Solution 4 to problem e3c2new
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem e3c2new
Expressions
The solution is given through the following expressions:
a33
a22=-----
2
b22=0
b33=0
c12=0
c13=0
c22=0
c23=0
2 2
- b31 - b32
c33=----------------
2*a33
n1=0
n2=0
m3=0
r6=0
r5=0
r4=0
r3=0
r2=0
r1=0
q20=0
q19=0
2 2 2 2
- 8*a33*b31*k1*m1*n3 + 8*b31 *k1*n3 + 8*b32 *k1*n3
q17=-------------------------------------------------------
3
a33 *b32
8*k1*m1*n3
q16=------------
2
a33
2 2 2 2 2 2 3
q15=(4*a33 *b31 *k1*m1 + 4*a33 *b32 *k1*m1 - 8*a33*b31 *k1*m1*n3
2 4 2 2 2 2
- 8*a33*b31*b32 *k1*m1*n3 + 4*b31 *k1*n3 + 8*b31 *b32 *k1*n3
4 2 4 2
+ 4*b32 *k1*n3 )/(a33 *b32 )
q14=0
q13=0
q12=0
q11=0
2 2 2 2 2 2 3
q10=(4*a33 *b31 *k1*m1 + 4*a33 *b32 *k1*m1 - 8*a33*b31 *k1*m1*n3
2 4 2 2 2 2
- 8*a33*b31*b32 *k1*m1*n3 + 4*b31 *k1*n3 + 8*b31 *b32 *k1*n3
4 2 4 2
+ 4*b32 *k1*n3 )/(a33 *b32 )
q9=0
q8=0
q7=0
q5=0
q4=0
2
- 4*k1*n3
q3=-------------
2
a33
q2=0
2
- 4*k1*n3
q1=-------------
2
a33
p56=0
3 2 4 2 2
p55=(4*a33*b31 *k1*m1 + 4*a33*b31*b32 *k1*m1 - 4*b31 *k1*n3 - 8*b31 *b32 *k1*n3
4 4
- 4*b32 *k1*n3)/(a33 *b32)
2 2
- 4*b31 *k1*m1 - 4*b32 *k1*m1
p54=--------------------------------
3
a33
2 2
- 4*b31 *k1*n3 - 4*b32 *k1*n3
p53=--------------------------------
3
a33
p52=0
p51=0
p50=0
p49=0
p48=0
2 2
8*b31 *k1*n3 + 8*b32 *k1*n3
p47=-----------------------------
3
a33
p46=0
p45=0
p44=0
p43=0
2 2
8*b31 *k1*n3 + 8*b32 *k1*n3
p42=-----------------------------
3
a33
p41=0
8*b32*k1*n3
p40=-------------
2
a33
8*b31*k1*n3
p39=-------------
2
a33
p38=0
p37=0
p36=0
p35=0
p34=0
p33=0
p32=0
p31=0
p30=0
p29=0
p28=0
p27=0
p26=0
p25=0
p24=0
2 2
4*a33*b31*k1*m1 - 4*b31 *k1*n3 - 4*b32 *k1*n3
p23=-----------------------------------------------
2
a33 *b32
- 8*k1*m1
p22=------------
a33
2 2
- 4*a33*b31*k1*m1 + 4*b31 *k1*n3 + 4*b32 *k1*n3
p21=--------------------------------------------------
2
a33 *b32
p20=0
p19=0
p18=0
p17=0
p16=0
p15=0
p14=0
4*k1*m1
p13=---------
a33
2 2
8*a33*b31*k1*m1 - 8*b31 *k1*n3 - 8*b32 *k1*n3
p12=-----------------------------------------------
2
a33 *b32
- 4*k1*m1
p11=------------
a33
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
k125=0
k124=0
k122=0
k121=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k120=-------------------------------------------
4
a33
