Solution 3 to problem over
Remaining equations |
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem over
Equations
The following unsolved equations remain:
2 2 2 2 2 2 2 2
0=a11 *n2 + 4*a13 *n1 + 4*a13 *n2 + b12 *kap*n2
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
- m3*n2*r494
r27=---------------
2
b12
- m3*n1*r494
r28=---------------
2
b12
1 2 1 2 1 2
- ---*kap*m3 *r494 + ---*n1 *r494 + ---*n2 *r494
2 2 2
r29=---------------------------------------------------
2
b12
r210=0
r211=0
r212=0
r213=0
1 2
---*n1 *r494
2
r214=--------------
2
b12
r215=0
r217=0
r218=0
- n1*n2*r494
r219=---------------
2
b12
1 2
---*n2 *r494
2
r220=--------------
2
b12
r30=0
r31=0
r32=0
r33=0
r34=0
r35=0
r36=0
r37=0
r38=0
r39=0
r312=0
r314=0
r315=0
2*a13*m3*n1*r494 + b12*n1*n2*r494
r317=-----------------------------------
2
b12 *n2
- 2*a13*m3*r494 - b12*n2*r494
r318=--------------------------------
2
b12
r319=0
r320=0
r321=0
r322=0
r323=0
r324=0
r325=0
- a11*m3*r494
r327=----------------
2
b12
r328=0
a11*n2*r494
r329=-------------
2
b12
r330=0
n1*r494
r331=---------
b12
r332=0
2*a13*n1*r494
r333=---------------
2
b12
r334=0
r335=0
r336=0
r337=0
r338=0
r339=0
r340=0
r342=0
- a11*m3*r494
r343=----------------
2
b12
a11*n1*r494
r344=-------------
2
b12
r345=0
- n2*r494
r346=------------
b12
n1*r494
r347=---------
b12
2 2
2*a13*n1 *r494 - 2*a13*n2 *r494
r348=---------------------------------
2
b12 *n2
r349=0
r350=0
r351=0
- n2*r494
r352=------------
b12
- 2*a13*n1*r494
r353=------------------
2
b12
r354=0
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
r418=0
r421=0
r423=0
r424=0
1
r427=---*r494
2
r429=0
1
r430=---*r494
2
2*a13*r494
r432=------------
b12
2*a13*n1*r494
r433=---------------
b12*n2
2 2 2 2 2 2 2 2
- a11 *n2 *r494 - 2*a13 *n1 *r494 - 2*a13 *n2 *r494 - b12 *kap*n2 *r494
r434=--------------------------------------------------------------------------
2 2
b12 *n2
r435=0
r436=0
r437=0
r438=0
r439=0
r440=0
r441=0
r442=0
r443=0
r444=0
r447=0
r449=0
r450=0
r452=0
- a11*r494
r453=-------------
b12
- 2*a11*a13*n1*r494
r454=----------------------
2
b12 *n2
r455=0
r456=0
1
r457=---*r494
2
r458=0
r459=0
r460=0
2*a13*r494
r462=------------
b12
r463=0
2 2 2 2 2 2 1 2 2
a11 *n2 *r494 + 2*a13 *n1 *r494 + 4*a13 *n2 *r494 + ---*b12 *kap*n2 *r494
2
r464=---------------------------------------------------------------------------
2 2
b12 *n2
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
r479=0
r482=0
r484=0
r485=0
a11*r494
r487=----------
b12
r488=0
2*a11*a13*r494
r489=----------------
2
b12
r490=0
r491=0
r492=0
r493=0
r495=0
2*a13*n1*r494
r497=---------------
b12*n2
2*a13*r494
r498=------------
b12
2
4*a13 *n1*r494
r499=----------------
2
b12 *n2
r4100=0
r4101=0
r4102=0
r4103=0
r4104=0
r4105=0
r4106=0
r4108=0
r4109=0
1
r4110=---*r494
2
r4112=0
2*a13*n1*r494
r4113=---------------
b12*n2
r4114
2 2 2 2 2 2 1 2 2
a11 *n2 *r494 + 4*a13 *n1 *r494 + 2*a13 *n2 *r494 + ---*b12 *kap*n2 *r494
2
=---------------------------------------------------------------------------
2 2
b12 *n2
r4115=0
r4116=0
r4117=0
r4118=0
r4119=0
r4120=0
r4121=0
r4122=0
r4123=0
r4124=0
r4125=0
m2=0
m1=0
n3=0
a33=0
- a13*n1
a23=-----------
n2
a22=a11
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, a13, a11, n2, n1, 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.
