Solution 5 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=m1 + m2
Expressions
The solution is given through the following expressions:
1
---*m2*n2*n3*r4114
2
r10=--------------------
3
a11
1 2 2 1 2 2 3 3 2
- ---*m1 *m2*n2 *r4114 - ---*m1 *m2*n3 *r4114 - ---*m2 *n2 *r4114
4 4 4
r11=--------------------------------------------------------------------
3 2
a11 *m1
3 2 2 1 2 2 1 2 2
- ---*m1 *n2 *r4114 - ---*m1 *n3 *r4114 - ---*m2 *n2 *r4114
4 4 4
r12=--------------------------------------------------------------
3
a11 *m1
3 2 2 1 2 3 3 2 2
- ---*m1 *n2 *n3*r4114 - ---*m1 *n3 *r4114 - ---*m2 *n2 *n3*r4114
2 4 2
r13=--------------------------------------------------------------------
3 2
a11 *m1
3 2 3 1 2 2 3 2 3
- ---*m1 *n2 *r4114 - ---*m1 *n2*n3 *r4114 - ---*m2 *n2 *r4114
2 4 2
r14=-----------------------------------------------------------------
3 2
a11 *m1
3 2 3 1 2 2 3 3 3
- ---*m1 *m2*n2 *r4114 - ---*m1 *m2*n2*n3 *r4114 - ---*m2 *n2 *r4114
2 4 2
r15=-----------------------------------------------------------------------
3 3
a11 *m1
1 2
---*m2 *r4114
4
r20=---------------
2
a11
r21=0
1 2 1 2
- ---*m1 *r4114 + ---*m2 *r4114
4 4
r22=----------------------------------
2
a11
r23=0
1
---*m1*m2*r4114
2
r24=-----------------
2
a11
1
---*m2*n2*r4114
2
r26=-----------------
2
a11
1
---*m2*n3*r4114
2
r27=-----------------
2
a11
1
---*m1*n3*r4114
2
r28=-----------------
2
a11
3 2 2 1 2 2 3 2 2
- ---*m1 *n2 *r4114 - ---*m1 *n3 *r4114 - ---*m2 *n2 *r4114
2 4 2
r29=--------------------------------------------------------------
2 2
a11 *m1
1
- ---*m2*n3*r4114
2
r210=--------------------
2
a11
3
---*m1*n2*r4114
2
r212=-----------------
2
a11
r213=0
r214=0
1
- ---*m1*n3*r4114
2
r215=--------------------
2
a11
3 2
---*m2 *n2*r4114
2
r216=------------------
2
a11 *m1
r217=0
r218=0
r219=0
r30=0
m2*r320
r31=---------
n2
r32=0
m2*r320
r33=---------
n2
m1*r320
r34=---------
n2
r35=0
m1*r320
r36=---------
n2
r37=0
m2*r320
r38=---------
n2
m1*r320
r39=---------
n2
n3*r320
r310=---------
n2
r311=0
n3*r320
r312=---------
n2
r313=0
r314=0
n3*r320
r315=---------
n2
r316=0
m2*r4114
r317=----------
a11
m1*r4114
r318=----------
a11
n3*r4114
r319=----------
a11
r321=0
r322=r320
r323=0
r324=0
r325=r320
r326=0
r327=0
r328=0
n2*r4114
r329=----------
a11
r330=0
m2*r4114
r331=----------
a11
r332=0
n3*r4114
r333=----------
a11
n2*r4114
r334=----------
a11
m2*r320
r335=---------
m1
r336=0
m2*r320
r337=---------
m1
r338=0
r339=0
m2*r320
r340=---------
m1
r341=0
r342=0
r343=0
m2*n2*r4114
r344=-------------
a11*m1
r345=0
m1*r4114
r346=----------
a11
m2*r4114
r347=----------
a11
r348=0
m2*n2*r4114
r349=-------------
a11*m1
r350=0
r351=0
m1*r4114
r352=----------
a11
n3*r4114
r353=----------
a11
n2*r4114
r354=----------
a11
m2*n2*r4114
r355=-------------
a11*m1
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
r422=0
r423=0
r424=0
a11*r320
r425=----------
n2
r426=0
2*a11*r320
r427=------------
n2
r428=0
r429=0
2*a11*r320
r430=------------
n2
r431=0
r432=0
r433=0
r434=r4114
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
- 2*a11*r320
r446=---------------
n2
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
r453=0
r454=0
a11*r320
r455=----------
n2
r456=0
r458=0
r459=0
a11*r320
r460=----------
n2
r461=0
r462=0
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*a11*r320
r483=---------------
n2
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
- 2*a11*r320
r494=---------------
n2
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
a11*r320
r4105=----------
n2
r4106=0
a11*r320
r4107=----------
n2
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
m3=0
m2*n2
n1=-------
m1
c33=0
c23=0
c22=0
c13=0
c12=0
b33=0
b32=0
b31=0
b22=0
a33=2*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:
r320, r4114, n3, m1, m2, n2, 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.
