Solution 3 to problem over
Expressions |
Parameters |
Inequalities |
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Expressions
The solution is given through the following expressions:
r10=0
r11=0
r12=0
r13=0
r14=0
r15=0
r20=0
r21=0
r22=0
r23=0
r24=0
r26=0
r27=0
r28=0
r29=0
r210=0
r212=0
r213=0
r214=0
r215=0
r216=0
r217=0
r218=0
r219=0
1
---*m3*r483
2
r30=-------------
a22
1
---*m2*r483
2
r31=-------------
a22
1
---*m3*r483
2
r32=-------------
a22
1
---*m2*r483
2
r33=-------------
a22
1
---*m1*r483
2
r34=-------------
a22
r35=0
1
---*m1*r483
2
r36=-------------
a22
1
---*m3*r483
2
r37=-------------
a22
1
---*m2*r483
2
r38=-------------
a22
1
---*m1*r483
2
r39=-------------
a22
1
---*n3*r483
2
r310=-------------
a22
1
- ---*n2*r483
2
r311=----------------
a22
1
---*n3*r483
2
r312=-------------
a22
r313=0
r314=0
1
---*n3*r483
2
r315=-------------
a22
r316=0
r317=0
r318=0
r319=0
1
---*n2*r483
2
r320=-------------
a22
r323=0
1
---*n2*r483
2
r325=-------------
a22
r326=0
r328=0
r329=0
r330=0
r332=0
r333=0
r334=0
1
---*n1*r483
2
r335=-------------
a22
r336=0
1
---*n1*r483
2
r337=-------------
a22
r338=0
1
- ---*n2*r483
2
r339=----------------
a22
1
---*n1*r483
2
r340=-------------
a22
r341=0
r342=0
r343=0
r344=0
r345=0
r347=0
r348=0
r349=0
r350=0
r351=0
r352=0
r353=0
r354=0
r355=0
1
---*c33*r483
2
r40=--------------
a22
c23*r483
r41=----------
a22
1 1
---*c22*r483 + ---*c33*r483
2 2
r42=-----------------------------
a22
c23*r483
r43=----------
a22
1
---*c22*r483
2
r44=--------------
a22
c13*r483
r45=----------
a22
c12*r483
r46=----------
a22
c13*r483
r47=----------
a22
c12*r483
r48=----------
a22
1
---*c33*r483
2
r49=--------------
a22
c23*r483
r410=----------
a22
1
---*c22*r483
2
r411=--------------
a22
c13*r483
r412=----------
a22
c12*r483
r413=----------
a22
1
---*b33*r483
2
r415=--------------
a22
1
---*b32*r483
2
r416=--------------
a22
1
---*b33*r483
2
r417=--------------
a22
1
---*b32*r483
2
r418=--------------
a22
1
---*b31*r483
2
r419=--------------
a22
1
- ---*b21*r483
2
r420=-----------------
a22
1
---*b31*r483
2
r421=--------------
a22
1
---*b33*r483
2
r422=--------------
a22
1
---*b32*r483
2
r423=--------------
a22
1
---*b31*r483
2
r424=--------------
a22
1
r425=---*r483
2
r426=0
r427=0
r428=0
r429=0
r430=0
r431=0
r432=0
r433=0
r434=0
r435=0
1
---*b21*r483
2
r439=--------------
a22
r442=0
1
---*b21*r483
2
r444=--------------
a22
r445=0
r448=0
r450=0
r451=0
r453=0
r454=0
1
r455=---*r483
2
r458=0
1
r460=---*r483
2
r461=0
r463=0
r464=0
r465=0
r467=0
r468=0
r469=0
r470=0
r471=0
r472=0
r473=0
1
---*b11*r483
2
r474=--------------
a22
r475=0
1
---*b11*r483
2
r476=--------------
a22
r477=0
1
- ---*b21*r483
2
r478=-----------------
a22
1
---*b11*r483
2
r479=--------------
a22
r480=0
r481=0
r482=0
r484=0
r485=0
r486=0
r487=0
r488=0
r489=0
r490=0
r493=0
r495=0
r496=0
r498=0
r499=0
r4100=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
c11=0
b23=0
b22=0
b13=0
b12=0
a33=0
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:
r483, m3, m1, c33, m2, c22, n3, n1, n2, b21, b33, b11,
c12, c23, c13, b32, b31, 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.
