Solution 6 to problem e3c2new
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem e3c2new
Expressions
The solution is given through the following expressions:
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
n1=0
n2=0
m1=0
m2=0
m3*q1
r6=-------
a22
r5=0
r4=0
r2=0
r1=0
q20=0
q19=0
q17=0
q16=0
2
- a22*c33*q1 - k1*m3
q15=------------------------
2
a22
q14=0
q13=0
- b33*q1 - m3*p5
q12=-------------------
a22
q11=0
2
- a22*c33*q1 - k1*m3
q10=------------------------
2
a22
q9=0
q8=0
- b33*q1 - m3*p5
q7=-------------------
a22
q5=0
q4=0
q3=q1
q2=0
2 2
a22 *k26*m3 + 2*a22*c33*k1*m3 + a22*c33*k10*m3 + b33 *k1*m3
p56=-------------------------------------------------------------
3
a22
p55=0
p54=0
2
a22 *p19 + a22*c33*p5 + 2*b33*k1*m3
p53=-------------------------------------
2
a22
p52=0
p51=0
2 2
a22 *k26*m3 + a22*c33*k10*m3 + b33 *k1*m3
p50=-------------------------------------------
3
a22
p49=0
p48=0
p47=0
p46=0
2 2
a22 *k26*m3 + a22*c33*k10*m3 + b33 *k1*m3
p45=-------------------------------------------
3
a22
p44=0
p43=0
p42=0
a22*p14 + b33*p5 + k10*m3
p41=---------------------------
a22
p40=0
p39=0
2*k1*m3
p38=---------
a22
p37=0
2*k1*m3
p36=---------
a22
p35=0
p34=0
p33=p19
p32=0
p31=0
p30=0
p29=0
p28=0
p27=0
p26=0
p25=p14
p24=0
p23=0
p22=0
p21=0
p20=0
p18=0
p17=0
p16=0
p15=0
p13=0
p12=0
p11=0
p9=0
p8=0
p7=p5
p6=0
p4=0
p3=0
p2=0
p1=0
k125=0
k124=0
k122=0
k121=0
2 2 2 2
- a22 *c33*k26 - 2*a22*c33 *k1 - a22*c33 *k10 - b33 *c33*k1
k120=--------------------------------------------------------------
3
a22
k119=0
k118=0
2 3
- a22 *b33*k26 - 2*a22*b33*c33*k1 - a22*b33*c33*k10 - b33 *k1
k117=----------------------------------------------------------------
3
a22
k116=0
2 2 2 2
- a22 *c33*k26 - 2*a22*c33 *k1 - a22*c33 *k10 - b33 *c33*k1
k115=--------------------------------------------------------------
3
a22
k114=0
k113=0
2 3
- a22 *b33*k26 - 2*a22*b33*c33*k1 - a22*b33*c33*k10 - b33 *k1
k112=----------------------------------------------------------------
3
a22
k110=0
k109=0
2 2
a22 *k26 + 2*a22*c33*k1 + a22*c33*k10 + b33 *k1
k108=-------------------------------------------------
2
a22
k107=0
2 2
a22 *k26 + 2*a22*c33*k1 + a22*c33*k10 + b33 *k1
k106=-------------------------------------------------
2
a22
k105=0
k104=0
- 2*b33*c33*k1
k103=-----------------
2
a22
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
2
- 2*a22*c33*k10 - 2*b33 *k1
