Solution 2 to problem e3null
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Parameters |
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Expressions
The solution is given through the following expressions:
b22=0
b31=0
b32=0
b33=0
c12=0
c13=0
c22=0
c23=0
c33=0
m1=0
m2=0
m3=0
r6=0
r5=0
r4=0
2 2 2 2
r3=(a22 *n3*q1 - 2*a22*a33*n3*q1 + a22*n1 *p10 - a22*n3 *p10 + a33 *n3*q1
2 2 2 3 3 2
- a33*n1 *p10 + a33*n3 *p10 - k1*n1 *n3 + k1*n3 )/(a22 - 3*a22 *a33
2 3
+ 3*a22*a33 - a33 )
2 2 2
r2=(a22 *n2*n3*q1 - 2*a22*a33*n2*n3*q1 + a22*n1 *n2*p10 - a22*n2*n3 *p10
2 2 2 2
+ a33 *n2*n3*q1 - a33*n1 *n2*p10 + a33*n2*n3 *p10 - k1*n1 *n2*n3
3 3 2 2 3
+ k1*n2*n3 )/(a22 *n3 - 3*a22 *a33*n3 + 3*a22*a33 *n3 - a33 *n3)
2 3 2
r1=(a22 *n1*n3*q1 - 2*a22*a33*n1*n3*q1 + a22*n1 *p10 - a22*n1*n3 *p10
2 3 2 3 3
+ a33 *n1*n3*q1 - a33*n1 *p10 + a33*n1*n3 *p10 - k1*n1 *n3 + k1*n1*n3 )/(
3 2 2 3
a22 *n3 - 3*a22 *a33*n3 + 3*a22*a33 *n3 - a33 *n3)
q19=0
q18=0
q16=0
q15=0
q14=0
q13=0
q12=0
q11=0
q10=0
q9=0
q8=0
q7=0
- 2*a22*n2*p10 + 2*a33*n2*p10 + 2*k1*n2*n3
q5=---------------------------------------------
2 2
a22 - 2*a22*a33 + a33
- 2*a22*n1*p10 + 2*a33*n1*p10 + 2*k1*n1*n3
q4=---------------------------------------------
2 2
a22 - 2*a22*a33 + a33
2 2 2 2
q3=(a22 *n3*q1 - 2*a22*a33*n3*q1 + a22*n1 *p10 - a22*n2 *p10 + a33 *n3*q1
2 2 2 2 2
- a33*n1 *p10 + a33*n2 *p10 - k1*n1 *n3 + k1*n2 *n3)/(a22 *n3
2
- 2*a22*a33*n3 + a33 *n3)
- 2*a22*n1*n2*p10 + 2*a33*n1*n2*p10 + 2*k1*n1*n2*n3
q2=------------------------------------------------------
2 2
a22 *n3 - 2*a22*a33*n3 + a33 *n3
p50=0
p49=0
p48=0
p47=0
p46=0
p45=0
p44=0
p43=0
p42=0
p41=0
p40=0
p39=0
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
p23=0
p22=0
p21=0
p20=0
p19=0
p18=0
p17=0
p16=0
p15=0
p14=0
p13=0
p12=0
p11=0
n2*p10
p9=--------
n3
n1*p10
p8=--------
n3
- a22*p10 + a33*p10 + 2*k1*n3
p7=--------------------------------
a22 - a33
p6=0
- a22*p10 + a33*p10 + 2*k1*n3
p5=--------------------------------
a22 - a33
- a22*n2*p10 + a33*n2*p10 + 2*k1*n2*n3
p4=-----------------------------------------