{r483,a22}
Relevance for the application:
The new Hamiltonian in form of a list of vanishing expressions:
{a11 + a22,
a12,
a13,
a23,
a33,
b12,
b13,
b22,
b23,
c11}$
The system of equations related to the Hamiltonian HAM:
2 2
HAM= - u1 *a22 + u1*v1*b11 + u1*n1 + u2 *a22 + u2*v1*b21 + u2*n2 + u3*v1*b31
+ u3*v2*b32 + u3*v3*b33 + u3*n3 + 2*v1*v2*c12 + 2*v1*v3*c13 + v1*m1
2 2
+ v2 *c22 + 2*v2*v3*c23 + v2*m2 + v3 *c33 + v3*m3
has apart from the Hamiltonian and Casimirs only the following first integral:
1 2 2 1 2 2 1 3
FI= - ---*u1 *v2 *a22 - ---*u1 *v3 *a22 + u1*u3*v1*v3*a22 + ---*u1*v1 *b11
2 2 2
1 2 1 2 1 2 1
- ---*u1*v1 *v2*b21 + ---*u1*v1 *n1 + ---*u1*v1*v2 *b11 - ---*u1*v1*v2*n2
2 2 2 2
1 2 1 2 1 2 1 2 2
+ ---*u1*v1*v3 *b11 + ---*u1*v2 *n1 + ---*u1*v3 *n1 + ---*u2 *v1 *a22
2 2 2 2
1 2 2 1 3 1 2 1 2
+ ---*u2 *v3 *a22 + ---*u2*v1 *b21 + ---*u2*v1 *n2 + ---*u2*v1*v3 *b21
2 2 2 2
1 2 1 2 2 1 3 1 2
+ ---*u2*v3 *n2 + ---*u3 *v3 *a22 + ---*u3*v1 *b31 + ---*u3*v1 *v2*b32
2 2 2 2
1 2 1 2 1 2
+ ---*u3*v1 *v3*b33 + ---*u3*v1 *n3 + ---*u3*v1*v2 *b31
2 2 2
1 1 2 1 3
- ---*u3*v1*v2*v3*b21 + ---*u3*v1*v3 *b31 + ---*u3*v2 *b32
2 2 2
1 2 1 2 1 2 1
+ ---*u3*v2 *v3*b33 + ---*u3*v2 *n3 + ---*u3*v2*v3 *b32 - ---*u3*v2*v3*n2
2 2 2 2
1 3 1 2 3 3 1 3
+ ---*u3*v3 *b33 + ---*u3*v3 *n3 + v1 *v2*c12 + v1 *v3*c13 + ---*v1 *m1
2 2 2
1 2 2 2 1 2 1 2 2
+ ---*v1 *v2 *c22 + v1 *v2*v3*c23 + ---*v1 *v2*m2 + ---*v1 *v3 *c33
2 2 2
1 2 3 2 1 2 2
+ ---*v1 *v3*m3 + v1*v2 *c12 + v1*v2 *v3*c13 + ---*v1*v2 *m1 + v1*v2*v3 *c12
2 2
3 1 2 1 4 3 1 3
+ v1*v3 *c13 + ---*v1*v3 *m1 + ---*v2 *c22 + v2 *v3*c23 + ---*v2 *m2
2 2 2
2 2 1 1 1 2 3 1 2
+ v2 *v3 *(---*c22 + ---*c33) + ---*v2 *v3*m3 + v2*v3 *c23 + ---*v2*v3 *m2
2 2 2 2
1 4 1 3
+ ---*v3 *c33 + ---*v3 *m3
2 2
1
= a product of the elements of: { - ---,
2
2 2 2 2 3 2
u1 *v2 *a22 + u1 *v3 *a22 - 2*u1*u3*v1*v3*a22 - u1*v1 *b11 + u1*v1 *v2*b21
2 2 2 2
- u1*v1 *n1 - u1*v1*v2 *b11 + u1*v1*v2*n2 - u1*v1*v3 *b11 - u1*v2 *n1
2 2 2 2 2 3 2
- u1*v3 *n1 - u2 *v1 *a22 - u2 *v3 *a22 - u2*v1 *b21 - u2*v1 *n2
2 2 2 2 3 2
- u2*v1*v3 *b21 - u2*v3 *n2 - u3 *v3 *a22 - u3*v1 *b31 - u3*v1 *v2*b32
2 2 2
- u3*v1 *v3*b33 - u3*v1 *n3 - u3*v1*v2 *b31 + u3*v1*v2*v3*b21
2 3 2 2 2
- u3*v1*v3 *b31 - u3*v2 *b32 - u3*v2 *v3*b33 - u3*v2 *n3 - u3*v2*v3 *b32
3 2 3 3
+ u3*v2*v3*n2 - u3*v3 *b33 - u3*v3 *n3 - 2*v1 *v2*c12 - 2*v1 *v3*c13
3 2 2 2 2 2 2
- v1 *m1 - v1 *v2 *c22 - 2*v1 *v2*v3*c23 - v1 *v2*m2 - v1 *v3 *c33
2 3 2 2 2
- v1 *v3*m3 - 2*v1*v2 *c12 - 2*v1*v2 *v3*c13 - v1*v2 *m1 - 2*v1*v2*v3 *c12