k119=0
2 3
- 4*b31 *b32*k1 - 4*b32 *k1
k118=------------------------------
3
a33
k117=0
k116=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k115=-------------------------------------------
4
a33
3 2
- 4*b31 *k1 - 4*b31*b32 *k1
k114=------------------------------
3
a33
k113=0
k112=0
k110=0
k109=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k108=--------------------------
2
a33
k107=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k106=--------------------------
2
a33
k105=0
k104=0
k103=0
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
2 2
8*b31 *k1 + 8*b32 *k1
k95=-----------------------
2
a33
k94=0
k93=0
k92=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
k85=0
2 2
8*b31 *k1 + 8*b32 *k1
k84=-----------------------
2
a33
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
4 2 2 4
- b31 *k1 - 2*b31 *b32 *k1 - b32 *k1
k70=---------------------------------------
4
a33
k69=0
k68=0
k67=0
k66=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k65=-------------------------------------------
4
a33
k64=0
k63=0
k62=0
k61=0
k60=0
k59=0
k58=0
k57=0
k56=0
k55=0
k54=0
k53=0
k52=0
k51=0
k50=0
k49=0
k48=0
k47=0
k46=0
k45=0
k44=0
k43=0
- 4*b32*k1
k42=-------------
a33
- 8*b31*k1
k41=-------------
a33
4*b32*k1
k40=----------
a33
k38=0
k37=0
k36=0
4 2 2 4
- b31 *k1 - 2*b31 *b32 *k1 - b32 *k1
k35=---------------------------------------
4
a33
k34=0
k33=0
k32=0
k30=0
k29=0
k28=0
k27=0
k26=0
k25=0
k24=0
k23=0
4*b31*k1
k22=----------
a33
- 8*b32*k1
k21=-------------
a33
- 4*b31*k1
k20=-------------
a33
k19=0
k18=0
k17=0
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
2 2
- a33*b31*m1 + b31 *n3 + b32 *n3
m2=-----------------------------------
a33*b32
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:
m1, n3, k1, b32, b31, 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 - 8*a33 *b31*w087 + 4*a33 *b31*w106
3 3 3 3
- 4*a33 *b31*w108 - 4*a33 *b32*w086 + 4*a33 *b32*w088 - 8*a33 *b32*w107
2 2 2 2 2 2 2 2
- 2*a33 *b31 *w020 - 2*a33 *b31 *w022 + 8*a33 *b31 *w033 + 8*a33 *b31 *w044
2 2 2 2 2 2 2 2
- 2*a33 *b32 *w020 - 2*a33 *b32 *w022 + 8*a33 *b32 *w033 + 8*a33 *b32 *w044
3 2 2
- 4*a33*b31 *w014 - 4*a33*b31 *b32*w010 - 4*a33*b31*b32 *w014
3 4 4 4 4
- 4*a33*b32 *w010 - 2*b31 *w008 - 2*b31 *w013 - b31 *w058 - 2*b31 *w063
4 2 2 2 2 2 2