k95=------------------------------
2
a22
k94=0
k93=0
k92=0
k91=0
k90=0
- 2*b33*c33*k1
k89=-----------------
2
a22
k88=0
k87=0
k86=0
k85=0
2
- 2*a22*c33*k10 - 2*b33 *k1
k84=------------------------------
2
a22
k83=0
k82=0
k81=0
a22*k23 + b33*k10
k80=-------------------
a22
k79=0
k78=0
2*b33*k1
k77=----------
a22
k76=0
2*b33*k1
k75=----------
a22
k74=0
k73=0
k72=0
k71=0
2 2 2 2
- a22 *c33*k26 - a22*c33 *k1 - a22*c33 *k10 - b33 *c33*k1
k70=------------------------------------------------------------
3
a22
k69=0
k68=0
2 3
- a22 *b33*k26 - 2*a22*b33*c33*k1 - a22*b33*c33*k10 - b33 *k1
k67=----------------------------------------------------------------
3
a22
k66=0
2 2 2 2
- 2*a22 *c33*k26 - 2*a22*c33 *k1 - 2*a22*c33 *k10 - 2*b33 *c33*k1
k65=--------------------------------------------------------------------
3
a22
k64=0
k63=0
2 3
- a22 *b33*k26 - 2*a22*b33*c33*k1 - a22*b33*c33*k10 - b33 *k1
k62=----------------------------------------------------------------
3
a22
k61=0
k60=0
k59=0
k58=k26
k57=0
2 2
a22 *k26 + a22*c33*k10 + b33 *k1
k56=----------------------------------
2
a22
k55=0
k54=0
2 3
- a22 *b33*k26 - 2*a22*b33*c33*k1 - a22*b33*c33*k10 - b33 *k1
k53=----------------------------------------------------------------
3
a22
k52=0
k51=0
k50=0
k49=0
k48=0
2
- 2*a22*c33*k10 - 2*b33 *k1
k47=------------------------------
2
a22
k46=0
k45=0
k44=k23
k43=0
k42=0
k41=0
k40=0
k38=0
k37=0
k36=0
2 2 2 2
- a22 *c33*k26 - a22*c33 *k1 - a22*c33 *k10 - b33 *c33*k1
k35=------------------------------------------------------------
3
a22
k34=0
k33=0
2 3
- a22 *b33*k26 - 2*a22*b33*c33*k1 - a22*b33*c33*k10 - b33 *k1
k32=----------------------------------------------------------------
3
a22
k30=0
k29=0
2 2
a22 *k26 + a22*c33*k10 + b33 *k1
k28=----------------------------------
2
a22
k27=0
k25=0
k24=0
k22=0
k21=0
k20=0
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
k12=k10
k11=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:
b33, c33, n3, m3, r3, q1, p19, p14, p10, p5, k26, k23,
k10, k1, a33, 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.
3 3 3 3 3
{a22 *k1*w123 + 2*a22 *k1*w125 + a22 *k1*w127 + a22 *k10*w116 + a22 *k10*w118
3 3 3 3
+ a22 *k23*w048 + a22 *k23*w084 + a22 *k23*w105 + a22 *k26*w020
3 3 3 3
+ a22 *k26*w022 + a22 *k26*w070 + a22 *k26*w072 + a22 *k26*w100
3 2 2 2
+ a22 *k26*w102 + 2*a22 *b33*k1*w051 + 2*a22 *b33*k1*w053 + a22 *b33*k10*w048
2 2 2
