Solution 5 to problem e3null
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem e3null
Expressions
The solution is given through the following expressions:
1
a22=---*a33
2
b22=0
b33=0
c12=0
c13=0
c22=0
c23=0
1 2 1 2
- ---*b31 - ---*b32
2 2
c33=------------------------
a33
n1=0
n2=0
2 2
- a33*b32*m2 + b31 *n3 + b32 *n3
m1=-----------------------------------
a33*b31
m3=0
r6=0
r5=0
r4=0
r3=0
r2=0
r1=0
q19=0
q18=0
q17=0
8*k1*m2*n3
q16=------------
2
a33
2 2 2 2
- 8*a33*b32*k1*m2*n3 + 8*b31 *k1*n3 + 8*b32 *k1*n3
q15=-------------------------------------------------------
3
a33 *b31
2 2 2 2 2 2 2
q14=(4*a33 *b31 *k1*m2 + 4*a33 *b32 *k1*m2 - 8*a33*b31 *b32*k1*m2*n3
3 4 2 2 2 2 4 2
- 8*a33*b32 *k1*m2*n3 + 4*b31 *k1*n3 + 8*b31 *b32 *k1*n3 + 4*b32 *k1*n3
4 2
)/(a33 *b31 )
q13=0
q12=0
q11=0
q10=0
2 2 2 2 2 2 2
q9=(4*a33 *b31 *k1*m2 + 4*a33 *b32 *k1*m2 - 8*a33*b31 *b32*k1*m2*n3
3 4 2 2 2 2 4 2
- 8*a33*b32 *k1*m2*n3 + 4*b31 *k1*n3 + 8*b31 *b32 *k1*n3 + 4*b32 *k1*n3 )
4 2
/(a33 *b31 )
q8=0
q7=0
q5=0
q4=0
2
- 4*k1*n3
q3=-------------
2
a33
q2=0
2
- 4*k1*n3
q1=-------------
2
a33
p50=0
2 2
- 4*b31 *k1*m2 - 4*b32 *k1*m2
p49=--------------------------------
3
a33
2 3 4 2 2
p48=(4*a33*b31 *b32*k1*m2 + 4*a33*b32 *k1*m2 - 4*b31 *k1*n3 - 8*b31 *b32 *k1*n3
4 4
- 4*b32 *k1*n3)/(a33 *b31)
2 2
- 12*b31 *k1*n3 - 12*b32 *k1*n3
p47=----------------------------------
3
a33
p46=0
p45=0
p44=0
p43=0
p42=0
p41=0
p40=0
p39=0
p38=0
p37=0
p36=0
8*a33*k1*m2 + 8*b32*k1*n3
p35=---------------------------
2
a33
2 2
- 4*a33*b32*k1*m2 + 12*b31 *k1*n3 + 4*b32 *k1*n3
p34=---------------------------------------------------
2
a33 *b31
p33=0
p32=0
p31=0
p30=0
p29=0
p28=0
p27=0
p26=0
p25=0
p24=0
p23=0
p22=0
p21=0
p20=0
4*k1*m2
p19=---------
a33
2 2
4*a33*b32*k1*m2 - 4*b31 *k1*n3 - 4*b32 *k1*n3
p18=-----------------------------------------------
2
a33 *b31
4*k1*m2
p17=---------
a33
p16=0
p15=0
p14=0
p13=0
p12=0
2 2
- 4*a33*b32*k1*m2 + 4*b31 *k1*n3 + 4*b32 *k1*n3
p11=--------------------------------------------------
2
a33 *b31
p10=0
p9=0
p8=0
- 4*k1*n3
p7=------------
a33
p6=0
- 4*k1*n3
p5=------------
a33
p4=0
p3=0
p2=0
p1=0
k104=0