- b31 *w093 - 4*b31 *b32 *w008 - 4*b31 *b32 *w013 - 2*b31 *b32 *w058
2 2 2 2 4 4 4
- 4*b31 *b32 *w063 - 2*b31 *b32 *w093 - 2*b32 *w008 - 2*b32 *w013 - b32 *w058
4 4
- 2*b32 *w063 - b32 *w093,
a33,
b32,
k1}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2 2 2
HAM=(a33 *b32*u1 + a33 *b32*u2 + 2*a33 *b32*u3 + 2*a33*b31*b32*u3*v1
2
- 2*a33*b31*m1*v2 + 2*a33*b32 *u3*v2 + 2*a33*b32*m1*v1 + 2*a33*b32*n3*u3
2 2 2 3 2 2
- b31 *b32*v3 + 2*b31 *n3*v2 - b32 *v3 + 2*b32 *n3*v2)/(2*a33*b32)
has apart from the Hamiltonian and Casimirs the following only first integral:
4 2 4 4 2 2 2 4 2 4
INT=a33 *b32 *u1 + 2*a33 *b32 *u1 *u2 + a33 *b32 *u2
3 2 2 3 2
- 4*a33 *b31*b32 *u1 *u3*v1 - 8*a33 *b31*b32 *u1*u2*u3*v2
3 2 2 3 2
+ 4*a33 *b31*b32 *u2 *u3*v1 - 4*a33 *b31*b32*m1*u1 *v2
3 3 2
+ 8*a33 *b31*b32*m1*u1*u2*v1 + 4*a33 *b31*b32*m1*u2 *v2
3 3 2 3 3 3 3 2
+ 4*a33 *b32 *u1 *u3*v2 - 8*a33 *b32 *u1*u2*u3*v1 - 4*a33 *b32 *u2 *u3*v2
3 2 2 3 2 3 2 2
- 4*a33 *b32 *m1*u1 *v1 - 8*a33 *b32 *m1*u1*u2*v2 + 4*a33 *b32 *m1*u2 *v1
3 2 2 3 2 2 2 2 2 2 2
- 4*a33 *b32 *n3*u1 *u3 - 4*a33 *b32 *n3*u2 *u3 - 2*a33 *b31 *b32 *u1 *v3
2 2 2 2 2 2 2 2
+ 8*a33 *b31 *b32 *u1*u3*v1*v3 - 2*a33 *b31 *b32 *u2 *v3
2 2 2 2 2 2
+ 8*a33 *b31 *b32 *u2*u3*v2*v3 + 4*a33 *b31 *b32*n3*u1 *v2
2 2 2 2 2
- 8*a33 *b31 *b32*n3*u1*u2*v1 - 4*a33 *b31 *b32*n3*u2 *v2
2 2 2 2 2 2 2 2 2 2
+ 4*a33 *b31 *m1 *v1 + 4*a33 *b31 *m1 *v2 + 8*a33 *b31*b32 *n3*u1*u3*v3
2 2 4 2 2
- 8*a33 *b31*b32*m1*n3*u2*v3 - 2*a33 *b32 *u1 *v3
2 4 2 4 2 2 2 4
+ 8*a33 *b32 *u1*u3*v1*v3 - 2*a33 *b32 *u2 *v3 + 8*a33 *b32 *u2*u3*v2*v3
2 3 2 2 3 2 3 2
+ 4*a33 *b32 *n3*u1 *v2 - 8*a33 *b32 *n3*u1*u2*v1 - 4*a33 *b32 *n3*u2 *v2
2 3 2 2 2 2 2 2 2 2
+ 8*a33 *b32 *n3*u2*u3*v3 + 4*a33 *b32 *m1 *v1 + 4*a33 *b32 *m1 *v2
2 2 2 2 2 2 2 2 2 2
+ 8*a33 *b32 *m1*n3*u1*v3 - 4*a33 *b32 *n3 *u1 - 4*a33 *b32 *n3 *u2
3 2 2 3 2
- 4*a33*b31 *b32 *u3*v1*v3 + 4*a33*b31 *b32*m1*v2*v3
3 2 3 2 2 3 2
- 8*a33*b31 *m1*n3*v1 - 8*a33*b31 *m1*n3*v2 - 4*a33*b31 *b32 *u3*v2*v3
2 2 2 2 2
- 4*a33*b31 *b32 *m1*v1*v3 + 8*a33*b31 *b32 *n3*u1*v1*v3
2 2 2 2 2
+ 8*a33*b31 *b32 *n3*u2*v2*v3 - 4*a33*b31 *b32 *n3*u3*v3
2 2 4 2
+ 8*a33*b31 *b32*n3 *u2*v3 - 4*a33*b31*b32 *u3*v1*v3
3 2 2 2
+ 4*a33*b31*b32 *m1*v2*v3 - 8*a33*b31*b32 *m1*n3*v1
2 2 5 2 4 2
- 8*a33*b31*b32 *m1*n3*v2 - 4*a33*b32 *u3*v2*v3 - 4*a33*b32 *m1*v1*v3
