r4114
4 4
1 2 1 4 2
- ---*a11*b31*b32 *m1*n3*r4114 + ----*b31 *n3 *r4114
4 16
3 2 2 2 1 4 2 4 2
+ ----*b31 *b32 *n3 *r4114 + ---*b32 *n3 *r4114)/(a11 *b32 )
16 8
r21=0
1 2 2 2 1 2 2 2
r22=(---*a11 *b31 *m1 *r4114 - ---*a11 *b32 *m1 *r4114
4 4
1 3 1 2
- ---*a11*b31 *m1*n3*r4114 - ---*a11*b31*b32 *m1*n3*r4114
4 4
1 4 2 1 2 2 2 1 4 2
+ ----*b31 *n3 *r4114 + ---*b31 *b32 *n3 *r4114 + ----*b32 *n3 *r4114)/(
16 8 16
4 2
a11 *b32 )
r23=0
1 2 1 2 1 2
- ---*a11*b31*m1 *r4114 + ---*b31 *m1*n3*r4114 + ---*b32 *m1*n3*r4114
2 4 4
r24=------------------------------------------------------------------------
3
a11 *b32
r26=0
1 1 2 2
- ---*a11*b31*m1*n3*r4114 + ---*b31 *n3 *r4114
2 4
r27=-------------------------------------------------
3
a11 *b32
1 1 2
---*a11*m1*n3*r4114 - ---*b31*n3 *r4114
2 4
r28=-----------------------------------------
3
a11
1 2
- ---*n3 *r4114
4
r29=------------------
2
a11
1 1 2 2 1 2 2
---*a11*b31*m1*n3*r4114 - ---*b31 *n3 *r4114 - ---*b32 *n3 *r4114
2 4 4
r210=-------------------------------------------------------------------
3
a11 *b32
r212=0
r213=0
r214=0
1
- ---*m1*n3*r4114
2
r215=--------------------
2
a11
r216=0
r217=0
r218=0
r219=0
r30=0
1 2 1 2
- a11*b31*m1*r455 + ---*b31 *n3*r455 + ---*b32 *n3*r455
2 2
r31=----------------------------------------------------------
2
a11 *b32
r32=0
1 2 1 2
- a11*b31*m1*r455 + ---*b31 *n3*r455 + ---*b32 *n3*r455
2 2
r33=----------------------------------------------------------
2
a11 *b32
m1*r455
r34=---------
a11
r35=0
m1*r455
r36=---------
a11
r37=0
1 2 1 2
- a11*b31*m1*r455 + ---*b31 *n3*r455 + ---*b32 *n3*r455
2 2
r38=----------------------------------------------------------
2
a11 *b32
m1*r455
r39=---------
a11
n3*r455
r310=---------
a11
r311=0
r312
2 1 1 2 1 2
a11 *n3*r455 - ---*a11*b31*m1*r4114 + ---*b31 *n3*r4114 + ---*b32 *n3*r4114
2 4 4
=-----------------------------------------------------------------------------
3
a11
r313=0
1 2 1 2 1 3
r314=( - ---*a11*b31 *m1*r4114 + ---*a11*b32 *m1*r4114 + ---*b31 *n3*r4114
2 2 4
1 2 3
+ ---*b31*b32 *n3*r4114)/(a11 *b32)
4
1
a11*n3*r455 + ---*b31*m1*r4114
2
r315=--------------------------------
2
a11
r316=0
1 2 2
- a11*b31*m1*r4114 + ---*b31 *n3*r4114 + b32 *n3*r4114
2
r317=---------------------------------------------------------
2
a11 *b32
1
a11*m1*r4114 + ---*b31*n3*r4114
2
r318=---------------------------------
2
a11
n3*r4114
r319=----------
a11
r320=0
1 2 1 2
- ---*b31 *n3*r4114 - ---*b32 *n3*r4114
4 4
r321=------------------------------------------
3
a11
r322=0
r323=0
r324=0
r325=0
1
- ---*b32*n3*r4114