k103=0
k102=0
k101=0
k100=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k99=-------------------------------------------
4
a33
k98=0
2 3
- 4*b31 *b32*k1 - 4*b32 *k1
k97=------------------------------
3
a33
k96=0
k95=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k94=-------------------------------------------
4
a33
3 2
- 4*b31 *k1 - 4*b31*b32 *k1
k93=------------------------------
3
a33
k92=0
2 2
- 8*b31 *k1 - 8*b32 *k1
k91=--------------------------
2
a33
k90=0
k89=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k88=--------------------------
2
a33
k87=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k86=--------------------------
2
a33
k85=0
k84=0
k83=0
k82=0
k81=0
k80=0
k79=0
k78=0
k77=0
k76=0
k75=0
k74=0
k73=0
k72=0
k71=0
k70=0
k69=0
k68=0
k67=0
k66=0
k65=0
8*b32*k1
k64=----------
a33
4*b31*k1
k63=----------
a33
k62=0
k61=0
k60=0
k59=0
k58=0
k57=0
k56=0
4 2 2 4
- b31 *k1 - 2*b31 *b32 *k1 - b32 *k1
k55=---------------------------------------
4
a33
k54=0
k53=0
k52=0
k51=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k50=-------------------------------------------
4
a33
k49=0
k48=0
k47=0
k46=0
k45=0
k44=0
k43=0
k42=0
k41=0
k40=0
k39=0
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
4*b32*k1
k32=----------
a33
- 4*b31*k1
k31=-------------
a33
4*b32*k1
k30=----------
a33
k29=0
k28=0
k27=0
k26=0
4 2 2 4
- b31 *k1 - 2*b31 *b32 *k1 - b32 *k1
k25=---------------------------------------
4
a33
k24=0
k23=0
k21=0
k20=0
k19=0
k18=0
4*b31*k1
k17=----------
a33
k16=0
k14=0
k13=0
k12=0
k11=0
k10=0
k9=0
k8=0
k7=0
k6=0
k5=k1
k4=0
k3=2*k1
k2=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:
k1, b32, b31, n3, m2, a33
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.
4 4 4 3 3
{a33 *w123 + 2*a33 *w125 + a33 *w127 + 4*a33 *b31*w065 - 4*a33 *b31*w097
3 3 3 3
+ 4*a33 *b31*w111 + 8*a33 *b32*w064 + 4*a33 *b32*w096 + 4*a33 *b32*w098
2 2 2 2 2 2 2 2
- 8*a33 *b31 *w037 - 2*a33 *b31 *w040 - 2*a33 *b31 *w042 - 8*a33 *b32 *w037
2 2 2 2 3 2
- 2*a33 *b32 *w040 - 2*a33 *b32 *w042 - 4*a33*b31 *w035 - 4*a33*b31 *b32*w031
2 3 4 4
- 4*a33*b31*b32 *w035 - 4*a33*b32 *w031 - 2*b31 *w029 - 2*b31 *w034