4 4 4 2
+ 8*a33*b32 *n3*u1*v1*v3 + 8*a33*b32 *n3*u2*v2*v3 - 4*a33*b32 *n3*u3*v3
3 2 4 2 4 4 2 2 2
+ 8*a33*b32 *n3 *u2*v3 - b31 *b32 *v1 - 2*b31 *b32 *v1 *v2
4 2 2 2 4 2 4 4 2 2 2
- 2*b31 *b32 *v1 *v3 - b31 *b32 *v2 - 2*b31 *b32 *v2 *v3
4 2 4 2 2 4 2 2 2 4 4
- 4*b31 *b32*n3*v2*v3 + 4*b31 *n3 *v1 + 4*b31 *n3 *v2 - 2*b31 *b32 *v1
2 4 2 2 2 4 2 2 2 4 4
- 4*b31 *b32 *v1 *v2 - 4*b31 *b32 *v1 *v3 - 2*b31 *b32 *v2
2 4 2 2 2 3 2 2 2 2 2
- 4*b31 *b32 *v2 *v3 - 8*b31 *b32 *n3*v2*v3 + 8*b31 *b32 *n3 *v1
2 2 2 2 6 4 6 2 2 6 2 2
+ 8*b31 *b32 *n3 *v2 - b32 *v1 - 2*b32 *v1 *v2 - 2*b32 *v1 *v3
6 4 6 2 2 5 2 4 2 2
- b32 *v2 - 2*b32 *v2 *v3 - 4*b32 *n3*v2*v3 + 4*b32 *n3 *v1
4 2 2
+ 4*b32 *n3 *v2
2 2 2 2 2 2 2 2 2
=((4*((v1 + v2 )*n3 - b32*v2*v3 )*n3 - (v1 + v2 + 2*v3 )*(v1 + v2 )*b32 )
2 2 2 2 4 2 2 2 2 2
*b32 + (u1 + u2 ) *a33 )*b32 - 4*(b31 + b32 )*(b31*b32 *u3*v1*v3
2 2 2 3 2
- b31*b32*m1*v2*v3 + 2*b31*m1*n3*v1 + 2*b31*m1*n3*v2 + b32 *u3*v2*v3
2 2 2 2
+ b32 *m1*v1*v3 - 2*b32 *n3*u1*v1*v3 - 2*b32 *n3*u2*v2*v3
2 2 2
+ b32 *n3*u3*v3 - 2*b32*n3 *u2*v3)*a33 +
2 2 2 2 2 2 2 2 2
(4*((v1 + v2 )*n3 - b32*v2*v3 )*n3 - (v1 + v2 + 2*v3 )*(v1 + v2 )*b32 )
2 2 2 2 2
*(b31 + 2*b32 )*b31 - 4*(((u1 *v1 + 2*u1*u2*v2 - u2 *v1)*m1
2 2 2 2
+ (u1 + u2 )*n3*u3 - (u1 *v2 - 2*u1*u2*v1 - u2 *v2)*b32*u3)*b32 + (
2 2
(u1 *v1 + 2*u1*u2*v2 - u2 *v1)*b32*u3
2 2 3
+ (u1 *v2 - 2*u1*u2*v1 - u2 *v2)*m1)*b31)*a33 *b32 - 2*(((
3
((u1*v3 - 4*u3*v1)*u1 + (u2*v3 - 4*u3*v2)*u2)*b32
- 4*(b32*u1*u3 - m1*u2)*b31*n3)*v3
2
- 2*((u1*v2 - 2*u2*v1)*u1 - (u2*v2 - 2*u3*v3)*u2)*b32 *n3
2 2 2 2 2
- 2*(((v1 + v2 )*m1 + 2*n3*u1*v3)*m1 - (u1 + u2 )*n3 )*b32)*b32 - (
2 2 2 2 2
2*((u1 *v2 - 2*u1*u2*v1 - u2 *v2)*b32*n3 + (v1 + v2 )*m1 )
2 2 2
- ((u1*v3 - 4*u3*v1)*u1 + (u2*v3 - 4*u3*v2)*u2)*b32 *v3)*b31 )*a33
And again in machine readable form:
HAM=(a33**2*b32*u1**2 + a33**2*b32*u2**2 + 2*a33**2*b32*u3**2 + 2*a33*b31*b32*u3
*v1 - 2*a33*b31*m1*v2 + 2*a33*b32**2*u3*v2 + 2*a33*b32*m1*v1 + 2*a33*b32*n3*u3 -
b31**2*b32*v3**2 + 2*b31**2*n3*v2 - b32**3*v3**2 + 2*b32**2*n3*v2)/(2*a33*b32)$
INT=a33**4*b32**2*u1**4 + 2*a33**4*b32**2*u1**2*u2**2 + a33**4*b32**2*u2**4 - 4*
a33**3*b31*b32**2*u1**2*u3*v1 - 8*a33**3*b31*b32**2*u1*u2*u3*v2 + 4*a33**3*b31*
b32**2*u2**2*u3*v1 - 4*a33**3*b31*b32*m1*u1**2*v2 + 8*a33**3*b31*b32*m1*u1*u2*v1