2
r326=---------------------
2
a11
r327=0
r328=0
r329=0
r330=0
1 2 1 2
- a11*b31*m1*r4114 + ---*b31 *n3*r4114 + ---*b32 *n3*r4114
2 2
r331=-------------------------------------------------------------
2
a11 *b32
r332=0
n3*r4114
r333=----------
a11
r334=0
r335=0
r336=0
r337=0
1 2 1 2
- ---*b31 *n3*r4114 - ---*b32 *n3*r4114
4 4
r338=------------------------------------------
3
a11
r339=0
r340=0
1
- ---*b31*n3*r4114
2
r341=---------------------
2
a11
r342=0
r343=0
r344=0
r345=0
m1*r4114
r346=----------
a11
1 2 1 2
- a11*b31*m1*r4114 + ---*b31 *n3*r4114 + ---*b32 *n3*r4114
2 2
r347=-------------------------------------------------------------
2
a11 *b32
r348=0
r349=0
r350=0
r351=0
m1*r4114
r352=----------
a11
n3*r4114
r353=----------
a11
r354=0
r355=0
1 2 1 2
- ---*b31 *r455 - ---*b32 *r455
4 4
r40=----------------------------------
2
a11
r41=0
1 2 1 2
- ---*b31 *r455 - ---*b32 *r455
4 4
r42=----------------------------------
2
a11
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
1 2 1 2
- ---*b31 *r455 - ---*b32 *r455
4 4
r49=----------------------------------
2
a11
r410=0
r411=0
r412=0
r413=0
r415=0
b32*r455
r416=----------
a11
r417=0
b32*r455
r418=----------
a11
b31*r455
r419=----------
a11
r420=0
b31*r455
r421=----------
a11
r422=0
b32*r455
r423=----------
a11
b31*r455
r424=----------
a11
2 1 2 1 2
a11 *r455 - ---*b31 *r4114 - ---*b32 *r4114
4 4
r425=---------------------------------------------
2
a11
r426=0
2 1 2
2*a11 *r455 + ---*b32 *r4114
4
r427=------------------------------
2
a11
r428=0
1
---*b31*b32*r4114
2
r429=-------------------
2
a11
2 1 2
2*a11 *r455 + ---*b31 *r4114
4
r430=------------------------------
2
a11
r431=0
b32*r4114
r432=-----------
a11
b31*r4114
r433=-----------
a11
r434=r4114
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
2 1 2 1 2
- 2*a11 *r455 - ---*b31 *r4114 - ---*b32 *r4114
2 2
r446=--------------------------------------------------
2
a11
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
r453=0
r454=0
r456=0
r458=0
r459=0
r460=r455
r461=0
b32*r4114
r462=-----------
a11
r463=0
r464=r4114
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
r478=0
r479=0
r480=0
r481=0
r482=0
2 1 2 1 2
- 2*a11 *r455 - ---*b31 *r4114 - ---*b32 *r4114
2 2
r483=--------------------------------------------------
2
a11
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494= - 2*r455
r495=0
r496=0
b31*r4114
r497=-----------
a11
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=r455
r4106=0
r4107=r455
r4108=0
r4109=0
r4111=0
r4112=0
b31*r4114
r4113=-----------
a11