- a22 *b33*k26*w011 - a22 *b33*k26*w016 - a22 *b33*k26*w061
2 2 2
- a22 *b33*k26*w066 - a22 *b33*k26*w075 - a22 *b33*k26*w096
2 2 2
+ 2*a22 *c33*k1*w020 + 2*a22 *c33*k1*w022 + a22 *c33*k10*w020
2 2 2
+ a22 *c33*k10*w022 - 2*a22 *c33*k10*w033 - 2*a22 *c33*k10*w044
2 2 2
+ a22 *c33*k10*w072 - 2*a22 *c33*k10*w081 + a22 *c33*k10*w100
2 2 2
- a22 *c33*k26*w008 - a22 *c33*k26*w013 - a22 *c33*k26*w058
2 2 2
- 2*a22 *c33*k26*w063 - a22 *c33*k26*w093 + a22*b33 *k1*w020
2 2 2
+ a22*b33 *k1*w022 - 2*a22*b33 *k1*w033 - 2*a22*b33 *k1*w044
2 2 2
+ a22*b33 *k1*w072 - 2*a22*b33 *k1*w081 + a22*b33 *k1*w100
- 2*a22*b33*c33*k1*w011 - 2*a22*b33*c33*k1*w016 - 2*a22*b33*c33*k1*w025
- 2*a22*b33*c33*k1*w039 - 2*a22*b33*c33*k1*w061 - 2*a22*b33*c33*k1*w066
- 2*a22*b33*c33*k1*w075 - 2*a22*b33*c33*k1*w096 - a22*b33*c33*k10*w011
- a22*b33*c33*k10*w016 - a22*b33*c33*k10*w061 - a22*b33*c33*k10*w066
2
- a22*b33*c33*k10*w075 - a22*b33*c33*k10*w096 - 2*a22*c33 *k1*w008
2 2 2
- 2*a22*c33 *k1*w013 - a22*c33 *k1*w058 - 2*a22*c33 *k1*w063
2 2 2 2
- a22*c33 *k1*w093 - a22*c33 *k10*w008 - a22*c33 *k10*w013 - a22*c33 *k10*w058
2 2 3 3
- 2*a22*c33 *k10*w063 - a22*c33 *k10*w093 - b33 *k1*w011 - b33 *k1*w016
3 3 3 3 2
- b33 *k1*w061 - b33 *k1*w066 - b33 *k1*w075 - b33 *k1*w096 - b33 *c33*k1*w008
2 2 2 2
- b33 *c33*k1*w013 - b33 *c33*k1*w058 - 2*b33 *c33*k1*w063 - b33 *c33*k1*w093,
a33,
a22 - a33,
a22}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2
HAM=a22*u1 + a22*u2 + a33*u3 + b33*u3*v3 + c33*v3 + m3*v3 + n3*u3
has apart from the Hamiltonian and Casimirs the following first integrals:
3 4 3 2 2 3 4 2 2
INT=a22 *u1 + 2*a22 *u1 *u2 + a22 *u2 + 2*a22 *b33*u1 *u3*v3
2 2 2 2 2 2 2 2
+ 2*a22 *b33*u2 *u3*v3 + 2*a22 *c33*u1 *v3 + 2*a22 *c33*u2 *v3
2 2 2 2 2 2 2 2 2 2
+ 2*a22 *m3*u1 *v3 + 2*a22 *m3*u2 *v3 + a22*b33 *u1 *v2 + a22*b33 *u1 *v3
2 2 2 2 2
- 2*a22*b33 *u1*u2*v1*v2 - 2*a22*b33 *u1*u3*v1*v3 + a22*b33 *u2 *v1
2 2 2 2 3
+ a22*b33 *u2 *v3 - 2*a22*b33 *u2*u3*v2*v3 - 2*a22*b33*c33*u1*v1
2 2
- 2*a22*b33*c33*u1*v1*v2 - 2*a22*b33*c33*u1*v1*v3
2 3 2
- 2*a22*b33*c33*u2*v1 *v2 - 2*a22*b33*c33*u2*v2 - 2*a22*b33*c33*u2*v2*v3
2 2 2
- 2*a22*b33*c33*u3*v1 *v3 - 2*a22*b33*c33*u3*v2 *v3 + 2*a22*b33*m3*u3*v3
2 4 2 2 2 2 2 2 2 4
- a22*c33 *v1 - 2*a22*c33 *v1 *v2 - 2*a22*c33 *v1 *v3 - a22*c33 *v2
2 2 2 3 2 2 2 2
- 2*a22*c33 *v2 *v3 + 2*a22*c33*m3*v3 - a22*m3 *v1 - a22*m3 *v2
3 3 3 2 3 2 3 2
