Solution 4 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r10=0
1 2
- ---*m2*n3 *r464
4
r11=--------------------
3
a22
1 2
- ---*m1*n3 *r464
4
r12=--------------------
3
a22
1 3
- ---*n3 *r464
4
r13=-----------------
3
a22
r14=0
r15=0
1 2 1 2
---*a22*m1 *r464 + ---*c22*n3 *r464
4 8
r20=-------------------------------------
3
a22
r21=0
r23=0
1 1 2
---*a22*m1*m2*r464 - ---*c12*n3 *r464
2 2
r24=---------------------------------------
3
a22
1 2 1 2 1 2
---*a22*m1 *r464 - ---*a22*m2 *r464 + ---*c22*n3 *r464
4 4 4
r25=--------------------------------------------------------
3
a22
r26=0
1
---*m2*n3*r464
2
r27=----------------
2
a22
1
---*m1*n3*r464
2
r28=----------------
2
a22
1 2
- ---*n3 *r464
4
r29=-----------------
2
a22
1
- ---*m2*n3*r464
2
r210=-------------------
2
a22
r212=0
r213=0
r214=0
1
- ---*m1*n3*r464
2
r215=-------------------
2
a22
r216=0
r217=0
r218=0
r219=0
r30=0
1
---*c12*m1*r464
2
r31=-----------------
2
a22
r32=0
r33=0
1 1
---*c12*m2*r464 - ---*c22*m1*r464
2 2
r34=-----------------------------------
2
a22
r35=0
c12*m2*r464
r36=-------------
2
a22
r37=0
1
c12*m1*r464 - ---*c22*m2*r464
2
r38=-------------------------------
2
a22
1
- ---*c22*m1*r464
2
r39=--------------------
2
a22
r310=0
r311=0
r312=0
r313=0
c12*n3*r464
r314=-------------
2
a22
1
- ---*c22*n3*r464
2
r315=--------------------
2
a22
r316=0
m2*r464
r317=---------
a22
m1*r464
r318=---------
a22
n3*r464
r319=---------
a22
r320=0
1
- ---*c12*n3*r464
2
r323=--------------------
2
a22
r325=0
- m2*r464
r326=------------
a22
r328=0
r329=0
r330=0
r332=0
n3*r464
r333=---------
a22
r334=0
r335=0
1
- ---*c12*n3*r464
2
r336=--------------------
2
a22
r337=0
1
---*c22*n3*r464
2
r338=-----------------
2
a22
r339=0
r340=0
- m1*r464
r341=------------
a22
r342=0
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
n3*r464
r353=---------
a22
r354=0
r355=0
1 2
- ---*c12 *r464
4
r40=------------------
2
a22
r41=0
r42=0
r43=0
r45=0
1
- ---*c12*c22*r464
2
r46=---------------------
2
a22
r47=0
r48=0
1 2
---*c22 *r464
4
r49=---------------
2
a22
r410=0
2
c12 *r464
r411=-----------
2
a22
r412=0
- c12*c22*r464
r413=-----------------
2
a22
1 2
---*c22 *r464
4
r414=---------------
2
a22
r415=0
r416=0
r417=0
r418=0
r419=0
r420=0
r421=0
r422=0
r423=0
r424=0
1
- ---*c22*r464
2
r425=-----------------
a22
r426=0
r427=0
r428=0
2*c12*r464
r429=------------
a22
- c22*r464
r430=-------------
a22
r431=0
r432=0
r433=0
r434=r464
r435=0
r439=0
r442=0
r444=0
r445=0
- c12*r464
r448=-------------
a22
r450=0
r451=0
r453=0
r454=0
1
- ---*c22*r464
2
r455=-----------------
a22
r458=0
1
- ---*c22*r464
2
r460=-----------------
a22
r461=0
r463=0
r465=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=0
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
- c12*r464
r490=-------------
a22
r493=0
- c12*r464
r495=-------------
a22
r496=0
r498=0
r499=0
r4100=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
c12*r464
r4109=----------
a22
1
- ---*c22*r464
2
r4110=-----------------
a22
r4111=0
r4112=0
r4113=0
r4114=r464
r4115=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
m3=0
n2=0
n1=0
1
c33=---*c22
2
c23=0
c13=0
c11=0
b33=0
b32=0
b31=0
b23=0
b22=0
b21=0
b13=0
b12=0
b11=0
a33=2*a22
a23=0
a13=0
a12=0
a11=a22
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:
r464, m1, m2, c22, n3, c12, a22
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.