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
m3=0
1 2 1 2
- a11*b31*m1 + ---*b31 *n3 + ---*b32 *n3
2 2
m2=-------------------------------------------
a11*b32
n2=0
n1=0
1 2 1 2
- ---*b31 - ---*b32
4 4
c33=------------------------
a11
c23=0
c22=0
c13=0
c12=0
b33=0
b22=0
a33=2*a11
b32*r4114
r498=-----------
a11
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:
r4114, r455, m1, n3, b31, b32, a11
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
{{n3,2*a11*b31*m1 - b31 *n3 - b32 *n3,m1},a11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2 2 2
HAM=(u1 *a11 *b32 + u2 *a11 *b32 + 2*u3 *a11 *b32 + u3*v1*a11*b31*b32
2
+ u3*v2*a11*b32 + u3*a11*b32*n3 + v1*a11*b32*m1
1 2 1 2
+ v2*( - a11*b31*m1 + ---*b31 *n3 + ---*b32 *n3)
2 2
2 1 2 1 3
+ v3 *( - ---*b31 *b32 - ---*b32 ))/(a11*b32)
4 4
has apart from the Hamiltonian and Casimirs the following 2 first integrals:
2 2 2 2 2 2 2
FI=u1 *v2 *a11 *b32 + u1 *v3 *a11 *b32 - 2*u1*u2*v1*v2*a11 *b32
2 2 2 2 2 2 2
- 2*u1*u3*v1*v3*a11 *b32 + u2 *v1 *a11 *b32 + u2 *v3 *a11 *b32
2 2 2 2 2 2 2
- 2*u2*u3*v2*v3*a11 *b32 + 2*u3 *v1 *a11 *b32 + 2*u3 *v2 *a11 *b32
2 2 2 3 2 2
+ u3 *v3 *a11 *b32 + u3*v1 *a11*b31*b32 + u3*v1 *v2*a11*b32
2 2 2
+ u3*v1 *a11*b32*n3 + u3*v1*v2 *a11*b31*b32 + u3*v1*v3 *a11*b31*b32
3 2 2 2 2
+ u3*v2 *a11*b32 + u3*v2 *a11*b32*n3 + u3*v2*v3 *a11*b32
2 3
+ u3*v3 *a11*b32*n3 + v1 *a11*b32*m1
2 1 2 1 2
+ v1 *v2*( - a11*b31*m1 + ---*b31 *n3 + ---*b32 *n3)
2 2
2 2 1 2 1 3 2
+ v1 *v3 *( - ---*b31 *b32 - ---*b32 ) + v1*v2 *a11*b32*m1
4 4
2 3 1 2 1 2
+ v1*v3 *a11*b32*m1 + v2 *( - a11*b31*m1 + ---*b31 *n3 + ---*b32 *n3)
2 2
2 2 1 2 1 3
+ v2 *v3 *( - ---*b31 *b32 - ---*b32 )
4 4
2 1 2 1 2
+ v2*v3 *( - a11*b31*m1 + ---*b31 *n3 + ---*b32 *n3)
2 2
4 1 2 1 3
+ v3 *( - ---*b31 *b32 - ---*b32 )
4 4
{HAM,FI} = 0
2 2 4 2 2 3 2 2 3 2
FI=u1 *u3 *a11 *b32 + u1 *u3*v1*a11 *b31*b32 + u1 *u3*a11 *b32 *n3
2 3 2 3 3 3 2
+ u1 *v1*a11 *b32 *m1 + u1*u2*u3*v1*a11 *b32 + u1*u2*u3*v2*a11 *b31*b32
3 1 2 2 1 2 3
+ u1*u2*v1*( - a11 *b31*b32*m1 + ---*a11 *b31 *b32*n3 + ---*a11 *b32 *n3)
2 2
3 2
+ u1*u2*v2*a11 *b32 *m1
1 2 2 2 1 2 4
+ u1*u3*v1*v3*( - ---*a11 *b31 *b32 - ---*a11 *b32 )
2 2
1 2 2
- ---*u1*u3*v3*a11 *b31*b32 *n3
2
1 2 2 1 4
+ u1*v1*v3*( - ---*a11*b31 *b32 *n3 - ---*a11*b32 *n3)
4 4
1 2 2 2 2 4 2 2 3 3
- ---*u1*v3*a11 *b32 *m1*n3 + u2 *u3 *a11 *b32 + u2 *u3*v2*a11 *b32
2
2 3 2
+ u2 *u3*a11 *b32 *n3
2 3 1 2 2 1 2 3
+ u2 *v2*( - a11 *b31*b32*m1 + ---*a11 *b31 *b32*n3 + ---*a11 *b32 *n3)
2 2
1 2 2 2 1 2 4
+ u2*u3*v2*v3*( - ---*a11 *b31 *b32 - ---*a11 *b32 )
2 2
1 2 3
