Solution 130 to problem e3null
Expressions |
Parameters |
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Back to problem e3null
Expressions
The solution is given through the following expressions:
1
a22=---*a33
3
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
3 2
---*b33
2
c33=----------
a33
n1=0
n2=0
m1=0
m2=0
2
- 3*m3 *p5
r6=-------------
a33*b33
r5=0
r4=0
r2=0
r1=0
q19=0
q18=0
q16=0
q15=0
9 2
---*b33*m3*p5 - 9*k1*m3
2
q14=--------------------------
2
a33
q13=0
q12=0
q11=0
q10=0
9 2
---*b33*m3*p5 - 9*k1*m3
2
q9=--------------------------
2
a33
q8=0
q7=0
q5=0
q4=0
- m3*p5
q3=----------
b33
q2=0
- m3*p5
q1=----------
b33
p50=0
p49=0
p48=0
2 9 2
a33 *p15 + ---*b33 *p5 + 18*b33*k1*m3
2
p47=---------------------------------------
2
a33
p46=0
p45=0
2
- 27*b33 *k1*m3
p44=------------------
3
a33
p43=0
p42=0
p41=0
p40=0
2
- 27*b33 *k1*m3
p39=------------------
3
a33
p38=0
p37=0
3*b33*p5
p36=----------
a33
p35=0
p34=0
6*k1*m3
p33=---------
a33
p32=0
6*k1*m3
p31=---------
a33
p30=0
p29=0
p28=p15
p27=0
p26=0
p25=0
p24=0
p23=0
p22=0
p21=0
p20=0
p19=0
p18=0
p17=0
p16=0
p14=0
p13=0
p12=0
p11=0
p9=0
p8=0
p7=p5
p6=0
p4=0
p3=0
p2=0
p1=0
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
3
- 27*b33 *k1
k83=---------------
3
a33
k82=0
k81=0
k80=0
k79=0
k78=0
k77=0
2
- 18*b33 *k1
k76=---------------
2
a33
k75=0
k74=0
k73=0
k72=0
k71=0
3
- 27*b33 *k1
k70=---------------
3
a33
k69=0
k68=0
k67=0
k66=0
k65=0
k64=0
k63=0
6*b33*k1
k62=----------
a33
k61=0
6*b33*k1
k60=----------
a33
k59=0
k58=0
k57=0
k56=0
81 4
----*b33 *k1
4
k55=--------------
4
a33
k54=0
k53=0
k52=0
k51=0
81 4
----*b33 *k1
2
k50=--------------
4
a33
k49=0
k48=0
k47=0
k46=0
k45=0
2
- 18*b33 *k1
k44=---------------
2
a33
k43=0
2
- 9*b33 *k1
k42=--------------
2
a33
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
81 4
----*b33 *k1
4
k25=--------------
4
a33
k24=0
k23=0
k21=0
2
- 9*b33 *k1
k20=--------------
2
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
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,m3,r3,p15,p10,p5,b33,k1,a33
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2 2 2 2
HAM=(2*a33 *u1 + 2*a33 *u2 + 6*a33 *u3 + 6*a33*b33*u3*v3 + 6*a33*m3*v3
2 2
+ 6*a33*n3*u3 + 9*b33 *v3 )/(6*a33)