3 2 4 3 3
- 2*v1*v3 *c13 - v1*v3 *m1 - v2 *c22 - 2*v2 *v3*c23 - v2 *m2
2 2 2 3 2 4
+ v2 *v3 *( - c22 - c33) - v2 *v3*m3 - 2*v2*v3 *c23 - v2*v3 *m2 - v3 *c33
3
- v3 *m3}
{HAM,FI} = {2,
u1*v1 + u2*v2 + u3*v3,
2 2 1
u1*u2*v3*a22 + u1*u3*v2*a22 + ---*u1*v1*v2*a22*b31
2
1 1 2
+ ---*u1*v1*v3*a22*b21 + ---*u1*v2 *a22*b32
2 2
1 1 1
+ u1*v2*v3*( - ---*a22*b11 + ---*a22*b33) + ---*u1*v2*a22*n3
2 2 2
1 1 2 1
+ ---*u1*v3*a22*n2 + ---*u2*v1 *a22*b31 + ---*u2*v1*v2*a22*b32
2 2 2
1 1 1
+ u2*v1*v3*( - ---*a22*b11 + ---*a22*b33) + ---*u2*v1*a22*n3
2 2 2
1 1 1
- ---*u2*v3*a22*n1 + u3*v1*v2*( - ---*a22*b11 + ---*a22*b33)
2 2 2
1 1 1 3
- ---*u3*v2*v3*a22*b31 - ---*u3*v2*a22*n1 + ---*v1 *b21*b31
2 2 4
2 1
+ v1 *v2*(a22*c13 + ---*b21*b32)
4
2 1 1
+ v1 *v3*( - ---*b11*b21 + ---*b21*b33)
4 4
2 1 1 2 1
+ v1 *(---*b21*n3 + ---*b31*n2) + v1*v2 *(a22*c23 - ---*b21*b31)
4 4 4
1 2 1 1
+ v1*v2*v3*(a22*c33 + ---*b21 ) + v1*v2*(---*a22*m3 + ---*b32*n2)
4 2 4
1 1 1 1
+ v1*v3*( - ---*b11*n2 - ---*b21*n1 + ---*b33*n2) + ---*v1*n2*n3
4 4 4 4
1 3 2 1
- ---*v2 *b21*b32 + v2 *v3*( - a22*c12 - ---*b21*b33)
4 4
1 2 2
- ---*v2 *b21*n3 - v2*v3 *a22*c13
4
1 1 1
+ v2*v3*( - ---*a22*m1 + ---*b21*n2) - ---*v3*n1*n2}
2 4 4
And again in machine readable form:
HAM= - u1**2*a22 + u1*v1*b11 + u1*n1 + u2**2*a22 + u2*v1*b21 + u2*n2 + u3*v1*b31
+ u3*v2*b32 + u3*v3*b33 + u3*n3 + 2*v1*v2*c12 + 2*v1*v3*c13 + v1*m1 + v2**2*c22
+ 2*v2*v3*c23 + v2*m2 + v3**2*c33 + v3*m3$
FI= - 1/2*u1**2*v2**2*a22 - 1/2*u1**2*v3**2*a22 + u1*u3*v1*v3*a22 + 1/2*u1*v1**3
*b11 - 1/2*u1*v1**2*v2*b21 + 1/2*u1*v1**2*n1 + 1/2*u1*v1*v2**2*b11 - 1/2*u1*v1*
v2*n2 + 1/2*u1*v1*v3**2*b11 + 1/2*u1*v2**2*n1 + 1/2*u1*v3**2*n1 + 1/2*u2**2*v1**
2*a22 + 1/2*u2**2*v3**2*a22 + 1/2*u2*v1**3*b21 + 1/2*u2*v1**2*n2 + 1/2*u2*v1*v3
**2*b21 + 1/2*u2*v3**2*n2 + 1/2*u3**2*v3**2*a22 + 1/2*u3*v1**3*b31 + 1/2*u3*v1**
2*v2*b32 + 1/2*u3*v1**2*v3*b33 + 1/2*u3*v1**2*n3 + 1/2*u3*v1*v2**2*b31 - 1/2*u3*
v1*v2*v3*b21 + 1/2*u3*v1*v3**2*b31 + 1/2*u3*v2**3*b32 + 1/2*u3*v2**2*v3*b33 + 1/
2*u3*v2**2*n3 + 1/2*u3*v2*v3**2*b32 - 1/2*u3*v2*v3*n2 + 1/2*u3*v3**3*b33 + 1/2*
u3*v3**2*n3 + v1**3*v2*c12 + v1**3*v3*c13 + 1/2*v1**3*m1 + 1/2*v1**2*v2**2*c22 +
v1**2*v2*v3*c23 + 1/2*v1**2*v2*m2 + 1/2*v1**2*v3**2*c33 + 1/2*v1**2*v3*m3 + v1*
v2**3*c12 + v1*v2**2*v3*c13 + 1/2*v1*v2**2*m1 + v1*v2*v3**2*c12 + v1*v3**3*c13 +
1/2*v1*v3**2*m1 + 1/2*v2**4*c22 + v2**3*v3*c23 + 1/2*v2**3*m2 + v2**2*v3**2*(1/
2*c22 + 1/2*c33) + 1/2*v2**2*v3*m3 + v2*v3**3*c23 + 1/2*v2*v3**2*m2 + 1/2*v3**4*
c33 + 1/2*v3**3*m3$