- ---*u2*u3*v3*a11 *b32 *n3
2
1 2 2 1 4
+ u2*v2*v3*( - ---*a11*b31 *b32 *n3 - ---*a11*b32 *n3)
4 4
1 2 1 2 2 1 3 2
+ u2*v3*(---*a11 *b31*b32*m1*n3 - ---*a11*b31 *b32*n3 - ---*a11*b32 *n3 )
2 4 4
4 4 2 3 3 2 3 3 3 3 3 2
+ u3 *a11 *b32 + u3 *v1*a11 *b31*b32 + u3 *v2*a11 *b32 + u3 *a11 *b32 *n3
1 2 2 2 2 2 1 2 2 3
+ ---*u3 *v1 *a11 *b31 *b32 + ---*u3 *v1*v2*a11 *b31*b32
4 2
2 3 2 1 2 2 1 2 2 2 4
+ u3 *v1*(a11 *b32 *m1 + ---*a11 *b31*b32 *n3) + ---*u3 *v2 *a11 *b32
2 4
2 3 1 2 2 2 3
+ u3 *v2*( - a11 *b31*b32*m1 + ---*a11 *b31 *b32*n3 + a11 *b32 *n3)
2
2 2 1 2 2 2 1 2 4 1 2 2 2 2
+ u3 *v3 *( - ---*a11 *b31 *b32 - ---*a11 *b32 ) - ---*u3 *a11 *b32 *n3
4 4 4
1 2 2 2 1 2 2
+ ---*u3*v1 *a11 *b31*b32 *m1 + u3*v1*v2*( - ---*a11 *b31 *b32*m1
2 2
1 2 3 1 3 1 3
+ ---*a11 *b32 *m1 + ---*a11*b31 *b32*n3 + ---*a11*b31*b32 *n3)
2 4 4
1 2 2 1 2 2
+ u3*v1*(---*a11 *b32 *m1*n3 - ---*a11*b31*b32 *n3 )
2 4
2 1 2 2 1 2 2 1 4
+ u3*v2 *( - ---*a11 *b31*b32 *m1 + ---*a11*b31 *b32 *n3 + ---*a11*b32 *n3)
2 4 4
1 2 1 2 2
+ u3*v2*( - ---*a11 *b31*b32*m1*n3 + ---*a11*b31 *b32*n3 )
2 4
1 2 3
- ---*u3*a11*b32 *n3 + v1*v2
4
1 2 2 1 2 1 3
*( - ---*a11 *b31*b32*m1 + ---*a11*b31 *b32*m1*n3 + ---*a11*b32 *m1*n3)
2 4 4
1 2 2 2 1 2 2 2 1 2 2 2
- ---*v1*a11*b32 *m1*n3 + v2 *(---*a11 *b31 *m1 - ---*a11 *b32 *m1
4 4 4
1 3 1 2 1 4 2
- ---*a11*b31 *m1*n3 - ---*a11*b31*b32 *m1*n3 + ----*b31 *n3
4 4 16
1 2 2 2 1 4 2
+ ---*b31 *b32 *n3 + ----*b32 *n3 )
8 16
1 2 1 2 3 1 3 3 2
+ v2*(---*a11*b31*b32*m1*n3 - ---*b31 *b32*n3 - ---*b32 *n3 ) + v3 *(
4 8 8
1 2 2 2 1 3 1 2
---*a11 *b31 *m1 - ---*a11*b31 *m1*n3 - ---*a11*b31*b32 *m1*n3
4 4 4
1 4 2 3 2 2 2 1 4 2
+ ----*b31 *n3 + ----*b31 *b32 *n3 + ---*b32 *n3 )
16 16 8
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a11**2*b32 + u2**2*a11**2*b32 + 2*u3**2*a11**2*b32 + u3*v1*a11*b31*
b32 + u3*v2*a11*b32**2 + u3*a11*b32*n3 + v1*a11*b32*m1 + v2*( - a11*b31*m1 + 1/2
*b31**2*n3 + 1/2*b32**2*n3) + v3**2*( - 1/4*b31**2*b32 - 1/4*b32**3))/(a11*b32)$
FI=u1**2*v2**2*a11**2*b32 + u1**2*v3**2*a11**2*b32 - 2*u1*u2*v1*v2*a11**2*b32 -
2*u1*u3*v1*v3*a11**2*b32 + u2**2*v1**2*a11**2*b32 + u2**2*v3**2*a11**2*b32 - 2*
u2*u3*v2*v3*a11**2*b32 + 2*u3**2*v1**2*a11**2*b32 + 2*u3**2*v2**2*a11**2*b32 +
u3**2*v3**2*a11**2*b32 + u3*v1**3*a11*b31*b32 + u3*v1**2*v2*a11*b32**2 + u3*v1**
2*a11*b32*n3 + u3*v1*v2**2*a11*b31*b32 + u3*v1*v3**2*a11*b31*b32 + u3*v2**3*a11*
b32**2 + u3*v2**2*a11*b32*n3 + u3*v2*v3**2*a11*b32**2 + u3*v3**2*a11*b32*n3 + v1
**3*a11*b32*m1 + v1**2*v2*( - a11*b31*m1 + 1/2*b31**2*n3 + 1/2*b32**2*n3) + v1**
2*v3**2*( - 1/4*b31**2*b32 - 1/4*b32**3) + v1*v2**2*a11*b32*m1 + v1*v3**2*a11*
