Solution 1 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2
0=4*a33 + b12 *kap
Expressions
The solution is given through the following expressions:
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
1
r427= - ---*r446
2
r428=0
1
r429= - ---*r448
2
1
r430= - ---*r494
2
- a33*r448 + b12*r452
r431=------------------------
b12
r432=0
r433=0
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r445=0
r447=0
r449=0
r450=0
r451=0
r454=0
r455=0
r456=0
1
r457=---*r446
2
r458=0
1
r459=---*r448
2
1 1
r460=---*r446 - ---*r494
2 2
- 2*a33*r448
r461=---------------
b12
r462=0
r463=0
1
- 2*a33*r453 - ---*b12*kap*r494
2
r464=----------------------------------
b12
r465=0
- a33*r448
r466=-------------
b12
- 2*a33*r446 + 2*a33*r494
r467=----------------------------
b12
r468=0
2 2
2*a33 *r446 - 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
- 2*a33*r446 + 2*a33*r494 - b12*r453
r487=---------------------------------------
b12
- 2*a33*r448 + b12*r452
r488=--------------------------
b12
r489=0
r490=0
r491=0
r492=0
r493=0
r495=r448
2*a33*r446 - 2*a33*r494
r496=-------------------------
b12
r497=0
r498=0
2
2*a33 *r448
r499=-------------
2
b12
r4100=0
2*a33*r446 - 2*a33*r494
r4101=-------------------------
b12
- 3*a33*r448
r4102=---------------
b12
r4103=0
2
2*a33 *r448
r4104=-------------
2
b12
r4105=0
r4106=0
r4108=0
1
r4109= - ---*r448
2
1
r4110=---*r494
2
r4111=0
r4112=0
r4113=0
1
- 2*a33*r453 + ---*b12*kap*r446 - b12*kap*r494
2
r4114=-------------------------------------------------
b12
r4115=0
a33*r448
r4116=----------
b12
r4117=0
r4118=0
2 2
2*a33 *r446 - 2*a33 *r494
r4119=---------------------------
2
b12
r4120=0
r4121=0
- a33*r448
r4122=-------------
b12
r4123=0
2
2*a33 *r448
r4124=-------------
2
b12
r4125=0
a23=0
a22=2*a33
a13=0
a11=2*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:
a33, r434, r453, r452, r446, r494, r448, 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.
{a11,b12,a33}
Relevance for the application:
Modulo the following equation:
2 2
0=4*a33 + b12 *kap
the system of equations related to the Hamiltonian HAM:
2 2 2
HAM=2*u1 *a33 + u1*v2*b12 + 2*u2 *a33 - u2*v1*b12 + u3 *a33
has apart from the Hamiltonian and Casimirs the following 6 first integrals:
3 2 3 2 1 2 2
FI=2*u1 *u2*a33 - u1 *v1*a33*b12 + u1 *u2*v2*a33*b12 - ---*u1 *v1*v2*b12
2
3 2 2 2 2 2 2
+ 2*u1*u2 *a33 - 3*u1*u2 *v1*a33*b12 + 2*u1*u2*u3 *a33 + u1*u2*v1 *b12
2 3 2
- 2*u1*u3 *v1*a33*b12 - u2 *v2*a33*b12 - 2*u2 *u3*v3*a33*b12
1 2 2 2 3
+ ---*u2 *v1*v2*b12 + u2*u3*v1*v3*b12 - u3 *v3*a33*b12
2
1 2 2
- ---*u3 *v1*v2*b12
2
1
= a product of the elements of: {---,
2
3 2 3 2 2 2
4*u1 *u2*a33 - 2*u1 *v1*a33*b12 + 2*u1 *u2*v2*a33*b12 - u1 *v1*v2*b12
3 2 2 2 2 2 2
+ 4*u1*u2 *a33 - 6*u1*u2 *v1*a33*b12 + 4*u1*u2*u3 *a33 + 2*u1*u2*v1 *b12
2 3 2
- 4*u1*u3 *v1*a33*b12 - 2*u2 *v2*a33*b12 - 4*u2 *u3*v3*a33*b12