{a23,a13,r494,a11,n2,b12,n1}
Relevance for the application:
Modulo the following equation:
2 2 2 2 2 2 2 2
0=a11 *n2 + 4*a13 *n1 + 4*a13 *n2 + b12 *kap*n2
the system of equations related to the Hamiltonian HAM:
2 2
HAM=(u1 *a11*n2 + 2*u1*u3*a13*n2 + u1*v2*b12*n2 + u1*n1*n2 + u2 *a11*n2
2
- 2*u2*u3*a13*n1 - u2*v1*b12*n2 + u2*n2 + v3*m3*n2)/n2
has apart from the Hamiltonian and Casimirs only the following first integral:
2 2 2 2 2 2 2 2 1 2 2
FI=u1 *u3 *(a11 *n2 + 4*a13 *n1 + 2*a13 *n2 + ---*b12 *kap*n2 )
2
2 2 2 1 2 2 2 2
+ 2*u1 *u3*v1*a13*b12*n1*n2 - 2*u1 *u3*a13*n1*n2 + ---*u1 *v1 *b12 *n2
2
2 3 1 2 4 2 2
- u1 *v1*b12*n2 + ---*u1 *n2 + 4*u1*u2*u3 *a13 *n1*n2
2
2
+ 2*u1*u2*u3*v1*a13*b12*n2 + 2*u1*u2*u3*v2*a13*b12*n1*n2
2 3 2 2
+ u1*u2*u3*(2*a13*n1 *n2 - 2*a13*n2 ) + u1*u2*v1*v2*b12 *n2
2 3 3
+ u1*u2*v1*b12*n1*n2 - u1*u2*v2*b12*n2 - u1*u2*n1*n2
3 2 2 2 2 2
+ 2*u1*u3 *a11*a13*n2 + u1*u3 *v2*a11*b12*n2 + u1*u3 *a11*n1*n2
2
- u1*u3*v1*a11*m3*n2
2 2 2 2 2 2 2 2 1 2 2
+ u2 *u3 *(a11 *n2 + 2*a13 *n1 + 4*a13 *n2 + ---*b12 *kap*n2 )
2
2 2 2 2 1 2 2 2 2
+ 2*u2 *u3*v2*a13*b12*n2 + 2*u2 *u3*a13*n1*n2 + ---*u2 *v2 *b12 *n2
2
2 2 1 2 2 2 3
+ u2 *v2*b12*n1*n2 + ---*u2 *n1 *n2 - 2*u2*u3 *a11*a13*n1*n2
2
2 2 2 3 2
- u2*u3 *v1*a11*b12*n2 + u2*u3 *a11*n2 - u2*u3*v2*a11*m3*n2
4 2 2 2 2 2 2 2 2
+ u3 *( - a11 *n2 - 2*a13 *n1 - 2*a13 *n2 - b12 *kap*n2 )
3 3 2 1 2 2 2 2
+ 2*u3 *v1*a13*b12*n1*n2 + 2*u3 *v2*a13*b12*n2 + ---*u3 *v1 *b12 *n2
2
2 2 3 1 2 2 2 2
+ u3 *v1*( - 2*a13*m3*n2 - b12*n2 ) + ---*u3 *v2 *b12 *n2
2
2 2
+ u3 *v2*(2*a13*m3*n1*n2 + b12*n1*n2 )
2 1 2 2 1 2 2 1 4 2
+ u3 *( - ---*kap*m3 *n2 + ---*n1 *n2 + ---*n2 ) - u3*v1*m3*n1*n2
2 2 2
3
- u3*v2*m3*n2
3 2 3 2 2 3
{HAM,FI} = u1 *u3*v1*a11 *b12*n2 + u1 *u2*u3*v2*a11 *b12*n2
2 2 3 2 2 2 3