a22*n3 - a33*n3
- a22*n1*p10 + a33*n1*p10 + 2*k1*n1*n3
p3=-----------------------------------------
a22*n3 - a33*n3
- a22*n2*p10 + a33*n2*p10 + 2*k1*n2*n3
p2=-----------------------------------------
a22*n3 - a33*n3
- a22*n1*p10 + a33*n1*p10 + 2*k1*n1*n3
p1=-----------------------------------------
a22*n3 - a33*n3
k104=0
k103=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
k95=0
k94=0
k93=0
k92=0
k90=0
k89=0
k88=0
k87=0
k86=0
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
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
k42=0
k41=0
k40=0
k39=0
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=0
k31=0
k30=0
k28=0
k27=0
k26=0
k25=0
k24=0
k23=0
k21=0
k20=0
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
a22*p10 - a33*p10
k12=-------------------
n3
k11=0
a22*p10 - a33*p10
k10=-------------------
n3
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:
n3,q1,k1,n2,p10,n1,a33,a22
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2
HAM=a22*u1 + a22*u2 + a33*u3 + n1*u1 + n2*u2 + n3*u3
4 2 2 4 2 2 3 2 2
INT=(a22 *p10*u1 *u3 + a22 *p10*u2 *u3 - 4*a22 *a33*p10*u1 *u3
3 2 2 3 4 3 2 2
- 4*a22 *a33*p10*u2 *u3 + a22 *k1*n3*u1 + 2*a22 *k1*n3*u1 *u2
3 4 3 3 3 2
+ a22 *k1*n3*u2 - a22 *n1*p10*u1 - a22 *n1*p10*u1*u2
3 2 3 2 3 3
+ a22 *n1*p10*u1*u3 - a22 *n2*p10*u1 *u2 - a22 *n2*p10*u2
3 2 3 2 3 2
+ a22 *n2*p10*u2*u3 - a22 *n3*p10*u1 *u3 - a22 *n3*p10*u2 *u3
3 3 3 2 3 2
+ a22 *n3*p10*u3 + a22 *n3*q1*u1 + a22 *n3*q1*u2
2 2 2 2 2 2 2 2 2 4
+ 6*a22 *a33 *p10*u1 *u3 + 6*a22 *a33 *p10*u2 *u3 - 3*a22 *a33*k1*n3*u1
2 2 2 2 4 2 3
- 6*a22 *a33*k1*n3*u1 *u2 - 3*a22 *a33*k1*n3*u2 + 3*a22 *a33*n1*p10*u1
2 2 2 2
+ 3*a22 *a33*n1*p10*u1*u2 - 3*a22 *a33*n1*p10*u1*u3
2 2 2 3
+ 3*a22 *a33*n2*p10*u1 *u2 + 3*a22 *a33*n2*p10*u2
2 2 2 2
- 3*a22 *a33*n2*p10*u2*u3 + 3*a22 *a33*n3*p10*u1 *u3
2 2 2 3 2 2
+ 3*a22 *a33*n3*p10*u2 *u3 - 3*a22 *a33*n3*p10*u3 - 3*a22 *a33*n3*q1*u1
2 2 2 3 2 2
- 3*a22 *a33*n3*q1*u2 + 2*a22 *k1*n1*n3*u1 + 2*a22 *k1*n1*n3*u1*u2
2 2 2 3 2 2 2
+ 2*a22 *k1*n2*n3*u1 *u2 + 2*a22 *k1*n2*n3*u2 + 2*a22 *k1*n3 *u1 *u3