4 4 4 2 2 2 2
- b31 *w073 - 2*b31 *w078 - b31 *w103 - 4*b31 *b32 *w029 - 4*b31 *b32 *w034
2 2 2 2 2 2 4
- 2*b31 *b32 *w073 - 4*b31 *b32 *w078 - 2*b31 *b32 *w103 - 2*b32 *w029
4 4 4 4
- 2*b32 *w034 - b32 *w073 - 2*b32 *w078 - b32 *w103,
k1,
b31,
a33}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2 2 2 2
HAM=(a33 *b31*u1 + a33 *b31*u2 + 2*a33 *b31*u3 + 2*a33*b31 *u3*v1
+ 2*a33*b31*b32*u3*v2 + 2*a33*b31*m2*v2 + 2*a33*b31*n3*u3
3 2 2 2 2 2
- 2*a33*b32*m2*v1 - b31 *v3 + 2*b31 *n3*v1 - b31*b32 *v3 + 2*b32 *n3*v1)
/(2*a33*b31)
has apart from the Hamiltonian and Casimirs the following only first integral:
4 2 4 4 2 2 2 4 2 4
INT=a33 *b31 *u1 + 2*a33 *b31 *u1 *u2 + a33 *b31 *u2
3 3 3 3 2 3 3 2
- 4*a33 *b31 *u1*u2*u3*v2 + 4*a33 *b31 *u1*u3 *v3 + 4*a33 *b31 *u2 *u3*v1
3 2 2 3 2 2
+ 4*a33 *b31 *b32*u1 *u3*v2 + 4*a33 *b31 *b32*u2 *u3*v2
3 2 2 3 2 2 3 2 2
+ 8*a33 *b31 *b32*u2*u3 *v3 + 4*a33 *b31 *m2*u1 *v2 + 4*a33 *b31 *m2*u2 *v2
3 2 3 2 2 3 2 2
+ 8*a33 *b31 *m2*u2*u3*v3 - 4*a33 *b31 *n3*u1 *u3 - 4*a33 *b31 *n3*u2 *u3
3 3
+ 4*a33 *b31*b32*m2*u1*u2*v2 - 4*a33 *b31*b32*m2*u1*u3*v3
3 2 2 4 2 2 2 4 2 2
- 4*a33 *b31*b32*m2*u2 *v1 - 2*a33 *b31 *u1 *v3 - 2*a33 *b31 *u2 *v3
2 4 2 2 2 3 2 3
- 8*a33 *b31 *u3 *v3 - 4*a33 *b31 *n3*u1*u2*v2 + 12*a33 *b31 *n3*u1*u3*v3
2 3 2 2 2 2 2 2
+ 4*a33 *b31 *n3*u2 *v1 - 2*a33 *b31 *b32 *u1 *v3
2 2 2 2 2 2 2 2 2 2
- 2*a33 *b31 *b32 *u2 *v3 - 8*a33 *b31 *b32 *u3 *v3
2 2 2 2 2 2 2 2 2 2
+ 8*a33 *b31 *b32*n3*u2*u3*v3 + 4*a33 *b31 *m2 *v1 + 4*a33 *b31 *m2 *v2
2 2 2 2 2 2 2 2 2 2
+ 8*a33 *b31 *m2*n3*u2*v3 - 4*a33 *b31 *n3 *u1 - 4*a33 *b31 *n3 *u2
2 2 2 2
- 4*a33 *b31*b32 *n3*u1*u2*v2 + 4*a33 *b31*b32 *n3*u1*u3*v3
2 2 2 2
+ 4*a33 *b31*b32 *n3*u2 *v1 - 8*a33 *b31*b32*m2*n3*u1*v3
2 2 2 2 2 2 2 2 5 2
+ 4*a33 *b32 *m2 *v1 + 4*a33 *b32 *m2 *v2 - 4*a33*b31 *u3*v1*v3
4 2 4 2 4 2
- 4*a33*b31 *b32*u3*v2*v3 - 4*a33*b31 *m2*v2*v3 - 12*a33*b31 *n3*u3*v3
3 2 2 3 2
- 4*a33*b31 *b32 *u3*v1*v3 + 4*a33*b31 *b32*m2*v1*v3
3 2 2 3 2
+ 8*a33*b31 *n3 *u1*v3 - 4*a33*b31 *b32 *u3*v2*v3
2 2 2 2 2 2
- 4*a33*b31 *b32 *m2*v2*v3 - 12*a33*b31 *b32 *n3*u3*v3
2 2 2 2