+ 4*a33**3*b31*b32*m1*u2**2*v2 + 4*a33**3*b32**3*u1**2*u3*v2 - 8*a33**3*b32**3*
u1*u2*u3*v1 - 4*a33**3*b32**3*u2**2*u3*v2 - 4*a33**3*b32**2*m1*u1**2*v1 - 8*a33
**3*b32**2*m1*u1*u2*v2 + 4*a33**3*b32**2*m1*u2**2*v1 - 4*a33**3*b32**2*n3*u1**2*
u3 - 4*a33**3*b32**2*n3*u2**2*u3 - 2*a33**2*b31**2*b32**2*u1**2*v3**2 + 8*a33**2
*b31**2*b32**2*u1*u3*v1*v3 - 2*a33**2*b31**2*b32**2*u2**2*v3**2 + 8*a33**2*b31**
2*b32**2*u2*u3*v2*v3 + 4*a33**2*b31**2*b32*n3*u1**2*v2 - 8*a33**2*b31**2*b32*n3*
u1*u2*v1 - 4*a33**2*b31**2*b32*n3*u2**2*v2 + 4*a33**2*b31**2*m1**2*v1**2 + 4*a33
**2*b31**2*m1**2*v2**2 + 8*a33**2*b31*b32**2*n3*u1*u3*v3 - 8*a33**2*b31*b32*m1*
n3*u2*v3 - 2*a33**2*b32**4*u1**2*v3**2 + 8*a33**2*b32**4*u1*u3*v1*v3 - 2*a33**2*
b32**4*u2**2*v3**2 + 8*a33**2*b32**4*u2*u3*v2*v3 + 4*a33**2*b32**3*n3*u1**2*v2 -
8*a33**2*b32**3*n3*u1*u2*v1 - 4*a33**2*b32**3*n3*u2**2*v2 + 8*a33**2*b32**3*n3*
u2*u3*v3 + 4*a33**2*b32**2*m1**2*v1**2 + 4*a33**2*b32**2*m1**2*v2**2 + 8*a33**2*
b32**2*m1*n3*u1*v3 - 4*a33**2*b32**2*n3**2*u1**2 - 4*a33**2*b32**2*n3**2*u2**2 -
4*a33*b31**3*b32**2*u3*v1*v3**2 + 4*a33*b31**3*b32*m1*v2*v3**2 - 8*a33*b31**3*
m1*n3*v1**2 - 8*a33*b31**3*m1*n3*v2**2 - 4*a33*b31**2*b32**3*u3*v2*v3**2 - 4*a33
*b31**2*b32**2*m1*v1*v3**2 + 8*a33*b31**2*b32**2*n3*u1*v1*v3 + 8*a33*b31**2*b32
**2*n3*u2*v2*v3 - 4*a33*b31**2*b32**2*n3*u3*v3**2 + 8*a33*b31**2*b32*n3**2*u2*v3
- 4*a33*b31*b32**4*u3*v1*v3**2 + 4*a33*b31*b32**3*m1*v2*v3**2 - 8*a33*b31*b32**
2*m1*n3*v1**2 - 8*a33*b31*b32**2*m1*n3*v2**2 - 4*a33*b32**5*u3*v2*v3**2 - 4*a33*
b32**4*m1*v1*v3**2 + 8*a33*b32**4*n3*u1*v1*v3 + 8*a33*b32**4*n3*u2*v2*v3 - 4*a33
*b32**4*n3*u3*v3**2 + 8*a33*b32**3*n3**2*u2*v3 - b31**4*b32**2*v1**4 - 2*b31**4*
b32**2*v1**2*v2**2 - 2*b31**4*b32**2*v1**2*v3**2 - b31**4*b32**2*v2**4 - 2*b31**
4*b32**2*v2**2*v3**2 - 4*b31**4*b32*n3*v2*v3**2 + 4*b31**4*n3**2*v1**2 + 4*b31**
4*n3**2*v2**2 - 2*b31**2*b32**4*v1**4 - 4*b31**2*b32**4*v1**2*v2**2 - 4*b31**2*
b32**4*v1**2*v3**2 - 2*b31**2*b32**4*v2**4 - 4*b31**2*b32**4*v2**2*v3**2 - 8*b31
**2*b32**3*n3*v2*v3**2 + 8*b31**2*b32**2*n3**2*v1**2 + 8*b31**2*b32**2*n3**2*v2
**2 - b32**6*v1**4 - 2*b32**6*v1**2*v2**2 - 2*b32**6*v1**2*v3**2 - b32**6*v2**4
- 2*b32**6*v2**2*v3**2 - 4*b32**5*n3*v2*v3**2 + 4*b32**4*n3**2*v1**2 + 4*b32**4*
n3**2*v2**2$