Solution 2 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=4*a23 + a33
Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
1
---*n3*r213
2
r13=-------------
a23
1
---*n2*r213
2
r14=-------------
a23
1
---*n1*r213
2
r15=-------------
a23
r20=0
r21=0
r23=0
r24=0
r25=0
r26=0
r27=0
r28=0
r210=0
r212=0
1
- ---*a33*r213
2
r214=-----------------
a23
r215=0
r216=0
r217=0
r218=0
r219=0
1
- ---*a33*r213
2
r220=-----------------
a23
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
1
---*n3*r450
2
r310=-------------
a23
r311=0
1
---*n3*r450
2
r312=-------------
a23
r313=0
r314=0
1
---*n3*r450
2
r315=-------------
a23
r316=0
r317=0
r318=0
r319=0
1
---*n2*r450
2
r320=-------------
a23
r321=0
1
---*n2*r450
2
r322=-------------
a23
r323=0
r324=0
1
---*n2*r450
2
r325=-------------
a23
r326=0
r327=0
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
1
---*n1*r450
2
r335=-------------
a23
r336=0
1
---*n1*r450
2
r337=-------------
a23
r338=0
r339=0
1
---*n1*r450
2
r340=-------------
a23
r341=0
r342=0
r343=0
r344=0
r345=0
r346=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
r40=0
r41=0
r42=0
r43=0
r45=0
r46=0
r47=0
r48=0
r49=0
r410=0
r411=0
r412=0
r413=0
r414=0
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
1
---*a33*r450
2
r425=--------------
a23
r426=0
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
r435=0
r436=0
r437=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=r450
a33*r450
r446=----------
a23
r447=r450
r448=0
r449=0
r451=0
r453=0
r454=0
1
- ---*a33*r450
2
r455=-----------------
a23
r456=0
r458=0
r459=0
1
- ---*a33*r450
2
r460=-----------------
a23
r461=0
r462=0
r463=0
r464=0
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
a33*r450
r483=----------
a23
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
a33*r450
r494=----------
a23
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
1
- ---*a33*r450
2
r4105=-----------------
a23
r4106=0
1
- ---*a33*r450
2
r4107=-----------------
a23
r4108=0
r4109=0
r4110=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m3=0
m2=0
m1=0
c33=0
c23=0
c22=0
c13=0
c12=0
b33=0
b23=0
b13=0
b12=0
b11=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:
r213, r450, n1, n3, n2, a33, a23
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,r450,a23}
Relevance for the application:
Modulo the following equation:
2 2
0=4*a23 + a33
the system of equations related to the Hamiltonian HAM:
2
HAM=u1*n1 + 2*u2*u3*a23 + u2*n2 + u3 *a33 + u3*n3
has apart from the Hamiltonian and Casimirs the following 2 first integrals:
1 2 2 1 2 2
FI= - ---*u1 *v2 *a33 - ---*u1 *v3 *a33 + u1*u2*v1*v2*a33 + u1*u3*v1*v3*a33
2 2
1 2 1 2 1 2 1 2 2
+ ---*u1*v1 *n1 + ---*u1*v2 *n1 + ---*u1*v3 *n1 - ---*u2 *v1 *a33
2 2 2 2
1 2 2 2 2
- ---*u2 *v3 *a33 + u2*u3*v1 *a23 + u2*u3*v2 *a23 + u2*u3*v2*v3*a33
2
2 1 2 1 2 1 2
+ u2*u3*v3 *a23 + ---*u2*v1 *n2 + ---*u2*v2 *n2 + ---*u2*v3 *n2
2 2 2
1 2 2 1 2 1 2 1 2
+ ---*u3 *v3 *a33 + ---*u3*v1 *n3 + ---*u3*v2 *n3 + ---*u3*v3 *n3
2 2 2 2
{HAM,FI} = 0
1 2 1 1 2 1 1
FI= - ---*u1 *a33 + ---*u1*n1 - ---*u2 *a33 + u2*u3*a23 + ---*u2*n2 + ---*u3*n3
2 2 2 2 2
{HAM,FI} = 0
And again in machine readable form:
HAM=u1*n1 + 2*u2*u3*a23 + u2*n2 + u3**2*a33 + u3*n3$
FI= - 1/2*u1**2*v2**2*a33 - 1/2*u1**2*v3**2*a33 + u1*u2*v1*v2*a33 + u1*u3*v1*v3*
a33 + 1/2*u1*v1**2*n1 + 1/2*u1*v2**2*n1 + 1/2*u1*v3**2*n1 - 1/2*u2**2*v1**2*a33
- 1/2*u2**2*v3**2*a33 + u2*u3*v1**2*a23 + u2*u3*v2**2*a23 + u2*u3*v2*v3*a33 + u2
*u3*v3**2*a23 + 1/2*u2*v1**2*n2 + 1/2*u2*v2**2*n2 + 1/2*u2*v3**2*n2 + 1/2*u3**2*
v3**2*a33 + 1/2*u3*v1**2*n3 + 1/2*u3*v2**2*n3 + 1/2*u3*v3**2*n3$
FI= - 1/2*u1**2*a33 + 1/2*u1*n1 - 1/2*u2**2*a33 + u2*u3*a23 + 1/2*u2*n2 + 1/2*u3
*n3$