Solution 1 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
r13=0
r14=0
r15=0
1 2
- ---*m3 *r494
2
r20=-----------------
2
b12
r21=0
1 2
- ---*m3 *r494
2
r22=-----------------
2
b12
r23=0
r24=0
m3*n2*r494
r27=------------
2
b12
r28=0
1 2
- ---*n2 *r494
2
r29=-----------------
2
b12
- m3*n2*r494
r210=---------------
2
b12
r211=0
r212=0
r213=0
1 2
- ---*n2 *r494
2
r214=-----------------
2
b12
r215=0
r217=0
r218=0
r219=0
1 2 1 2
- ---*kap*m3 *r494 - ---*n2 *r494
2 2
r220=------------------------------------
2
b12
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r310=0
r311=0
r312=0
r313=0
- m3*r494
r314=------------
b12
r315=0
- 2*a33*m3*r494
r316=------------------
2
b12
r317=0
n2*r494
r318=---------
b12
r319=0
r320=0
r321=0
r322=0
m3*r494
r323=---------
b12
r324=0
r325=0
r326=0
r327=0
r328=0
- 2*a33*n2*r494
r329=------------------
2
b12
- 2*a33*m3*r494
r330=------------------
2
b12
r331=0
n2*r494
r332=---------
b12
r333=0
- 2*a33*n2*r494
r334=------------------
2
b12
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
- n2*r494
r341=------------
b12
r342=0
- 2*a33*m3*r494
r343=------------------
2
b12
r344=0
r345=0
- n2*r494
r346=------------
b12
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
- 2*a33*n2*r494
r354=------------------
2
b12
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
r423=0
r424=0
r426=0
r427=0
r428=0
r429=0
1
r430= - ---*r494
2
r431=0
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
r445=0
r446=0
r447=0
r448=0
r449=0
r450=0
r451=0
r452=0
2*a33*r494
r453=------------
b12
r454=0
r455=0
r456=0
r457=0
r458=0
r459=0
1
r460= - ---*r494
2
r461=0
r462=0
r463=0
2
- 2*a33 *r494
r464=----------------
2
b12
r465=0
r466=0
2*a33*r494
r467=------------
b12
r468=0
2
- 2*a33 *r494
r469=----------------
2
b12
r470=0
r471=0
r472=0
r473=0
r474=0
r475=0
r476=0
r477=0
r479=0
r480=0
r482=0
r483=r494
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r491=0
r492=0
r493=0
r495=0
- 2*a33*r494
r496=---------------
b12
r497=0
r498=0
r499=0
r4100=0
- 2*a33*r494
r4101=---------------
b12
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
1
r4110=---*r494
2
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4116=0
r4117=0
r4118=0
2
- 2*a33 *r494
r4119=----------------
2
b12
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
n3=0
n1=0
a23=0
a22=2*a33
a13=0
2 1 2
a33 - ---*b12 *kap
4
a11=---------------------
a33
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:
r494, m3, a33, n2, 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.