+ 4*u1 *u3 *v1*a11*a13*b12*n2 + u1 *u3 *v3*a11 *b12*n2
2 2 3
+ 2*u1 *u3*v1*v2*a11*b12 *n2
2 2 3
+ u1 *u3*v1*(2*a11*a13*m3*n1*n2 + 2*a11*b12*n1*n2 )
2 4 2 2 3
- u1 *v1*a11*m3*n2 + u1*u2 *u3*v1*a11 *b12*n2
2 2 2 3
- 4*u1*u2*u3 *v1*a11*a13*b12*n1*n2 + 4*u1*u2*u3 *v2*a11*a13*b12*n2
2 2 3
- 2*u1*u2*u3*v1 *a11*b12 *n2
3 4
+ u1*u2*u3*v1*(2*a11*a13*m3*n2 + 2*a11*b12*n2 )
2 2 3
+ 2*u1*u2*u3*v2 *a11*b12 *n2
2 3
+ u1*u2*u3*v2*(2*a11*a13*m3*n1*n2 + 2*a11*b12*n1*n2 )
3 4
+ u1*u2*v1*a11*m3*n1*n2 - u1*u2*v2*a11*m3*n2
3 2 3 3 3
+ u1*u3 *v1*( - a11 *b12*n2 - b12 *kap*n2 )
3 3 2 2 2 2
+ 4*u1*u3 *v3*a11*a13*b12*n2 + 4*u1*u3 *v1 *a13*b12 *n1*n2
2 2 3 2 2 3
+ 4*u1*u3 *v1*v2*a13*b12 *n2 + 2*u1*u3 *v2*v3*a11*b12 *n2
2 2 3
+ u1*u3 *v3*(2*a11*a13*m3*n1*n2 + 2*a11*b12*n1*n2 )
3 3 3 2 3 2 4
+ u1*u3*v1 *b12 *n2 + u1*u3*v1 *( - 2*a13*b12*m3*n2 - 2*b12 *n2 )
2 3 3
+ u1*u3*v1*v2 *b12 *n2
2 2 3
+ u1*u3*v1*v2*(2*a13*b12*m3*n1*n2 + 2*b12 *n1*n2 )
2 3 5 4
+ u1*u3*v1*(b12*n1 *n2 + b12*n2 ) - u1*u3*v3*a11*m3*n2
2 3 4
- u1*v1 *b12*m3*n1*n2 - u1*v1*v2*b12*m3*n2
3 2 3 2 2 2
+ u2 *u3*v2*a11 *b12*n2 - 4*u2 *u3 *v2*a11*a13*b12*n1*n2
2 2 2 3 2 2 3
+ u2 *u3 *v3*a11 *b12*n2 - 2*u2 *u3*v1*v2*a11*b12 *n2
2 3 4
+ u2 *u3*v2*(2*a11*a13*m3*n2 + 2*a11*b12*n2 )
2 3 3 2 3 3 3
+ u2 *v2*a11*m3*n1*n2 + u2*u3 *v2*( - a11 *b12*n2 - b12 *kap*n2 )
3 2 2 2 2
- 4*u2*u3 *v3*a11*a13*b12*n1*n2 + 4*u2*u3 *v1*v2*a13*b12 *n1*n2
2 2 3 2 2 2 3
- 2*u2*u3 *v1*v3*a11*b12 *n2 + 4*u2*u3 *v2 *a13*b12 *n2
2 3 4
+ u2*u3 *v3*(2*a11*a13*m3*n2 + 2*a11*b12*n2 )
2 3 3
+ u2*u3*v1 *v2*b12 *n2
3 2 4
+ u2*u3*v1*v2*( - 2*a13*b12*m3*n2 - 2*b12 *n2 )
3 3 3
+ u2*u3*v2 *b12 *n2
2 2 2 3
+ u2*u3*v2 *(2*a13*b12*m3*n1*n2 + 2*b12 *n1*n2 )
2 3 5 3
+ u2*u3*v2*(b12*n1 *n2 + b12*n2 ) + u2*u3*v3*a11*m3*n1*n2
3 2 4
- u2*v1*v2*b12*m3*n1*n2 - u2*v2 *b12*m3*n2
4 2 3 3 3