- b33 *u1*v1 - b33 *u1*v1*v2 - b33 *u1*v1*v3 - b33 *u2*v1 *v2
3 3 3 2 2 4 2 2 2
- b33 *u2*v2 - b33 *u2*v2*v3 - b33 *c33*v1 - 2*b33 *c33*v1 *v2
2 2 2 2 4 2 2 2 2 2
- b33 *c33*v1 *v3 - b33 *c33*v2 - b33 *c33*v2 *v3 + b33 *m3*v1 *v3
2 2 2 3
+ b33 *m3*v2 *v3 + b33 *m3*v3
2 2 2 2 2
= - ((2*(((v2 + v3 + v1 )*(u1*v1 + u2*v2) + (v1 + v2 )*u3*v3)*c33
2 2 2 3
- m3*u3*v3 )*b33 + ((v1 + v2 )*m3 - 2*c33*v3 )*m3
2 2 2 2 2 2 2 2 2
+ (v1 + v2 + 2*v3 )*(v1 + v2 )*c33 - ((v2 + v3 )*u1
2 2 2 2
- 2*u2*u3*v2*v3 + (v1 + v3 )*u2 - 2*(u2*v2 + u3*v3)*u1*v1)*b33 )
2 2 2 2 2 2
*a22 + (c33*v1 + c33*v2 - m3*v3)*(v1 + v2 + v3 )*b33
2 2 2 3 2 2 2 3
- (u1 + u2 ) *a22 + (v2 + v3 + v1 )*(u1*v1 + u2*v2)*b33
2 2 2
- 2*(c33*v3 + m3 + b33*u3)*(u1 + u2 )*a22 *v3)
2 2 2 2 2 2 3 2 2
INT=a22 *u1 *u3 + a22 *u2 *u3 + a22*b33*u3 *v3 + a22*c33*u1 *v2
2 2
+ a22*c33*u1 *v3 - 2*a22*c33*u1*u2*v1*v2 - 2*a22*c33*u1*u3*v1*v3
2 2 2 2 2
+ a22*c33*u2 *v1 + a22*c33*u2 *v3 - 2*a22*c33*u2*u3*v2*v3 + a22*m3*u3 *v3
3 2 2
- b33*c33*u1*v1 - b33*c33*u1*v1*v2 - b33*c33*u1*v1*v3
2 3 2 2 4
- b33*c33*u2*v1 *v2 - b33*c33*u2*v2 - b33*c33*u2*v2*v3 - c33 *v1
2 2 2 2 2 2 2 4 2 2 2 2
- 2*c33 *v1 *v2 - c33 *v1 *v3 - c33 *v2 - c33 *v2 *v3 + c33*m3*v1 *v3
2 3
+ c33*m3*v2 *v3 + c33*m3*v3
2 2 2 2 2 2
=(((v2 + v3 )*u1 - 2*u2*u3*v2*v3 + (v1 + v3 )*u2
2
- 2*(u2*v2 + u3*v3)*u1*v1)*c33 + (b33*u3 + m3)*u3 *v3)*a22
2 2 2 2 2 2 2 2 2 2
+ (u1 + u2 )*a22 *u3 - (v1 + v2 + v3 )*(v1 + v2 )*c33
2 2 2
- ((u1*v1 + u2*v2)*b33 - m3*v3)*(v2 + v3 + v1 )*c33
2
INT=u3 *(u1*v1 + u2*v2 + u3*v3)
2
=(u2*v2 + u3*v3 + u1*v1)*u3
2 2 2 2 2 2 2 2 2 2
INT=a22*u1 *v1 + a22*u1 *v2 + a22*u1 *v3 + a22*u2 *v1 + a22*u2 *v2
2 2 3 2 2 2
+ a22*u2 *v3 - b33*u1*v1 - b33*u1*v1*v2 - b33*u1*v1*v3 - b33*u2*v1 *v2
3 2 4 2 2 2 2
- b33*u2*v2 - b33*u2*v2*v3 - c33*v1 - 2*c33*v1 *v2 - c33*v1 *v3
4 2 2 2 2 3
- c33*v2 - c33*v2 *v3 + m3*v1 *v3 + m3*v2 *v3 + m3*v3
2 2 2 2
=(a22*u1 + a22*u2 - b33*u1*v1 - b33*u2*v2 - c33*v1 - c33*v2 + m3*v3)
2 2 2
*(v1 + v2 + v3 )
2 2 2 2
INT=a22*u1 *u3 + a22*u2 *u3 + b33*u3 *v3 + c33*u3*v3 - m3*u1*v1 - m3*u2*v2
2 2
=(b33*u3 + c33*v3)*u3*v3 - (u1*v1 + u2*v2)*m3 + (u1 + u2 )*a22*u3
3
INT=u3
3
=u3
INT=u3*(u1*v1 + u2*v2 + u3*v3)
=(u2*v2 + u3*v3 + u1*v1)*u3
2 2 2
INT=u3*(v1 + v2 + v3 )
2 2 2
=(v2 + v3 + v1 )*u3
2 2 2 2
INT=a22*u1 + a22*u2 - b33*u1*v1 - b33*u2*v2 - c33*v1 - c33*v2 + m3*v3
2 2 2 2
= - ((v1 + v2 )*c33 - m3*v3 + (u1*v1 + u2*v2)*b33 - (u1 + u2 )*a22)