- 8*a33*b31 *b32*m2*n3*v1 - 8*a33*b31 *b32*m2*n3*v2
3 2 2 2
+ 4*a33*b31*b32 *m2*v1*v3 + 8*a33*b31*b32 *n3 *u1*v3
3 2 3 2 6 4 6 2 2
- 8*a33*b32 *m2*n3*v1 - 8*a33*b32 *m2*n3*v2 - b31 *v1 - 2*b31 *v1 *v2
6 2 2 6 4 6 2 2 5 2
- 2*b31 *v1 *v3 - b31 *v2 - 2*b31 *v2 *v3 - 4*b31 *n3*v1*v3
4 2 4 4 2 2 2 4 2 2 2
- 2*b31 *b32 *v1 - 4*b31 *b32 *v1 *v2 - 4*b31 *b32 *v1 *v3
4 2 4 4 2 2 2 4 2 2 4 2 2
- 2*b31 *b32 *v2 - 4*b31 *b32 *v2 *v3 + 4*b31 *n3 *v1 + 4*b31 *n3 *v2
3 2 2 2 4 4 2 4 2 2
- 8*b31 *b32 *n3*v1*v3 - b31 *b32 *v1 - 2*b31 *b32 *v1 *v2
2 4 2 2 2 4 4 2 4 2 2
- 2*b31 *b32 *v1 *v3 - b31 *b32 *v2 - 2*b31 *b32 *v2 *v3
2 2 2 2 2 2 2 2 4 2
+ 8*b31 *b32 *n3 *v1 + 8*b31 *b32 *n3 *v2 - 4*b31*b32 *n3*v1*v3
4 2 2 4 2 2
+ 4*b32 *n3 *v1 + 4*b32 *n3 *v2
2 2 2 2 2 2 4
= - (4*((b31 + b32 ) *b31*v1*v3 - (v1 + v2 )*b32 *n3)*n3
2 2 2 2 2 6 2 2 2 4 2
+ (v1 + v2 + 2*v3 )*(v1 + v2 )*b31 - (u1 + u2 ) *a33 *b31 + 4*(
3 2 2 2 2 2
b31 *u3*v1*v3 + b31 *b32*u3*v2*v3 + b31 *m2*v2*v3
2 2 2 2
+ 3*b31 *n3*u3*v3 - b31*b32*m2*v1*v3 - 2*b31*n3 *u1*v3
2 2 2 2
+ 2*b32*m2*n3*v1 + 2*b32*m2*n3*v2 )*(b31 + b32 )*a33
2 2 2 2 2 2 2 2 2
+ ((v1 + v2 + 2*v3 )*b32 - 8*n3 )*(v1 + v2 )*b31 *b32
2 2 2 2 2 2 2 4
+ 2*((v1 + v2 + 2*v3 )*b32 - 2*n3 )*(v1 + v2 )*b31 - 4*(
2 2 2
(((u2*v2 + 2*u3*v3)*u2 + u1 *v2)*(b32*u3 + m2) - (u1 + u2 )*n3*u3)
2 2 3
*b31 - ((u2*v2 - u3*v3)*u1 - u2 *v1)*(b31 *u3 - b32*m2))*a33 *b31 + 2
2 3 2 2 2 2
*(2*(((u2*v2 - 3*u3*v3)*u1 - u2 *v1)*b31 *n3 - (v1 + v2 )*b32 *m2 )
2 2 2 4 2
+ (u2 + 4*u3 + u1 )*b31 *v3
2
+ 2*(((u2*v2 - u3*v3)*u1 - u2 *v1)*b32 + 2*m2*u1*v3)*b31*b32*n3 - (
2 2 2 2 2 2
2*(2*(b32*u3 + m2)*n3*u2*v3 + (v1 + v2 )*m2 - (u1 + u2 )*n3 )
2 2 2 2 2 2 2
- (u2 + 4*u3 + u1 )*b32 *v3 )*b31 )*a33 )
And again in machine readable form:
HAM=(a33**2*b31*u1**2 + a33**2*b31*u2**2 + 2*a33**2*b31*u3**2 + 2*a33*b31**2*u3*
v1 + 2*a33*b31*b32*u3*v2 + 2*a33*b31*m2*v2 + 2*a33*b31*n3*u3 - 2*a33*b32*m2*v1 -
b31**3*v3**2 + 2*b31**2*n3*v1 - b31*b32**2*v3**2 + 2*b32**2*n3*v1)/(2*a33*b31)$
INT=a33**4*b31**2*u1**4 + 2*a33**4*b31**2*u1**2*u2**2 + a33**4*b31**2*u2**4 - 4*
a33**3*b31**3*u1*u2*u3*v2 + 4*a33**3*b31**3*u1*u3**2*v3 + 4*a33**3*b31**3*u2**2*