2 2 2 2 2 2 2
+ 2*a22 *k1*n3 *u2 *u3 + a22 *n1 *p10*u2 - 2*a22 *n1*n2*p10*u1*u2
2 2 2 2 2
- 2*a22 *n1*n3*p10*u1*u3 + a22 *n1*n3*q1*u1 - a22 *n2 *p10*u2
2 2 2 2
- 2*a22 *n2*n3*p10*u2*u3 + a22 *n2*n3*q1*u2 + a22 *n3 *q1*u3
3 2 2 3 2 2 2 4
- 4*a22*a33 *p10*u1 *u3 - 4*a22*a33 *p10*u2 *u3 + 3*a22*a33 *k1*n3*u1
2 2 2 2 4 2 3
+ 6*a22*a33 *k1*n3*u1 *u2 + 3*a22*a33 *k1*n3*u2 - 3*a22*a33 *n1*p10*u1
2 2 2 2
- 3*a22*a33 *n1*p10*u1*u2 + 3*a22*a33 *n1*p10*u1*u3
2 2 2 3
- 3*a22*a33 *n2*p10*u1 *u2 - 3*a22*a33 *n2*p10*u2
2 2 2 2
+ 3*a22*a33 *n2*p10*u2*u3 - 3*a22*a33 *n3*p10*u1 *u3
2 2 2 3 2 2
- 3*a22*a33 *n3*p10*u2 *u3 + 3*a22*a33 *n3*p10*u3 + 3*a22*a33 *n3*q1*u1
2 2 3
+ 3*a22*a33 *n3*q1*u2 - 4*a22*a33*k1*n1*n3*u1
2 2
- 4*a22*a33*k1*n1*n3*u1*u2 - 4*a22*a33*k1*n2*n3*u1 *u2
3 2 2
- 4*a22*a33*k1*n2*n3*u2 - 4*a22*a33*k1*n3 *u1 *u3
2 2 2 2
- 4*a22*a33*k1*n3 *u2 *u3 - 2*a22*a33*n1 *p10*u2
+ 4*a22*a33*n1*n2*p10*u1*u2 + 4*a22*a33*n1*n3*p10*u1*u3
2 2
- 2*a22*a33*n1*n3*q1*u1 + 2*a22*a33*n2 *p10*u2
2
+ 4*a22*a33*n2*n3*p10*u2*u3 - 2*a22*a33*n2*n3*q1*u2 - 2*a22*a33*n3 *q1*u3
2 2 2
- a22*k1*n1 *n3*u2 + 2*a22*k1*n1*n2*n3*u1*u2 + 2*a22*k1*n1*n3 *u1*u3
2 2 2 3
+ a22*k1*n2 *n3*u2 + 2*a22*k1*n2*n3 *u2*u3 + a22*n1 *p10*u1
2 2 2
+ a22*n1 *n2*p10*u2 + a22*n1 *n3*p10*u3 - a22*n1*n3 *p10*u1
2 3 4 2 2 4 2 2
- a22*n2*n3 *p10*u2 - a22*n3 *p10*u3 + a33 *p10*u1 *u3 + a33 *p10*u2 *u3
3 4 3 2 2 3 4 3 3
- a33 *k1*n3*u1 - 2*a33 *k1*n3*u1 *u2 - a33 *k1*n3*u2 + a33 *n1*p10*u1
3 2 3 2 3 2
+ a33 *n1*p10*u1*u2 - a33 *n1*p10*u1*u3 + a33 *n2*p10*u1 *u2
3 3 3 2 3 2
+ a33 *n2*p10*u2 - a33 *n2*p10*u2*u3 + a33 *n3*p10*u1 *u3
3 2 3 3 3 2 3 2
+ a33 *n3*p10*u2 *u3 - a33 *n3*p10*u3 - a33 *n3*q1*u1 - a33 *n3*q1*u2
2 3 2 2 2 2
+ 2*a33 *k1*n1*n3*u1 + 2*a33 *k1*n1*n3*u1*u2 + 2*a33 *k1*n2*n3*u1 *u2
2 3 2 2 2 2 2 2
+ 2*a33 *k1*n2*n3*u2 + 2*a33 *k1*n3 *u1 *u3 + 2*a33 *k1*n3 *u2 *u3
2 2 2 2 2
+ a33 *n1 *p10*u2 - 2*a33 *n1*n2*p10*u1*u2 - 2*a33 *n1*n3*p10*u1*u3
2 2 2 2 2
+ a33 *n1*n3*q1*u1 - a33 *n2 *p10*u2 - 2*a33 *n2*n3*p10*u2*u3
2 2 2 2 2
+ a33 *n2*n3*q1*u2 + a33 *n3 *q1*u3 + a33*k1*n1 *n3*u2