4 4 4 2 2 4 4
INT=(4*a33 *b33*k1*u1 + 8*a33 *b33*k1*u1 *u2 + 4*a33 *b33*k1*u2
4 3 4 2 4 2
+ 4*a33 *b33*p10*u3 + 4*a33 *b33*p15*u3*v1 + 4*a33 *b33*p15*u3*v2
4 2 4 2 4 2
+ 4*a33 *b33*p15*u3*v3 + 4*a33 *b33*p5*u1 *u3 + 4*a33 *b33*p5*u2 *u3
4 4 2 4 2
+ 4*a33 *b33*r3*u3 - 4*a33 *m3*p5*u1 - 4*a33 *m3*p5*u2
3 2 2 3 2 2
+ 24*a33 *b33 *k1*u1 *u3*v3 + 24*a33 *b33 *k1*u2 *u3*v3
3 2 2 3 2
+ 12*a33 *b33 *p5*u3 *v3 + 24*a33 *b33*k1*m3*u1 *v3
3 2 3 2 2 3 2 2
+ 24*a33 *b33*k1*m3*u2 *v3 - 12*a33 *m3 *p5*v3 - 36*a33 *b33 *k1*u1 *v2
2 3 2 2 2 3 2 2
- 36*a33 *b33 *k1*u2 *v1 - 72*a33 *b33 *k1*u2 *v2
2 3 2 3 2
- 72*a33 *b33 *k1*u2*u3*v2*v3 + 18*a33 *b33 *p5*u3*v3
2 2 2 2 2 2
+ 72*a33 *b33 *k1*m3*u3*v3 + 18*a33 *b33 *m3*p5*v1
2 2 2 2 2 2 2 2 2
+ 18*a33 *b33 *m3*p5*v2 - 36*a33 *b33*k1*m3 *v1 - 36*a33 *b33*k1*m3 *v2
4 2 4 2
- 108*a33*b33 *k1*u3*v1 *v3 - 108*a33*b33 *k1*u3*v2 *v3
3 2 3 2 5 4
- 108*a33*b33 *k1*m3*v1 *v3 - 108*a33*b33 *k1*m3*v2 *v3 + 81*b33 *k1*v1
5 2 2 5 4 4
+ 162*b33 *k1*v1 *v2 + 81*b33 *k1*v2 )/(4*a33 *b33)
And again in machine readable form:
HAM=(2*a33**2*u1**2 + 2*a33**2*u2**2 + 6*a33**2*u3**2 + 6*a33*b33*u3*v3 + 6*a33*
m3*v3 + 6*a33*n3*u3 + 9*b33**2*v3**2)/(6*a33)$
INT=(4*a33**4*b33*k1*u1**4 + 8*a33**4*b33*k1*u1**2*u2**2 + 4*a33**4*b33*k1*u2**4
+ 4*a33**4*b33*p10*u3**3 + 4*a33**4*b33*p15*u3*v1**2 + 4*a33**4*b33*p15*u3*v2**
2 + 4*a33**4*b33*p15*u3*v3**2 + 4*a33**4*b33*p5*u1**2*u3 + 4*a33**4*b33*p5*u2**2
*u3 + 4*a33**4*b33*r3*u3 - 4*a33**4*m3*p5*u1**2 - 4*a33**4*m3*p5*u2**2 + 24*a33
**3*b33**2*k1*u1**2*u3*v3 + 24*a33**3*b33**2*k1*u2**2*u3*v3 + 12*a33**3*b33**2*
p5*u3**2*v3 + 24*a33**3*b33*k1*m3*u1**2*v3 + 24*a33**3*b33*k1*m3*u2**2*v3 - 12*
a33**3*m3**2*p5*v3 - 36*a33**2*b33**3*k1*u1**2*v2**2 - 36*a33**2*b33**3*k1*u2**2
*v1**2 - 72*a33**2*b33**3*k1*u2**2*v2**2 - 72*a33**2*b33**3*k1*u2*u3*v2*v3 + 18*
a33**2*b33**3*p5*u3*v3**2 + 72*a33**2*b33**2*k1*m3*u3*v3**2 + 18*a33**2*b33**2*
m3*p5*v1**2 + 18*a33**2*b33**2*m3*p5*v2**2 - 36*a33**2*b33*k1*m3**2*v1**2 - 36*
a33**2*b33*k1*m3**2*v2**2 - 108*a33*b33**4*k1*u3*v1**2*v3 - 108*a33*b33**4*k1*u3
*v2**2*v3 - 108*a33*b33**3*k1*m3*v1**2*v3 - 108*a33*b33**3*k1*m3*v2**2*v3 + 81*
b33**5*k1*v1**4 + 162*b33**5*k1*v1**2*v2**2 + 81*b33**5*k1*v2**4)/(4*a33**4*b33)
$