b32*m1 + v2**3*( - a11*b31*m1 + 1/2*b31**2*n3 + 1/2*b32**2*n3) + v2**2*v3**2*( -
1/4*b31**2*b32 - 1/4*b32**3) + v2*v3**2*( - a11*b31*m1 + 1/2*b31**2*n3 + 1/2*
b32**2*n3) + v3**4*( - 1/4*b31**2*b32 - 1/4*b32**3)$
FI=u1**2*u3**2*a11**4*b32**2 + u1**2*u3*v1*a11**3*b31*b32**2 + u1**2*u3*a11**3*
b32**2*n3 + u1**2*v1*a11**3*b32**2*m1 + u1*u2*u3*v1*a11**3*b32**3 + u1*u2*u3*v2*
a11**3*b31*b32**2 + u1*u2*v1*( - a11**3*b31*b32*m1 + 1/2*a11**2*b31**2*b32*n3 +
1/2*a11**2*b32**3*n3) + u1*u2*v2*a11**3*b32**2*m1 + u1*u3*v1*v3*( - 1/2*a11**2*
b31**2*b32**2 - 1/2*a11**2*b32**4) - 1/2*u1*u3*v3*a11**2*b31*b32**2*n3 + u1*v1*
v3*( - 1/4*a11*b31**2*b32**2*n3 - 1/4*a11*b32**4*n3) - 1/2*u1*v3*a11**2*b32**2*
m1*n3 + u2**2*u3**2*a11**4*b32**2 + u2**2*u3*v2*a11**3*b32**3 + u2**2*u3*a11**3*
b32**2*n3 + u2**2*v2*( - a11**3*b31*b32*m1 + 1/2*a11**2*b31**2*b32*n3 + 1/2*a11
**2*b32**3*n3) + u2*u3*v2*v3*( - 1/2*a11**2*b31**2*b32**2 - 1/2*a11**2*b32**4) -
1/2*u2*u3*v3*a11**2*b32**3*n3 + u2*v2*v3*( - 1/4*a11*b31**2*b32**2*n3 - 1/4*a11
*b32**4*n3) + u2*v3*(1/2*a11**2*b31*b32*m1*n3 - 1/4*a11*b31**2*b32*n3**2 - 1/4*
a11*b32**3*n3**2) + u3**4*a11**4*b32**2 + u3**3*v1*a11**3*b31*b32**2 + u3**3*v2*
a11**3*b32**3 + u3**3*a11**3*b32**2*n3 + 1/4*u3**2*v1**2*a11**2*b31**2*b32**2 +
1/2*u3**2*v1*v2*a11**2*b31*b32**3 + u3**2*v1*(a11**3*b32**2*m1 + 1/2*a11**2*b31*
b32**2*n3) + 1/4*u3**2*v2**2*a11**2*b32**4 + u3**2*v2*( - a11**3*b31*b32*m1 + 1/
2*a11**2*b31**2*b32*n3 + a11**2*b32**3*n3) + u3**2*v3**2*( - 1/4*a11**2*b31**2*
b32**2 - 1/4*a11**2*b32**4) - 1/4*u3**2*a11**2*b32**2*n3**2 + 1/2*u3*v1**2*a11**
2*b31*b32**2*m1 + u3*v1*v2*( - 1/2*a11**2*b31**2*b32*m1 + 1/2*a11**2*b32**3*m1 +
1/4*a11*b31**3*b32*n3 + 1/4*a11*b31*b32**3*n3) + u3*v1*(1/2*a11**2*b32**2*m1*n3
- 1/4*a11*b31*b32**2*n3**2) + u3*v2**2*( - 1/2*a11**2*b31*b32**2*m1 + 1/4*a11*
b31**2*b32**2*n3 + 1/4*a11*b32**4*n3) + u3*v2*( - 1/2*a11**2*b31*b32*m1*n3 + 1/4
*a11*b31**2*b32*n3**2) - 1/4*u3*a11*b32**2*n3**3 + v1*v2*( - 1/2*a11**2*b31*b32*
m1**2 + 1/4*a11*b31**2*b32*m1*n3 + 1/4*a11*b32**3*m1*n3) - 1/4*v1*a11*b32**2*m1*
n3**2 + v2**2*(1/4*a11**2*b31**2*m1**2 - 1/4*a11**2*b32**2*m1**2 - 1/4*a11*b31**
3*m1*n3 - 1/4*a11*b31*b32**2*m1*n3 + 1/16*b31**4*n3**2 + 1/8*b31**2*b32**2*n3**2
+ 1/16*b32**4*n3**2) + v2*(1/4*a11*b31*b32*m1*n3**2 - 1/8*b31**2*b32*n3**3 - 1/
8*b32**3*n3**3) + v3**2*(1/4*a11**2*b31**2*m1**2 - 1/4*a11*b31**3*m1*n3 - 1/4*
a11*b31*b32**2*m1*n3 + 1/16*b31**4*n3**2 + 3/16*b31**2*b32**2*n3**2 + 1/8*b32**4
*n3**2)$