2 2 2
- 2*a33*k1*n1*n2*n3*u1*u2 - 2*a33*k1*n1*n3 *u1*u3 - a33*k1*n2 *n3*u2
2 3 2
- 2*a33*k1*n2*n3 *u2*u3 - a33*n1 *p10*u1 - a33*n1 *n2*p10*u2
2 2 2
- a33*n1 *n3*p10*u3 + a33*n1*n3 *p10*u1 + a33*n2*n3 *p10*u2
3 3 2 2 2
+ a33*n3 *p10*u3 - k1*n1 *n3*u1 - k1*n1 *n2*n3*u2 - k1*n1 *n3 *u3
3 3 4
+ k1*n1*n3 *u1 + k1*n2*n3 *u2 + k1*n3 *u3)/(n3
3 2 2 3
*(a22 - 3*a22 *a33 + 3*a22*a33 - a33 ))
And again in machine readable form:
HAM=a22*u1**2 + a22*u2**2 + a33*u3**2 + n1*u1 + n2*u2 + n3*u3$
INT=(a22**4*p10*u1**2*u3**2 + a22**4*p10*u2**2*u3**2 - 4*a22**3*a33*p10*u1**2*u3
**2 - 4*a22**3*a33*p10*u2**2*u3**2 + a22**3*k1*n3*u1**4 + 2*a22**3*k1*n3*u1**2*
u2**2 + a22**3*k1*n3*u2**4 - a22**3*n1*p10*u1**3 - a22**3*n1*p10*u1*u2**2 + a22
**3*n1*p10*u1*u3**2 - a22**3*n2*p10*u1**2*u2 - a22**3*n2*p10*u2**3 + a22**3*n2*
p10*u2*u3**2 - a22**3*n3*p10*u1**2*u3 - a22**3*n3*p10*u2**2*u3 + a22**3*n3*p10*
u3**3 + a22**3*n3*q1*u1**2 + a22**3*n3*q1*u2**2 + 6*a22**2*a33**2*p10*u1**2*u3**
2 + 6*a22**2*a33**2*p10*u2**2*u3**2 - 3*a22**2*a33*k1*n3*u1**4 - 6*a22**2*a33*k1
*n3*u1**2*u2**2 - 3*a22**2*a33*k1*n3*u2**4 + 3*a22**2*a33*n1*p10*u1**3 + 3*a22**
2*a33*n1*p10*u1*u2**2 - 3*a22**2*a33*n1*p10*u1*u3**2 + 3*a22**2*a33*n2*p10*u1**2
*u2 + 3*a22**2*a33*n2*p10*u2**3 - 3*a22**2*a33*n2*p10*u2*u3**2 + 3*a22**2*a33*n3
*p10*u1**2*u3 + 3*a22**2*a33*n3*p10*u2**2*u3 - 3*a22**2*a33*n3*p10*u3**3 - 3*a22
**2*a33*n3*q1*u1**2 - 3*a22**2*a33*n3*q1*u2**2 + 2*a22**2*k1*n1*n3*u1**3 + 2*a22
**2*k1*n1*n3*u1*u2**2 + 2*a22**2*k1*n2*n3*u1**2*u2 + 2*a22**2*k1*n2*n3*u2**3 + 2
*a22**2*k1*n3**2*u1**2*u3 + 2*a22**2*k1*n3**2*u2**2*u3 + a22**2*n1**2*p10*u2**2
- 2*a22**2*n1*n2*p10*u1*u2 - 2*a22**2*n1*n3*p10*u1*u3 + a22**2*n1*n3*q1*u1 - a22
**2*n2**2*p10*u2**2 - 2*a22**2*n2*n3*p10*u2*u3 + a22**2*n2*n3*q1*u2 + a22**2*n3
**2*q1*u3 - 4*a22*a33**3*p10*u1**2*u3**2 - 4*a22*a33**3*p10*u2**2*u3**2 + 3*a22*
a33**2*k1*n3*u1**4 + 6*a22*a33**2*k1*n3*u1**2*u2**2 + 3*a22*a33**2*k1*n3*u2**4 -
3*a22*a33**2*n1*p10*u1**3 - 3*a22*a33**2*n1*p10*u1*u2**2 + 3*a22*a33**2*n1*p10*