2 2
{4*a33 + b12 *kap,b12,a33,r494}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 1 2 2 2
HAM=(u1 *(a33 - ---*b12 *kap) + u1*v2*a33*b12 + 2*u2 *a33 - u2*v1*a33*b12
4
2 2
+ u2*a33*n2 + u3 *a33 + v3*a33*m3)/a33
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 1 2 2 2
FI= - 2*u1 *u2 *a33 - 2*u1 *u2*a33*n2 + ---*u1 *v1 *b12
2
2 1 2 1 2 2
+ u1 *( - ---*kap*m3 - ---*n2 ) - 2*u1*u2 *v2*a33*b12
2 2
2
- 2*u1*u2*u3*v3*a33*b12 + u1*u2*v1*v2*b12 - u1*u2*v2*b12*n2
2 4 2
+ u1*u3*v1*v3*b12 - 2*u1*u3*v1*a33*m3 - u1*u3*v3*b12*n2 - 2*u2 *a33
3 3 2 2 2 1 2 2 2
+ 2*u2 *v1*a33*b12 - 2*u2 *a33*n2 - 2*u2 *u3 *a33 - ---*u2 *v1 *b12
2
2 2 1 2 2 2
+ u2 *v1*b12*n2 - 2*u2 *v3*a33*m3 - ---*u2 *n2 + 2*u2*u3 *v1*a33*b12
2
2 1 2 2 2
- 2*u2*u3 *a33*n2 + u2*v1*v3*b12*m3 - u2*v3*m3*n2 - ---*u3 *v1 *b12
2
2 2 1 2 2
+ u3 *v1*b12*n2 - 2*u3 *v3*a33*m3 - ---*u3 *n2 - u3*v1*v2*b12*m3
2
1 2 2 1 2 2
+ u3*v2*m3*n2 - ---*v2 *m3 - ---*v3 *m3
2 2
1
= a product of the elements of: { - ---,
2
2 2 2 2 2 2 2 2 2 2
4*u1 *u2 *a33 + 4*u1 *u2*a33*n2 - u1 *v1 *b12 + u1 *(kap*m3 + n2 )
2 2
+ 4*u1*u2 *v2*a33*b12 + 4*u1*u2*u3*v3*a33*b12 - 2*u1*u2*v1*v2*b12
2
+ 2*u1*u2*v2*b12*n2 - 2*u1*u3*v1*v3*b12 + 4*u1*u3*v1*a33*m3
4 2 3 3
+ 2*u1*u3*v3*b12*n2 + 4*u2 *a33 - 4*u2 *v1*a33*b12 + 4*u2 *a33*n2
2 2 2 2 2 2 2 2
+ 4*u2 *u3 *a33 + u2 *v1 *b12 - 2*u2 *v1*b12*n2 + 4*u2 *v3*a33*m3
2 2 2 2
+ u2 *n2 - 4*u2*u3 *v1*a33*b12 + 4*u2*u3 *a33*n2 - 2*u2*v1*v3*b12*m3
2 2 2 2 2 2 2
+ 2*u2*v3*m3*n2 + u3 *v1 *b12 - 2*u3 *v1*b12*n2 + 4*u3 *v3*a33*m3 + u3 *n2
2 2 2 2
+ 2*u3*v1*v2*b12*m3 - 2*u3*v2*m3*n2 + v2 *m3 + v3 *m3 }
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*(a33**2 - 1/4*b12**2*kap) + u1*v2*a33*b12 + 2*u2**2*a33**2 - u2*v1*
a33*b12 + u2*a33*n2 + u3**2*a33**2 + v3*a33*m3)/a33$
FI= - 2*u1**2*u2**2*a33**2 - 2*u1**2*u2*a33*n2 + 1/2*u1**2*v1**2*b12**2 + u1**2*
( - 1/2*kap*m3**2 - 1/2*n2**2) - 2*u1*u2**2*v2*a33*b12 - 2*u1*u2*u3*v3*a33*b12 +
u1*u2*v1*v2*b12**2 - u1*u2*v2*b12*n2 + u1*u3*v1*v3*b12**2 - 2*u1*u3*v1*a33*m3 -
u1*u3*v3*b12*n2 - 2*u2**4*a33**2 + 2*u2**3*v1*a33*b12 - 2*u2**3*a33*n2 - 2*u2**
2*u3**2*a33**2 - 1/2*u2**2*v1**2*b12**2 + u2**2*v1*b12*n2 - 2*u2**2*v3*a33*m3 -
1/2*u2**2*n2**2 + 2*u2*u3**2*v1*a33*b12 - 2*u2*u3**2*a33*n2 + u2*v1*v3*b12*m3 -
u2*v3*m3*n2 - 1/2*u3**2*v1**2*b12**2 + u3**2*v1*b12*n2 - 2*u3**2*v3*a33*m3 - 1/2
*u3**2*n2**2 - u3*v1*v2*b12*m3 + u3*v2*m3*n2 - 1/2*v2**2*m3**2 - 1/2*v3**2*m3**2
$