Solution 4 to problem over
Expressions |
Parameters |
Inequalities |
Relevance |
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Expressions
The solution is given through the following expressions:
2
b33*n1 *n3*r434
r10=-----------------
4
a11
r11=0
1 2 1 3 1 2
- ---*a11 *b33*n1*r29 + ---*b33*n1 *r434 + ---*b33*n1*n3 *r434
2 2 2
r12=-----------------------------------------------------------------
4
a11
2 2 3
- a11 *n3*r29 - n1 *n3*r434 + n3 *r434
r13=-----------------------------------------
3
a11
r14=0
2 3 2
- a11 *n1*r29 - n1 *r434 + n1*n3 *r434
r15=-----------------------------------------
3
a11
r20=0
r21=0
1 2 2
- ---*b33 *n1 *r434
4
r22=----------------------
4
a11
r23=0
r24=0
2
- b33*n1 *r434
r26=-----------------
3
a11
r27=0
b33*n1*n3*r434
r28=----------------
3
a11
r210=0
r212=0
r213=0
2
- n1 *r434
r214=-------------
2
a11
r215=0
r216=0
2
b33*n1 *r434
r217=--------------
3
a11
2*n1*n3*r434
r218=--------------
2
a11
r219=0
1
---*b33*n3*r483
2
r30=-----------------
2
a11
r31=0
1
---*b33*n3*r483
2
r32=-----------------
2
a11
r33=0
1 1
---*a11*b33*r335 + ---*b33*n1*r483
2 2
r34=------------------------------------
2
a11
r35=0
1 1
---*a11*b33*r335 + ---*b33*n1*r483
2 2
r36=------------------------------------
2
a11
1
---*b33*n3*r483
2
r37=-----------------
2
a11
r38=0
1 1
---*a11*b33*r335 + ---*b33*n1*r483
2 2
r39=------------------------------------
2
a11
n3*r335
r310=---------
n1
r311=0
n3*r335
r312=---------
n1
1
- ---*b33*n1*r488
2
r313=--------------------
2
a11
r314=0
n3*r335
r315=---------
n1
- n3*r488
r316=------------
a11
r317=0
- b33*n1*r434
r318=----------------
2
a11
- 2*n3*r434
r319=--------------
a11
r320=0
r321=0
r322=0
r323=0
1
- ---*b33*n1*r488
2
r324=--------------------
2
a11
r325=0
r326=0
- n3*r488
r327=------------
a11
r328=0
r329=0
r330=0
r331=0
r332=0
r333=0
r334=0
r336=0
r337=r335
r338=0
r339=0
2 1
a11 *r335 - ---*b33*n1*r488
2
r340=-----------------------------
2
a11
- n1*r488
r341=------------
a11
r342=0
- n3*r488
r343=------------
a11
- 2*n1*r434
r344=--------------
a11
r345=0
- n1*r488
r346=------------
a11
r347=0
r348=0
r349=0
r350=0
r351=0
- n1*r488
r352=------------
a11
r353=0
r354=0
r355=0
1 2
---*b33 *r483
8
r40=---------------
2
a11
r41=0
1 2
---*b33 *r483
8
r42=---------------
2
a11
r43=0
r44=0
r45=0
r46=0
r47=0
r48=0
1 2
---*b33 *r483
8
r49=---------------
2
a11
r410=0
r411=0
r412=0
r413=0
1
- ---*b33*r483
2
r415=-----------------
a11
r416=0
1
- ---*b33*r483
2
r417=-----------------
a11
r418=0
r419=0
r420=0
r421=0
1
- ---*b33*r483
2
r422=-----------------
a11
r423=0
r424=0
- a11*r335
r425=-------------
n1
r426=0
1
- a11*r335 - ---*n1*r483
2
r427=---------------------------
n1
r428=0
r429=0
1
- a11*r335 - ---*n1*r483
2
r430=---------------------------
n1
r431=r488
r432=0
r433=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r444=0
r445=0
r446=r483
r447=0
r448=0
r449=0
r450=0
r451=0
r452=r488
r453=0
r454=0
1
r455= - ---*r483
2
r456=0
r458=0
r459=0
1
r460= - ---*r483
2
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
r484=0
r485=0
r486=0
r487=0
r489=0
r490=0
r491=0
r492=0
r493=0
r494=r483
r495=0
r496=0
r497=0
r498=0
r499=0
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
1
r4105= - ---*r483
2
r4106=0
1
r4107= - ---*r483
2
r4108=0
r4109=0
r4111=0
r4112=0
r4113=0
r4114=0
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
- b33*n3
m3=-----------
a11
m2=0
1
- ---*b33*n1
2
m1=---------------
a11
n2=0
1 2
- ---*b33
4
c33=-------------
a11
c23=0
c22=0
c13=0
c12=0
b32=0
b31=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:
r29, r335, r488, r483, r434, n3, b33, n1, 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.