+ u3 *v3*( - a11 *b12*n2 - b12 *kap*n2 )
3 2 2 3 2 3
+ 4*u3 *v1*v3*a13*b12 *n1*n2 + 4*u3 *v2*v3*a13*b12 *n2
2 2 3 3 2 3 2 4
+ u3 *v1 *v3*b12 *n2 + u3 *v1*v3*( - 2*a13*b12*m3*n2 - 2*b12 *n2 )
2 2 3 3
+ u3 *v2 *v3*b12 *n2
2 2 2 3
+ u3 *v2*v3*(2*a13*b12*m3*n1*n2 + 2*b12 *n1*n2 )
2 2 3 5 3
+ u3 *v3*(b12*n1 *n2 + b12*n2 ) - u3*v1*v3*b12*m3*n1*n2
4
- u3*v2*v3*b12*m3*n2
And again in machine readable form:
HAM=(u1**2*a11*n2 + 2*u1*u3*a13*n2 + u1*v2*b12*n2 + u1*n1*n2 + u2**2*a11*n2 - 2*
u2*u3*a13*n1 - u2*v1*b12*n2 + u2*n2**2 + v3*m3*n2)/n2$
FI=u1**2*u3**2*(a11**2*n2**2 + 4*a13**2*n1**2 + 2*a13**2*n2**2 + 1/2*b12**2*kap*
n2**2) + 2*u1**2*u3*v1*a13*b12*n1*n2 - 2*u1**2*u3*a13*n1*n2**2 + 1/2*u1**2*v1**2
*b12**2*n2**2 - u1**2*v1*b12*n2**3 + 1/2*u1**2*n2**4 + 4*u1*u2*u3**2*a13**2*n1*
n2 + 2*u1*u2*u3*v1*a13*b12*n2**2 + 2*u1*u2*u3*v2*a13*b12*n1*n2 + u1*u2*u3*(2*a13
*n1**2*n2 - 2*a13*n2**3) + u1*u2*v1*v2*b12**2*n2**2 + u1*u2*v1*b12*n1*n2**2 - u1
*u2*v2*b12*n2**3 - u1*u2*n1*n2**3 + 2*u1*u3**3*a11*a13*n2**2 + u1*u3**2*v2*a11*
b12*n2**2 + u1*u3**2*a11*n1*n2**2 - u1*u3*v1*a11*m3*n2**2 + u2**2*u3**2*(a11**2*
n2**2 + 2*a13**2*n1**2 + 4*a13**2*n2**2 + 1/2*b12**2*kap*n2**2) + 2*u2**2*u3*v2*
a13*b12*n2**2 + 2*u2**2*u3*a13*n1*n2**2 + 1/2*u2**2*v2**2*b12**2*n2**2 + u2**2*
v2*b12*n1*n2**2 + 1/2*u2**2*n1**2*n2**2 - 2*u2*u3**3*a11*a13*n1*n2 - u2*u3**2*v1
*a11*b12*n2**2 + u2*u3**2*a11*n2**3 - u2*u3*v2*a11*m3*n2**2 + u3**4*( - a11**2*
n2**2 - 2*a13**2*n1**2 - 2*a13**2*n2**2 - b12**2*kap*n2**2) + 2*u3**3*v1*a13*b12
*n1*n2 + 2*u3**3*v2*a13*b12*n2**2 + 1/2*u3**2*v1**2*b12**2*n2**2 + u3**2*v1*( -
2*a13*m3*n2**2 - b12*n2**3) + 1/2*u3**2*v2**2*b12**2*n2**2 + u3**2*v2*(2*a13*m3*
n1*n2 + b12*n1*n2**2) + u3**2*( - 1/2*kap*m3**2*n2**2 + 1/2*n1**2*n2**2 + 1/2*n2
**4) - u3*v1*m3*n1*n2**2 - u3*v2*m3*n2**3$