u3*v1 + 4*a33**3*b31**2*b32*u1**2*u3*v2 + 4*a33**3*b31**2*b32*u2**2*u3*v2 + 8*
a33**3*b31**2*b32*u2*u3**2*v3 + 4*a33**3*b31**2*m2*u1**2*v2 + 4*a33**3*b31**2*m2
*u2**2*v2 + 8*a33**3*b31**2*m2*u2*u3*v3 - 4*a33**3*b31**2*n3*u1**2*u3 - 4*a33**3
*b31**2*n3*u2**2*u3 + 4*a33**3*b31*b32*m2*u1*u2*v2 - 4*a33**3*b31*b32*m2*u1*u3*
v3 - 4*a33**3*b31*b32*m2*u2**2*v1 - 2*a33**2*b31**4*u1**2*v3**2 - 2*a33**2*b31**
4*u2**2*v3**2 - 8*a33**2*b31**4*u3**2*v3**2 - 4*a33**2*b31**3*n3*u1*u2*v2 + 12*
a33**2*b31**3*n3*u1*u3*v3 + 4*a33**2*b31**3*n3*u2**2*v1 - 2*a33**2*b31**2*b32**2
*u1**2*v3**2 - 2*a33**2*b31**2*b32**2*u2**2*v3**2 - 8*a33**2*b31**2*b32**2*u3**2
*v3**2 + 8*a33**2*b31**2*b32*n3*u2*u3*v3 + 4*a33**2*b31**2*m2**2*v1**2 + 4*a33**
2*b31**2*m2**2*v2**2 + 8*a33**2*b31**2*m2*n3*u2*v3 - 4*a33**2*b31**2*n3**2*u1**2
- 4*a33**2*b31**2*n3**2*u2**2 - 4*a33**2*b31*b32**2*n3*u1*u2*v2 + 4*a33**2*b31*
b32**2*n3*u1*u3*v3 + 4*a33**2*b31*b32**2*n3*u2**2*v1 - 8*a33**2*b31*b32*m2*n3*u1
*v3 + 4*a33**2*b32**2*m2**2*v1**2 + 4*a33**2*b32**2*m2**2*v2**2 - 4*a33*b31**5*
u3*v1*v3**2 - 4*a33*b31**4*b32*u3*v2*v3**2 - 4*a33*b31**4*m2*v2*v3**2 - 12*a33*
b31**4*n3*u3*v3**2 - 4*a33*b31**3*b32**2*u3*v1*v3**2 + 4*a33*b31**3*b32*m2*v1*v3
**2 + 8*a33*b31**3*n3**2*u1*v3 - 4*a33*b31**2*b32**3*u3*v2*v3**2 - 4*a33*b31**2*
b32**2*m2*v2*v3**2 - 12*a33*b31**2*b32**2*n3*u3*v3**2 - 8*a33*b31**2*b32*m2*n3*
v1**2 - 8*a33*b31**2*b32*m2*n3*v2**2 + 4*a33*b31*b32**3*m2*v1*v3**2 + 8*a33*b31*
b32**2*n3**2*u1*v3 - 8*a33*b32**3*m2*n3*v1**2 - 8*a33*b32**3*m2*n3*v2**2 - b31**
6*v1**4 - 2*b31**6*v1**2*v2**2 - 2*b31**6*v1**2*v3**2 - b31**6*v2**4 - 2*b31**6*
v2**2*v3**2 - 4*b31**5*n3*v1*v3**2 - 2*b31**4*b32**2*v1**4 - 4*b31**4*b32**2*v1
**2*v2**2 - 4*b31**4*b32**2*v1**2*v3**2 - 2*b31**4*b32**2*v2**4 - 4*b31**4*b32**
2*v2**2*v3**2 + 4*b31**4*n3**2*v1**2 + 4*b31**4*n3**2*v2**2 - 8*b31**3*b32**2*n3
*v1*v3**2 - b31**2*b32**4*v1**4 - 2*b31**2*b32**4*v1**2*v2**2 - 2*b31**2*b32**4*
v1**2*v3**2 - b31**2*b32**4*v2**4 - 2*b31**2*b32**4*v2**2*v3**2 + 8*b31**2*b32**
2*n3**2*v1**2 + 8*b31**2*b32**2*n3**2*v2**2 - 4*b31*b32**4*n3*v1*v3**2 + 4*b32**
4*n3**2*v1**2 + 4*b32**4*n3**2*v2**2$