{n1,a11}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2 1
HAM=(u1 *a11 + u1*a11*n1 + u2 *a11 + u3*v3*a11*b33 + u3*a11*n3 - ---*v1*b33*n1
2
1 2 2
- ---*v3 *b33 - v3*b33*n3)/a11
4
has apart from the Hamiltonian and Casimirs the following 5 first integrals:
2 3 2 2
FI= - 2*u1*u3 *a11 *n1 + 2*u1*u3*a11 *n1*n3 + u1*v1*a11*b33*n1
3 2 2 2 2 4 4 3 3
+ u1*( - a11*n1 + a11*n1*n3 ) - u2 *a11 *n1 + u3 *a11 - 2*u3 *a11 *n3
2 2 2
- u3 *v1*a11 *b33*n1 + u3*v1*a11*b33*n1*n3 - u3*v3*a11*b33*n1
2 3 1 3 1 2
+ u3*( - a11*n1 *n3 + a11*n3 ) + v1*(---*b33*n1 + ---*b33*n1*n3 )
2 2
1 2 2 2 2
- ---*v2 *b33 *n1 + v3*b33*n1 *n3
4
{HAM,FI} = 0
1 2 2 2 1 2 2 2 2 2
FI= - ---*u1 *v2 *a11 - ---*u1 *v3 *a11 + u1*u2*v1*v2*a11 + u1*u3*v1*v3*a11
2 2
1 2 2 2 1 2 2 2 2 1 2 2 2
- ---*u2 *v1 *a11 - ---*u2 *v3 *a11 + u2*u3*v2*v3*a11 - ---*u3 *v1 *a11
2 2 2
1 2 2 2 1 2 1 2
- ---*u3 *v2 *a11 - ---*u3*v1 *v3*a11*b33 - ---*u3*v2 *v3*a11*b33
2 2 2
1 3 1 3 1 2 2 2 1 2
- ---*u3*v3 *a11*b33 + ---*v1 *b33*n1 + ---*v1 *v3 *b33 + ---*v1 *v3*b33*n3
2 2 8 2
1 2 1 2 1 2 2 2
+ ---*v1*v2 *b33*n1 + ---*v1*v3 *b33*n1 + ---*v2 *v3 *b33
2 2 8
1 2 1 4 2 1 3
+ ---*v2 *v3*b33*n3 + ---*v3 *b33 + ---*v3 *b33*n3
2 8 2
{HAM,FI} = 0
2 2 2
FI= - u1 *v1*a11*n1 - u1*u2*v2*a11*n1 + u1*u3 *v1*a11 - u1*u3*v1*a11*n3
1 2 2 2
- u1*u3*v3*a11*n1 - ---*u1*v1 *b33*n1 + u2*u3 *v2*a11 - u2*u3*v2*a11*n3
2
1 3 2 2 1
- ---*u2*v1*v2*b33*n1 + u3 *v3*a11 - u3 *v3*a11*n3 - ---*u3*v1*v3*b33*n1
2 2
{HAM,FI} = 0
2 2 2 2 2 2 2 2 2
FI=u1*v1 *a11*n1 + u1*v2 *a11*n1 + u1*v3 *a11*n1 - u3 *v1 *a11 - u3 *v2 *a11
2 2 2 2 2 2
- u3 *v3 *a11 + u3*v1 *a11*n3 + u3*v2 *a11*n3 + u3*v3 *a11*n3
1 3 1 2 1 2
+ ---*v1 *b33*n1 + ---*v1*v2 *b33*n1 + ---*v1*v3 *b33*n1