2 2 2 3 2 2
+ u2 *v1*v2*b12 + 2*u2*u3*v1*v3*b12 - 2*u3 *v3*a33*b12 - u3 *v1*v2*b12 }
{HAM,FI} = 0
2 2 2 2 2 2 1 2 2 2 2
FI= - 2*u1 *u2 *a33 - u1 *u3 *b12 *kap + ---*u1 *v1 *b12 - 2*u1*u2 *v2*a33*b12
2
2 2
- 2*u1*u2*u3*v3*a33*b12 + u1*u2*v1*v2*b12 + 2*u1*u3 *v2*a33*b12
2 4 2 3 1 2 2 2
+ u1*u3*v1*v3*b12 - 2*u2 *a33 + 2*u2 *v1*a33*b12 - ---*u2 *u3 *b12 *kap
2
1 2 2 2 1 2 2 2
- ---*u2 *v1 *b12 - ---*u3 *v1 *b12
2 2
1
= a product of the elements of: { - ---,
2
2 2 2 2 2 2 2 2 2 2
4*u1 *u2 *a33 + 2*u1 *u3 *b12 *kap - u1 *v1 *b12 + 4*u1*u2 *v2*a33*b12
2 2
+ 4*u1*u2*u3*v3*a33*b12 - 2*u1*u2*v1*v2*b12 - 4*u1*u3 *v2*a33*b12
2 4 2 3 2 2 2
- 2*u1*u3*v1*v3*b12 + 4*u2 *a33 - 4*u2 *v1*a33*b12 + u2 *u3 *b12 *kap
2 2 2 2 2 2
+ u2 *v1 *b12 + u3 *v1 *b12 }
{HAM,FI} = 0
2 2 2 1 2 2 2 2
FI=2*u1 *u2 *a33 + ---*u1 *u3 *b12 *kap + 2*u1*u2 *v2*a33*b12
2
2 4 2
+ 2*u1*u2*u3*v3*a33*b12 - 2*u1*u3 *v2*a33*b12 + 2*u2 *a33
3 1 2 2 2 1 2 2 2 2
- 2*u2 *v1*a33*b12 + ---*u2 *v1 *b12 + ---*u2 *v2 *b12 + u2*u3*v2*v3*b12
2 2
1 2 2 2
- ---*u3 *v2 *b12
2
1
= a product of the elements of: {---,
2
2 2 2 2 2 2 2
4*u1 *u2 *a33 + u1 *u3 *b12 *kap + 4*u1*u2 *v2*a33*b12
2 4 2
+ 4*u1*u2*u3*v3*a33*b12 - 4*u1*u3 *v2*a33*b12 + 4*u2 *a33
3 2 2 2 2 2 2 2
- 4*u2 *v1*a33*b12 + u2 *v1 *b12 + u2 *v2 *b12 + 2*u2*u3*v2*v3*b12
2 2 2
- u3 *v2 *b12 }
{HAM,FI} = 0
2 2 3
FI=u1*u3 *v1 + u2*u3 *v2 + u3 *v3
= a product of the elements of: {u3,
u3,
u1*v1 + u2*v2 + u3*v3}
{HAM,FI} = 0
2 2 2 2 2 2
FI= - 2*u1 *u3 *a33 - u1*u3 *v2*b12 - 2*u2 *u3 *a33 + u2*u3 *v1*b12
= a product of the elements of: { - u3,
u3,
2 2
2*u1 *a33 + u1*v2*b12 + 2*u2 *a33 - u2*v1*b12}
{HAM,FI} = 0
4
FI=u3
= a product of the elements of: {u3,u3,u3,u3}
{HAM,FI} = 0
And again in machine readable form:
HAM=2*u1**2*a33 + u1*v2*b12 + 2*u2**2*a33 - u2*v1*b12 + u3**2*a33$
FI=2*u1**3*u2*a33**2 - u1**3*v1*a33*b12 + u1**2*u2*v2*a33*b12 - 1/2*u1**2*v1*v2*
b12**2 + 2*u1*u2**3*a33**2 - 3*u1*u2**2*v1*a33*b12 + 2*u1*u2*u3**2*a33**2 + u1*
u2*v1**2*b12**2 - 2*u1*u3**2*v1*a33*b12 - u2**3*v2*a33*b12 - 2*u2**2*u3*v3*a33*
b12 + 1/2*u2**2*v1*v2*b12**2 + u2*u3*v1*v3*b12**2 - u3**3*v3*a33*b12 - 1/2*u3**2
*v1*v2*b12**2$
FI= - 2*u1**2*u2**2*a33**2 - u1**2*u3**2*b12**2*kap + 1/2*u1**2*v1**2*b12**2 - 2
*u1*u2**2*v2*a33*b12 - 2*u1*u2*u3*v3*a33*b12 + u1*u2*v1*v2*b12**2 + 2*u1*u3**2*
v2*a33*b12 + u1*u3*v1*v3*b12**2 - 2*u2**4*a33**2 + 2*u2**3*v1*a33*b12 - 1/2*u2**
2*u3**2*b12**2*kap - 1/2*u2**2*v1**2*b12**2 - 1/2*u3**2*v1**2*b12**2$
FI=2*u1**2*u2**2*a33**2 + 1/2*u1**2*u3**2*b12**2*kap + 2*u1*u2**2*v2*a33*b12 + 2
*u1*u2*u3*v3*a33*b12 - 2*u1*u3**2*v2*a33*b12 + 2*u2**4*a33**2 - 2*u2**3*v1*a33*
b12 + 1/2*u2**2*v1**2*b12**2 + 1/2*u2**2*v2**2*b12**2 + u2*u3*v2*v3*b12**2 - 1/2
*u3**2*v2**2*b12**2$
FI=u1*u3**2*v1 + u2*u3**2*v2 + u3**3*v3$
FI= - 2*u1**2*u3**2*a33 - u1*u3**2*v2*b12 - 2*u2**2*u3**2*a33 + u2*u3**2*v1*b12$
FI=u3**4$