INT=u3
=u3
And again in machine readable form:
HAM=a22*u1**2 + a22*u2**2 + a33*u3**2 + b33*u3*v3 + c33*v3**2 + m3*v3 + n3*u3$
INT=a22**3*u1**4 + 2*a22**3*u1**2*u2**2 + a22**3*u2**4 + 2*a22**2*b33*u1**2*u3*
v3 + 2*a22**2*b33*u2**2*u3*v3 + 2*a22**2*c33*u1**2*v3**2 + 2*a22**2*c33*u2**2*v3
**2 + 2*a22**2*m3*u1**2*v3 + 2*a22**2*m3*u2**2*v3 + a22*b33**2*u1**2*v2**2 + a22
*b33**2*u1**2*v3**2 - 2*a22*b33**2*u1*u2*v1*v2 - 2*a22*b33**2*u1*u3*v1*v3 + a22*
b33**2*u2**2*v1**2 + a22*b33**2*u2**2*v3**2 - 2*a22*b33**2*u2*u3*v2*v3 - 2*a22*
b33*c33*u1*v1**3 - 2*a22*b33*c33*u1*v1*v2**2 - 2*a22*b33*c33*u1*v1*v3**2 - 2*a22
*b33*c33*u2*v1**2*v2 - 2*a22*b33*c33*u2*v2**3 - 2*a22*b33*c33*u2*v2*v3**2 - 2*
a22*b33*c33*u3*v1**2*v3 - 2*a22*b33*c33*u3*v2**2*v3 + 2*a22*b33*m3*u3*v3**2 -
a22*c33**2*v1**4 - 2*a22*c33**2*v1**2*v2**2 - 2*a22*c33**2*v1**2*v3**2 - a22*c33
**2*v2**4 - 2*a22*c33**2*v2**2*v3**2 + 2*a22*c33*m3*v3**3 - a22*m3**2*v1**2 -
a22*m3**2*v2**2 - b33**3*u1*v1**3 - b33**3*u1*v1*v2**2 - b33**3*u1*v1*v3**2 -
b33**3*u2*v1**2*v2 - b33**3*u2*v2**3 - b33**3*u2*v2*v3**2 - b33**2*c33*v1**4 - 2
*b33**2*c33*v1**2*v2**2 - b33**2*c33*v1**2*v3**2 - b33**2*c33*v2**4 - b33**2*c33
*v2**2*v3**2 + b33**2*m3*v1**2*v3 + b33**2*m3*v2**2*v3 + b33**2*m3*v3**3$
INT=a22**2*u1**2*u3**2 + a22**2*u2**2*u3**2 + a22*b33*u3**3*v3 + a22*c33*u1**2*
v2**2 + a22*c33*u1**2*v3**2 - 2*a22*c33*u1*u2*v1*v2 - 2*a22*c33*u1*u3*v1*v3 +
a22*c33*u2**2*v1**2 + a22*c33*u2**2*v3**2 - 2*a22*c33*u2*u3*v2*v3 + a22*m3*u3**2
*v3 - b33*c33*u1*v1**3 - b33*c33*u1*v1*v2**2 - b33*c33*u1*v1*v3**2 - b33*c33*u2*
v1**2*v2 - b33*c33*u2*v2**3 - b33*c33*u2*v2*v3**2 - c33**2*v1**4 - 2*c33**2*v1**
2*v2**2 - c33**2*v1**2*v3**2 - c33**2*v2**4 - c33**2*v2**2*v3**2 + c33*m3*v1**2*
v3 + c33*m3*v2**2*v3 + c33*m3*v3**3$
INT=u3**2*(u1*v1 + u2*v2 + u3*v3)$
INT=a22*u1**2*v1**2 + a22*u1**2*v2**2 + a22*u1**2*v3**2 + a22*u2**2*v1**2 + a22*
u2**2*v2**2 + a22*u2**2*v3**2 - b33*u1*v1**3 - b33*u1*v1*v2**2 - b33*u1*v1*v3**2
- b33*u2*v1**2*v2 - b33*u2*v2**3 - b33*u2*v2*v3**2 - c33*v1**4 - 2*c33*v1**2*v2
**2 - c33*v1**2*v3**2 - c33*v2**4 - c33*v2**2*v3**2 + m3*v1**2*v3 + m3*v2**2*v3
+ m3*v3**3$
INT=a22*u1**2*u3 + a22*u2**2*u3 + b33*u3**2*v3 + c33*u3*v3**2 - m3*u1*v1 - m3*u2
*v2$
INT=u3**3$
INT=u3*(u1*v1 + u2*v2 + u3*v3)$
INT=u3*(v1**2 + v2**2 + v3**2)$
INT=a22*u1**2 + a22*u2**2 - b33*u1*v1 - b33*u2*v2 - c33*v1**2 - c33*v2**2 + m3*
v3$
INT=u3$