2 2 2
{HAM,FI} = 0
2 2 1
FI= - u1*a11*n1 + u3 *a11 - u3*a11*n3 - ---*v1*b33*n1
2
{HAM,FI} = 0
And again in machine readable form:
HAM=(u1**2*a11**2 + u1*a11*n1 + u2**2*a11**2 + u3*v3*a11*b33 + u3*a11*n3 - 1/2*
v1*b33*n1 - 1/4*v3**2*b33**2 - v3*b33*n3)/a11$
FI= - 2*u1*u3**2*a11**3*n1 + 2*u1*u3*a11**2*n1*n3 + u1*v1*a11*b33*n1**2 + u1*( -
a11*n1**3 + a11*n1*n3**2) - u2**2*a11**2*n1**2 + u3**4*a11**4 - 2*u3**3*a11**3*
n3 - u3**2*v1*a11**2*b33*n1 + u3*v1*a11*b33*n1*n3 - u3*v3*a11*b33*n1**2 + u3*( -
a11*n1**2*n3 + a11*n3**3) + v1*(1/2*b33*n1**3 + 1/2*b33*n1*n3**2) - 1/4*v2**2*
b33**2*n1**2 + v3*b33*n1**2*n3$
FI= - 1/2*u1**2*v2**2*a11**2 - 1/2*u1**2*v3**2*a11**2 + u1*u2*v1*v2*a11**2 + u1*
u3*v1*v3*a11**2 - 1/2*u2**2*v1**2*a11**2 - 1/2*u2**2*v3**2*a11**2 + u2*u3*v2*v3*
a11**2 - 1/2*u3**2*v1**2*a11**2 - 1/2*u3**2*v2**2*a11**2 - 1/2*u3*v1**2*v3*a11*
b33 - 1/2*u3*v2**2*v3*a11*b33 - 1/2*u3*v3**3*a11*b33 + 1/2*v1**3*b33*n1 + 1/8*v1
**2*v3**2*b33**2 + 1/2*v1**2*v3*b33*n3 + 1/2*v1*v2**2*b33*n1 + 1/2*v1*v3**2*b33*
n1 + 1/8*v2**2*v3**2*b33**2 + 1/2*v2**2*v3*b33*n3 + 1/8*v3**4*b33**2 + 1/2*v3**3
*b33*n3$
FI= - u1**2*v1*a11*n1 - u1*u2*v2*a11*n1 + u1*u3**2*v1*a11**2 - u1*u3*v1*a11*n3 -
u1*u3*v3*a11*n1 - 1/2*u1*v1**2*b33*n1 + u2*u3**2*v2*a11**2 - u2*u3*v2*a11*n3 -
1/2*u2*v1*v2*b33*n1 + u3**3*v3*a11**2 - u3**2*v3*a11*n3 - 1/2*u3*v1*v3*b33*n1$
FI=u1*v1**2*a11*n1 + u1*v2**2*a11*n1 + u1*v3**2*a11*n1 - u3**2*v1**2*a11**2 - u3
**2*v2**2*a11**2 - u3**2*v3**2*a11**2 + u3*v1**2*a11*n3 + u3*v2**2*a11*n3 + u3*
v3**2*a11*n3 + 1/2*v1**3*b33*n1 + 1/2*v1*v2**2*b33*n1 + 1/2*v1*v3**2*b33*n1$
FI= - u1*a11*n1 + u3**2*a11**2 - u3*a11*n3 - 1/2*v1*b33*n1$