u1*u3**2 - 3*a22*a33**2*n2*p10*u1**2*u2 - 3*a22*a33**2*n2*p10*u2**3 + 3*a22*a33
**2*n2*p10*u2*u3**2 - 3*a22*a33**2*n3*p10*u1**2*u3 - 3*a22*a33**2*n3*p10*u2**2*
u3 + 3*a22*a33**2*n3*p10*u3**3 + 3*a22*a33**2*n3*q1*u1**2 + 3*a22*a33**2*n3*q1*
u2**2 - 4*a22*a33*k1*n1*n3*u1**3 - 4*a22*a33*k1*n1*n3*u1*u2**2 - 4*a22*a33*k1*n2
*n3*u1**2*u2 - 4*a22*a33*k1*n2*n3*u2**3 - 4*a22*a33*k1*n3**2*u1**2*u3 - 4*a22*
a33*k1*n3**2*u2**2*u3 - 2*a22*a33*n1**2*p10*u2**2 + 4*a22*a33*n1*n2*p10*u1*u2 +
4*a22*a33*n1*n3*p10*u1*u3 - 2*a22*a33*n1*n3*q1*u1 + 2*a22*a33*n2**2*p10*u2**2 +
4*a22*a33*n2*n3*p10*u2*u3 - 2*a22*a33*n2*n3*q1*u2 - 2*a22*a33*n3**2*q1*u3 - a22*
k1*n1**2*n3*u2**2 + 2*a22*k1*n1*n2*n3*u1*u2 + 2*a22*k1*n1*n3**2*u1*u3 + a22*k1*
n2**2*n3*u2**2 + 2*a22*k1*n2*n3**2*u2*u3 + a22*n1**3*p10*u1 + a22*n1**2*n2*p10*
u2 + a22*n1**2*n3*p10*u3 - a22*n1*n3**2*p10*u1 - a22*n2*n3**2*p10*u2 - a22*n3**3
*p10*u3 + a33**4*p10*u1**2*u3**2 + a33**4*p10*u2**2*u3**2 - a33**3*k1*n3*u1**4 -
2*a33**3*k1*n3*u1**2*u2**2 - a33**3*k1*n3*u2**4 + a33**3*n1*p10*u1**3 + a33**3*
n1*p10*u1*u2**2 - a33**3*n1*p10*u1*u3**2 + a33**3*n2*p10*u1**2*u2 + a33**3*n2*
p10*u2**3 - a33**3*n2*p10*u2*u3**2 + a33**3*n3*p10*u1**2*u3 + a33**3*n3*p10*u2**
2*u3 - a33**3*n3*p10*u3**3 - a33**3*n3*q1*u1**2 - a33**3*n3*q1*u2**2 + 2*a33**2*
k1*n1*n3*u1**3 + 2*a33**2*k1*n1*n3*u1*u2**2 + 2*a33**2*k1*n2*n3*u1**2*u2 + 2*a33
**2*k1*n2*n3*u2**3 + 2*a33**2*k1*n3**2*u1**2*u3 + 2*a33**2*k1*n3**2*u2**2*u3 +
a33**2*n1**2*p10*u2**2 - 2*a33**2*n1*n2*p10*u1*u2 - 2*a33**2*n1*n3*p10*u1*u3 +
a33**2*n1*n3*q1*u1 - a33**2*n2**2*p10*u2**2 - 2*a33**2*n2*n3*p10*u2*u3 + a33**2*
n2*n3*q1*u2 + a33**2*n3**2*q1*u3 + a33*k1*n1**2*n3*u2**2 - 2*a33*k1*n1*n2*n3*u1*
u2 - 2*a33*k1*n1*n3**2*u1*u3 - a33*k1*n2**2*n3*u2**2 - 2*a33*k1*n2*n3**2*u2*u3 -
a33*n1**3*p10*u1 - a33*n1**2*n2*p10*u2 - a33*n1**2*n3*p10*u3 + a33*n1*n3**2*p10
*u1 + a33*n2*n3**2*p10*u2 + a33*n3**3*p10*u3 - k1*n1**3*n3*u1 - k1*n1**2*n2*n3*
u2 - k1*n1**2*n3**2*u3 + k1*n1*n3**3*u1 + k1*n2*n3**3*u2 + k1*n3**4*u3)/(n3*(a22
**3 - 3*a22**2*a33 + 3*a22*a33**2 - a33**3))$