{a33,m2,a11,r4114,n2,m1}
Relevance for the application:
Modulo the following equation:
2 2
0=m1 + m2
the system of equations related to the Hamiltonian HAM:
2 2 2
HAM=(u1 *a11*m1 + u1*m2*n2 + u2 *a11*m1 + u2*m1*n2 + 2*u3 *a11*m1 + u3*m1*n3
2
+ v1*m1 + v2*m1*m2)/m1
has apart from the Hamiltonian and Casimirs the following 2 first integrals:
3 2 2 2 2 3 2 2 3 3
FI=u1 *a11 *m1 *m2*n2 + u1 *u2*a11 *m1 *n2 + u1 *u3 *a11 *m1
2 2 3 2 2 4 2 2 2
+ u1 *u3*a11 *m1 *n3 + u1 *v1*a11 *m1 + u1*u2 *a11 *m1 *m2*n2
2 3 2 4 2 2 2
+ u1*u2*v1*a11 *m1 *m2 + u1*u2*v2*a11 *m1 + u1*u3 *a11 *m1 *m2*n2
3 2 2 1 4
+ ---*u1*v2*a11*m1 *m2 *n2 - ---*u1*v3*a11*m1 *n3
2 2
3 2 3 1 2 2 3 3 3 3 2 3
+ u1*( - ---*m1 *m2*n2 - ---*m1 *m2*n2*n3 - ---*m2 *n2 ) + u2 *a11 *m1 *n2
2 4 2
2 2 3 3 2 2 3 2 2 3
+ u2 *u3 *a11 *m1 + u2 *u3*a11 *m1 *n3 + u2 *v2*a11 *m1 *m2
2 2 3 3 4 1 3
+ u2*u3 *a11 *m1 *n2 + ---*u2*v1*a11*m1 *n2 - ---*u2*v3*a11*m1 *m2*n3
2 2
3 3 3 1 3 2 3 2 3 4 3 3
+ u2*( - ---*m1 *n2 - ---*m1 *n2*n3 - ---*m1*m2 *n2 ) + u3 *a11 *m1
2 4 2
3 2 3 2 2 4 2 2 3
+ u3 *a11 *m1 *n3 + u3 *v1*a11 *m1 + u3 *v2*a11 *m1 *m2
2 3 3 2 1 3 2 3 2 2
+ u3 *( - ---*a11*m1 *n2 - ---*a11*m1 *n3 - ---*a11*m1*m2 *n2 )
2 4 2
1 4 1 3 1 3
+ ---*u3*v1*a11*m1 *n3 + ---*u3*v2*a11*m1 *m2*n3 + ---*u3*v3*a11*m1 *m2*n2
2 2 2
3 3 2 1 3 3 3 2 2
+ u3*( - ---*m1 *n2 *n3 - ---*m1 *n3 - ---*m1*m2 *n2 *n3)
2 4 2
1 4 3 4 2 1 4 2 1 2 2 2
+ ---*v1*v2*a11*m1 *m2 + v1*( - ---*m1 *n2 - ---*m1 *n3 - ---*m1 *m2 *n2 )
2 4 4 4
2 1 5 1 3 2
+ v2 *( - ---*a11*m1 + ---*a11*m1 *m2 )
4 4
1 3 2 1 3 2 3 3 2
+ v2*( - ---*m1 *m2*n2 - ---*m1 *m2*n3 - ---*m1*m2 *n2 )
4 4 4
1 2 3 2 1 3
+ ---*v3 *a11*m1 *m2 + ---*v3*m1 *m2*n2*n3
4 2
{HAM,FI} = 0
2 2 2 2
FI=u1 *v2 *a11*m1 + u1 *v3 *a11*m1 - 2*u1*u2*v1*v2*a11*m1 - 2*u1*u3*v1*v3*a11*m1
2 2 2 2 2
+ u1*v1 *m2*n2 + u1*v2 *m2*n2 + u1*v3 *m2*n2 + u2 *v1 *a11*m1
2 2 2 2
+ u2 *v3 *a11*m1 - 2*u2*u3*v2*v3*a11*m1 + u2*v1 *m1*n2 + u2*v2 *m1*n2
2 2 2 2 2 2 2
+ u2*v3 *m1*n2 + 2*u3 *v1 *a11*m1 + 2*u3 *v2 *a11*m1 + u3 *v3 *a11*m1
