Solution 6 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
r10=0
1 2
- ---*m2*n3 *r4114
4
r11=---------------------
3
a11
1 2
- ---*m1*n3 *r4114
4
r12=---------------------
3
a11
1 3
- ---*n3 *r4114
4
r13=------------------
3
a11
r14=0
r15=0
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
r26=0
1
---*m2*n3*r4114
2
r27=-----------------
2
a11
1
---*m1*n3*r4114
2
r28=-----------------
2
a11
1 2
- ---*n3 *r4114
4
r29=------------------
2
a11
1
- ---*m2*n3*r4114
2
r210=--------------------
2
a11
r212=0
r213=0
r214=0
1
- ---*m1*n3*r4114
2
r215=--------------------
2
a11
r216=0
r217=0
r218=0
r219=0
r30=0
m2*r455
r31=---------
a11
r32=0
m2*r455
r33=---------
a11
m1*r455
r34=---------
a11
r35=0
m1*r455
r36=---------
a11
r37=0
m2*r455
r38=---------
a11
m1*r455
r39=---------
a11
n3*r455
r310=---------
a11
r311=0
n3*r455
r312=---------
a11
r313=0
r314=0
n3*r455
r315=---------
a11
r316=0
m2*r4114
r317=----------
a11
m1*r4114
r318=----------
a11
n3*r4114
r319=----------
a11
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
r326=0
r327=0
r328=0
r329=0
r330=0
m2*r4114
r331=----------
a11
r332=0
n3*r4114
r333=----------
a11
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r341=0
r342=0
r343=0
r344=0
r345=0
m1*r4114
r346=----------
a11
m2*r4114
r347=----------
a11
r348=0
r349=0
r350=0
r351=0
m1*r4114
r352=----------
a11
n3*r4114
r353=----------
a11
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
r422=0
r423=0
r424=0
r425=r455
r426=0
r427=2*r455
r428=0
r429=0
r430=2*r455
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
r446= - 2*r455
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
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
r483= - 2*r455
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
r497=0
r498=0
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
r4113=0
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
m3=0
n2=0
n1=0
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:
r455, r4114, m1, n3, m2, 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.
{m2,a11,r4114}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2
HAM=u1 *a11 + u2 *a11 + 2*u3 *a11 + u3*n3 + v1*m1 + v2*m2
has apart from the Hamiltonian and Casimirs the following 2 first integrals:
2 2 3 2 2 2 2 2
FI=u1 *u3 *a11 + u1 *u3*a11 *n3 + u1 *v1*a11 *m1 + u1*u2*v1*a11 *m2
2 1 2 2 3 2 2
+ u1*u2*v2*a11 *m1 - ---*u1*v3*a11*m1*n3 + u2 *u3 *a11 + u2 *u3*a11 *n3
2
2 2 1 4 3 3 2
+ u2 *v2*a11 *m2 - ---*u2*v3*a11*m2*n3 + u3 *a11 + u3 *a11 *n3
2
2 2 2 2 1 2 2 1
+ u3 *v1*a11 *m1 + u3 *v2*a11 *m2 - ---*u3 *a11*n3 + ---*u3*v1*a11*m1*n3
4 2
1 1 3 1 1 2
+ ---*u3*v2*a11*m2*n3 - ---*u3*n3 + ---*v1*v2*a11*m1*m2 - ---*v1*m1*n3
2 4 2 4
2 1 2 1 2 1 2 1 2 2
+ v2 *( - ---*a11*m1 + ---*a11*m2 ) - ---*v2*m2*n3 + ---*v3 *a11*m2
4 4 4 4
{HAM,FI} = 0
2 2 2 2
FI=u1 *v2 *a11 + u1 *v3 *a11 - 2*u1*u2*v1*v2*a11 - 2*u1*u3*v1*v3*a11
2 2 2 2 2 2
+ u2 *v1 *a11 + u2 *v3 *a11 - 2*u2*u3*v2*v3*a11 + 2*u3 *v1 *a11
2 2 2 2 2 2 2 3
+ 2*u3 *v2 *a11 + u3 *v3 *a11 + u3*v1 *n3 + u3*v2 *n3 + u3*v3 *n3 + v1 *m1
2 2 2 3 2
+ v1 *v2*m2 + v1*v2 *m1 + v1*v3 *m1 + v2 *m2 + v2*v3 *m2
{HAM,FI} = 0
And again in machine readable form:
HAM=u1**2*a11 + u2**2*a11 + 2*u3**2*a11 + u3*n3 + v1*m1 + v2*m2$
FI=u1**2*u3**2*a11**3 + u1**2*u3*a11**2*n3 + u1**2*v1*a11**2*m1 + u1*u2*v1*a11**
2*m2 + u1*u2*v2*a11**2*m1 - 1/2*u1*v3*a11*m1*n3 + u2**2*u3**2*a11**3 + u2**2*u3*
a11**2*n3 + u2**2*v2*a11**2*m2 - 1/2*u2*v3*a11*m2*n3 + u3**4*a11**3 + u3**3*a11
**2*n3 + u3**2*v1*a11**2*m1 + u3**2*v2*a11**2*m2 - 1/4*u3**2*a11*n3**2 + 1/2*u3*
v1*a11*m1*n3 + 1/2*u3*v2*a11*m2*n3 - 1/4*u3*n3**3 + 1/2*v1*v2*a11*m1*m2 - 1/4*v1
*m1*n3**2 + v2**2*( - 1/4*a11*m1**2 + 1/4*a11*m2**2) - 1/4*v2*m2*n3**2 + 1/4*v3
**2*a11*m2**2$
FI=u1**2*v2**2*a11 + u1**2*v3**2*a11 - 2*u1*u2*v1*v2*a11 - 2*u1*u3*v1*v3*a11 +
u2**2*v1**2*a11 + u2**2*v3**2*a11 - 2*u2*u3*v2*v3*a11 + 2*u3**2*v1**2*a11 + 2*u3
**2*v2**2*a11 + u3**2*v3**2*a11 + u3*v1**2*n3 + u3*v2**2*n3 + u3*v3**2*n3 + v1**
3*m1 + v1**2*v2*m2 + v1*v2**2*m1 + v1*v3**2*m1 + v2**3*m2 + v2*v3**2*m2$