{r464,a22,a11}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11 - a22,
a12,
a13,
a23,
- 2*a22 + a33,
b11,
b12,
b13,
b21,
b22,
b23,
b31,
b32,
b33,
c11,
c13,
c23,
- 1/2*c22 + c33,
n1,
n2,
m3}$
The system of equations related to the Hamiltonian HAM:
2 2 2 2
HAM=u1 *a22 + u2 *a22 + 2*u3 *a22 + u3*n3 + 2*v1*v2*c12 + v1*m1 + v2 *c22
1 2
+ v2*m2 + ---*v3 *c22
2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 3 2 2 1 2 2 2 2 2
FI=u1 *u3 *a22 + u1 *u3*a22 *n3 - ---*u1 *v1 *a22 *c22 + u1 *v1*v2*a22 *c12
2
2 2 2 2 2
- u1*u2*v1 *a22 *c12 - u1*u2*v3 *a22 *c12 - u1*u3*v3*a22 *m1
1 1 1
+ ---*u1*v1*v3*a22*c22*n3 - ---*u1*v2*v3*a22*c12*n3 - ---*u1*v3*a22*m1*n3
2 2 2
2 2 3 2 2 1 2 2 2
+ u2 *u3 *a22 + u2 *u3*a22 *n3 - ---*u2 *v1 *a22 *c22
2
1 2 2 2 2 2
- ---*u2 *v3 *a22 *c22 - u2*u3*v1*v3*a22 *c12 - u2*u3*v3*a22 *m2
2
1 1 4 3 3 2
- ---*u2*v1*v3*a22*c12*n3 - ---*u2*v3*a22*m2*n3 + u3 *a22 + u3 *a22 *n3
2 2
2 2 2 2 2 2 2 2 2
- u3 *v1 *a22 *c22 + 2*u3 *v1*v2*a22 *c12 + u3 *v1*a22 *m1 + u3 *v2*a22 *m2
1 2 2 2 1 2 2 1 2
- ---*u3 *v3 *a22 *c22 - ---*u3 *a22*n3 - ---*u3*v1 *a22*c22*n3
2 4 2
1 1
+ u3*v1*v2*a22*c12*n3 + ---*u3*v1*a22*m1*n3 + ---*u3*v2*a22*m2*n3
2 2
1 3 1 4 2 3 1 3
- ---*u3*n3 + ---*v1 *a22*c22 - v1 *v2*a22*c12*c22 - ---*v1 *a22*c22*m1
4 4 2
2 2 2 2 1
+ v1 *v2 *a22*c12 + v1 *v2*(a22*c12*m1 - ---*a22*c22*m2)
2
1 2 2 2 2 1 2 1 2 1 2
+ ---*v1 *v3 *a22*c22 + v1 *(---*a22*m1 - ---*a22*m2 + ---*c22*n3 )
4 4 4 4
2 1 2
+ v1*v2 *a22*c12*m2 - ---*v1*v2*v3 *a22*c12*c22
2
1 1 2
+ v1*v2*(---*a22*m1*m2 - ---*c12*n3 )
2 2
2 1 1 1 2
+ v1*v3 *(---*a22*c12*m2 - ---*a22*c22*m1) - ---*v1*m1*n3
2 2 4
1 2 1 2 1 4 2
+ ---*v2*v3 *a22*c12*m1 - ---*v2*m2*n3 - ---*v3 *a22*c12
2 4 4
2 1 2 1 2
+ v3 *(---*a22*m1 + ---*c22*n3 )
4 8
which the program can not factorize further.
{HAM,FI} = {2,
a22,
a22,
u1*v1 + u2*v2 + u3*v3,
2
u1 *v3*a22*c12 + u1*u2*v3*a22*c22 + u1*u3*v1*a22*c12 + u1*u3*a22*m2
1
+ u1*v1*c12*n3 + ---*u1*m2*n3 - 2*u2*u3*v2*a22*c12 - u2*u3*a22*m1
2
1 1
+ ---*u2*v1*c22*n3 - u2*v2*c12*n3 - ---*u2*m1*n3
2 2
1 2 2 1
+ ---*v1 *v3*c12*c22 - v1*v2*v3*c12 - ---*v1*v3*c12*m1}
2 2
And again in machine readable form:
HAM=u1**2*a22 + u2**2*a22 + 2*u3**2*a22 + u3*n3 + 2*v1*v2*c12 + v1*m1 + v2**2*
c22 + v2*m2 + 1/2*v3**2*c22$
FI=u1**2*u3**2*a22**3 + u1**2*u3*a22**2*n3 - 1/2*u1**2*v1**2*a22**2*c22 + u1**2*
v1*v2*a22**2*c12 - u1*u2*v1**2*a22**2*c12 - u1*u2*v3**2*a22**2*c12 - u1*u3*v3*
a22**2*m1 + 1/2*u1*v1*v3*a22*c22*n3 - 1/2*u1*v2*v3*a22*c12*n3 - 1/2*u1*v3*a22*m1
*n3 + u2**2*u3**2*a22**3 + u2**2*u3*a22**2*n3 - 1/2*u2**2*v1**2*a22**2*c22 - 1/2
*u2**2*v3**2*a22**2*c22 - u2*u3*v1*v3*a22**2*c12 - u2*u3*v3*a22**2*m2 - 1/2*u2*
v1*v3*a22*c12*n3 - 1/2*u2*v3*a22*m2*n3 + u3**4*a22**3 + u3**3*a22**2*n3 - u3**2*
v1**2*a22**2*c22 + 2*u3**2*v1*v2*a22**2*c12 + u3**2*v1*a22**2*m1 + u3**2*v2*a22
**2*m2 - 1/2*u3**2*v3**2*a22**2*c22 - 1/4*u3**2*a22*n3**2 - 1/2*u3*v1**2*a22*c22
*n3 + u3*v1*v2*a22*c12*n3 + 1/2*u3*v1*a22*m1*n3 + 1/2*u3*v2*a22*m2*n3 - 1/4*u3*
n3**3 + 1/4*v1**4*a22*c22**2 - v1**3*v2*a22*c12*c22 - 1/2*v1**3*a22*c22*m1 + v1
**2*v2**2*a22*c12**2 + v1**2*v2*(a22*c12*m1 - 1/2*a22*c22*m2) + 1/4*v1**2*v3**2*
a22*c22**2 + v1**2*(1/4*a22*m1**2 - 1/4*a22*m2**2 + 1/4*c22*n3**2) + v1*v2**2*
a22*c12*m2 - 1/2*v1*v2*v3**2*a22*c12*c22 + v1*v2*(1/2*a22*m1*m2 - 1/2*c12*n3**2)
+ v1*v3**2*(1/2*a22*c12*m2 - 1/2*a22*c22*m1) - 1/4*v1*m1*n3**2 + 1/2*v2*v3**2*
a22*c12*m1 - 1/4*v2*m2*n3**2 - 1/4*v3**4*a22*c12**2 + v3**2*(1/4*a22*m1**2 + 1/8
*c22*n3**2)$