2 2 2 3 2 2
+ u3*v1 *m1*n3 + u3*v2 *m1*n3 + u3*v3 *m1*n3 + v1 *m1 + v1 *v2*m1*m2
2 2 2 2 3 2
+ v1*v2 *m1 + v1*v3 *m1 + v2 *m1*m2 + v2*v3 *m1*m2
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a11*m1 + u1*m2*n2 + u2**2*a11*m1 + u2*m1*n2 + 2*u3**2*a11*m1 + u3*m1*
n3 + v1*m1**2 + v2*m1*m2)/m1$
FI=u1**3*a11**2*m1**2*m2*n2 + u1**2*u2*a11**2*m1**3*n2 + u1**2*u3**2*a11**3*m1**
3 + u1**2*u3*a11**2*m1**3*n3 + u1**2*v1*a11**2*m1**4 + u1*u2**2*a11**2*m1**2*m2*
n2 + u1*u2*v1*a11**2*m1**3*m2 + u1*u2*v2*a11**2*m1**4 + u1*u3**2*a11**2*m1**2*m2
*n2 + 3/2*u1*v2*a11*m1**2*m2**2*n2 - 1/2*u1*v3*a11*m1**4*n3 + u1*( - 3/2*m1**2*
m2*n2**3 - 1/4*m1**2*m2*n2*n3**2 - 3/2*m2**3*n2**3) + u2**3*a11**2*m1**3*n2 + u2
**2*u3**2*a11**3*m1**3 + u2**2*u3*a11**2*m1**3*n3 + u2**2*v2*a11**2*m1**3*m2 +
u2*u3**2*a11**2*m1**3*n2 + 3/2*u2*v1*a11*m1**4*n2 - 1/2*u2*v3*a11*m1**3*m2*n3 +
u2*( - 3/2*m1**3*n2**3 - 1/4*m1**3*n2*n3**2 - 3/2*m1*m2**2*n2**3) + u3**4*a11**3
*m1**3 + u3**3*a11**2*m1**3*n3 + u3**2*v1*a11**2*m1**4 + u3**2*v2*a11**2*m1**3*
m2 + u3**2*( - 3/2*a11*m1**3*n2**2 - 1/4*a11*m1**3*n3**2 - 3/2*a11*m1*m2**2*n2**
2) + 1/2*u3*v1*a11*m1**4*n3 + 1/2*u3*v2*a11*m1**3*m2*n3 + 1/2*u3*v3*a11*m1**3*m2
*n2 + u3*( - 3/2*m1**3*n2**2*n3 - 1/4*m1**3*n3**3 - 3/2*m1*m2**2*n2**2*n3) + 1/2
*v1*v2*a11*m1**4*m2 + v1*( - 3/4*m1**4*n2**2 - 1/4*m1**4*n3**2 - 1/4*m1**2*m2**2
*n2**2) + v2**2*( - 1/4*a11*m1**5 + 1/4*a11*m1**3*m2**2) + v2*( - 1/4*m1**3*m2*
n2**2 - 1/4*m1**3*m2*n3**2 - 3/4*m1*m2**3*n2**2) + 1/4*v3**2*a11*m1**3*m2**2 + 1
/2*v3*m1**3*m2*n2*n3$
FI=u1**2*v2**2*a11*m1 + u1**2*v3**2*a11*m1 - 2*u1*u2*v1*v2*a11*m1 - 2*u1*u3*v1*
v3*a11*m1 + u1*v1**2*m2*n2 + u1*v2**2*m2*n2 + u1*v3**2*m2*n2 + u2**2*v1**2*a11*
m1 + u2**2*v3**2*a11*m1 - 2*u2*u3*v2*v3*a11*m1 + u2*v1**2*m1*n2 + u2*v2**2*m1*n2
+ u2*v3**2*m1*n2 + 2*u3**2*v1**2*a11*m1 + 2*u3**2*v2**2*a11*m1 + u3**2*v3**2*
a11*m1 + u3*v1**2*m1*n3 + u3*v2**2*m1*n3 + u3*v3**2*m1*n3 + v1**3*m1**2 + v1**2*
v2*m1*m2 + v1*v2**2*m1**2 + v1*v3**2*m1**2 + v2**3*m1*m2 + v2*v3**2*m1*m2$