Solution 17 to problem e3null
Expressions |
Parameters |
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Back to problem e3null
Expressions
The solution is given through the following expressions:
a22= - a33
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
1 2
---*b33
4
c33=----------
a33
n1=0
n3=0
m1=0
m2=0
m3=0
r6=0
r5=0
r4=0
r3=0
r2=0
r1=0
q19=0
q18=0
q16=0
q15=0
7 2 2
----*b33 *k1*n2
16
q14=------------------
4
a33
q13=0
q12=0
1 2
---*b33*k1*n2
4
q11=----------------
3
a33
q10=0
q9=0
q8=0
q7=0
q5=0
q4=0
5 2
- ---*k1*n2
4
q3=---------------
2
a33
q2=0
q1=0
p50=0
3 3
---*b33 *k1*n2
8
p49=----------------
4
a33
p48=0
p47=0
3 2
- ---*b33 *k1*n2
4
p46=-------------------
3
a33
p45=0
p44=0
p43=0
2
2*b33 *k1*n2
p42=--------------
3
a33
p41=0
p40=0
p39=0
p38=0
p37=0
p36=0
- 2*b33*k1*n2
p35=----------------
2
a33
p34=0
p33=0
p32=0
p31=0
1 3
- ---*b33 *k1*n2
8
p30=-------------------
4
a33
p29=0
p28=0
1 2
- ---*b33 *k1*n2
4
p27=-------------------
3
a33
p26=0
1 3
- ---*b33 *k1*n2
8
p25=-------------------
4
a33
p24=0
p23=0
3
- ---*b33*k1*n2
2
p22=------------------
2
a33
p21=0
p20=0
- 2*b33*k1*n2
p19=----------------
2
a33
p18=0
- 2*b33*k1*n2
p17=----------------
2
a33
p16=0
p15=0
1 2
- ---*b33 *k1*n2
4
p14=-------------------
3
a33
p13=0
p12=0
p11=0
p10=0
- 3*k1*n2
p9=------------
a33
p8=0
p7=0
p6=0
p5=0
2*k1*n2
p4=---------
a33
p3=0
2*k1*n2
p2=---------
a33
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
1 3
- ---*b33 *k1
2
k83=----------------
3
a33
k82=0
k81=0
k80=0
k79=0
k78=0
k77=0
2
b33 *k1
k76=---------
2
a33
k75=0
k74=0
k73=0
k72=0
k71=0
1 3
- ---*b33 *k1
2
k70=----------------
3
a33
k69=0
k68=0
k67=0
k66=0
- 6*b33*k1
k65=-------------
a33
k64=0
k63=0
- 2*b33*k1
k62=-------------
a33
k61=0
- 2*b33*k1
k60=-------------
a33
k59=0
k58=0
k57=0
k56=0
1 4
----*b33 *k1
16
k55=--------------
4
a33
k54=0
k53=0
k52=0
k51=0
1 4
---*b33 *k1
8
k50=-------------
4
a33
k49=0
k48=0
k47=0
k46=0
k45=0
2
b33 *k1
k44=---------
2
a33
k43=0
1 2
---*b33 *k1
2
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
1 4
----*b33 *k1
16
k25=--------------
4
a33
k24=0
k23=0
k21=0
1 2
---*b33 *k1
2
k20=-------------
2
a33
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
k12=6*k1
k11=0
k10=6*k1
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,b33,n2,a33
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2 2 2 2
HAM=( - 4*a33 *u1 - 4*a33 *u2 + 4*a33 *u3 + 4*a33*b33*u3*v3 + 4*a33*n2*u2
2 2
+ b33 *v3 )/(4*a33)
4 4 4 2 2 4 2 2 4 4
INT=(k1*(16*a33 *u1 + 32*a33 *u1 *u2 + 96*a33 *u1 *u3 + 16*a33 *u2
4 2 2 3 2 3 2
+ 96*a33 *u2 *u3 - 32*a33 *b33*u1 *u3*v3 - 32*a33 *b33*u2 *u3*v3
3 3 3 2 3 3
- 96*a33 *b33*u3 *v3 + 32*a33 *n2*u1 *u2 + 32*a33 *n2*u2
3 2 2 2 2 2 2 2 2 2
- 48*a33 *n2*u2*u3 + 8*a33 *b33 *u1 *v2 + 8*a33 *b33 *u2 *v1
2 2 2 2 2 2
+ 16*a33 *b33 *u2 *v2 + 16*a33 *b33 *u2*u3*v2*v3
2 2 2 2
- 32*a33 *b33*n2*u1 *v2 - 32*a33 *b33*n2*u2 *v2
2 2 2 2 2 2
- 32*a33 *b33*n2*u2*u3*v3 - 24*a33 *b33*n2*u3 *v2 - 20*a33 *n2 *u2
3 2 3 2 2 2
- 8*a33*b33 *u3*v1 *v3 - 8*a33*b33 *u3*v2 *v3 - 4*a33*b33 *n2*u2*v1
2 2 2 2
- 4*a33*b33 *n2*u2*v2 - 12*a33*b33 *n2*u2*v3
2 2 4 4
+ 32*a33*b33 *n2*u3*v2*v3 + 4*a33*b33*n2 *u2*v2 + b33 *v1
4 2 2 4 4 3 2 3 3
+ 2*b33 *v1 *v2 + b33 *v2 - 2*b33 *n2*v1 *v2 - 2*b33 *n2*v2
3 2 2 2 2 4
+ 6*b33 *n2*v2*v3 + 7*b33 *n2 *v2 ))/(16*a33 )
And again in machine readable form:
HAM=( - 4*a33**2*u1**2 - 4*a33**2*u2**2 + 4*a33**2*u3**2 + 4*a33*b33*u3*v3 + 4*
a33*n2*u2 + b33**2*v3**2)/(4*a33)$
INT=(k1*(16*a33**4*u1**4 + 32*a33**4*u1**2*u2**2 + 96*a33**4*u1**2*u3**2 + 16*
a33**4*u2**4 + 96*a33**4*u2**2*u3**2 - 32*a33**3*b33*u1**2*u3*v3 - 32*a33**3*b33
*u2**2*u3*v3 - 96*a33**3*b33*u3**3*v3 + 32*a33**3*n2*u1**2*u2 + 32*a33**3*n2*u2
**3 - 48*a33**3*n2*u2*u3**2 + 8*a33**2*b33**2*u1**2*v2**2 + 8*a33**2*b33**2*u2**
2*v1**2 + 16*a33**2*b33**2*u2**2*v2**2 + 16*a33**2*b33**2*u2*u3*v2*v3 - 32*a33**
2*b33*n2*u1**2*v2 - 32*a33**2*b33*n2*u2**2*v2 - 32*a33**2*b33*n2*u2*u3*v3 - 24*
a33**2*b33*n2*u3**2*v2 - 20*a33**2*n2**2*u2**2 - 8*a33*b33**3*u3*v1**2*v3 - 8*
a33*b33**3*u3*v2**2*v3 - 4*a33*b33**2*n2*u2*v1**2 - 4*a33*b33**2*n2*u2*v2**2 -
12*a33*b33**2*n2*u2*v3**2 + 32*a33*b33**2*n2*u3*v2*v3 + 4*a33*b33*n2**2*u2*v2 +
b33**4*v1**4 + 2*b33**4*v1**2*v2**2 + b33**4*v2**4 - 2*b33**3*n2*v1**2*v2 - 2*
b33**3*n2*v2**3 + 6*b33**3*n2*v2*v3**2 + 7*b33**2*n2**2*v2**2))/(16*a33**4)$