Solution 12 to problem e3c2new
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Parameters |
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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
4
c33=-------------
a33
3*b33*n1
m1=----------
a33
3*b33*n2
m2=----------
a33
3
---*b33*n3
2
m3=------------
a33
9
---*b33*n3*q1
2
r6=---------------
2
a33
9
---*b33*n2*q1
4
r5=---------------
2
a33
9
---*b33*n1*q1
4
r4=---------------
2
a33
3
- ---*n3*q1
2
r3=--------------
a33
3
- ---*n2*q1
2
r2=--------------
a33
3
- ---*n1*q1
2
r1=--------------
a33
q20=0
q19=0
q17=0
q16=0
9 2
---*b33 *q1
4
q15=-------------
2
a33
q14=0
q13=0
- 3*b33*q1
q12=-------------
a33
q11=0
9 2
---*b33 *q1
4
q10=-------------
2
a33
q9=0
q8=0
- 3*b33*q1
q7=-------------
a33
q5=0
q4=0
q3=q1
q2=0
9
---*b33*k26*n3
2
p56=----------------
2
a33
9
---*b33*k26*n2
4
p55=----------------
2
a33
9
---*b33*k26*n1
4
p54=----------------
2
a33
3
- ---*k26*n3
2
p53=---------------
a33
3
- ---*k26*n2
2
p52=---------------
a33
3
- ---*k26*n1
2
p51=---------------
a33
9
---*b33*k26*n3
2
p50=----------------
2
a33
p49=0
p48=0
p47=0
p46=0
9
---*b33*k26*n3
2
p45=----------------
2
a33
p44=0
p43=0
p42=0
p41=0
p40=0
p39=0
p38=0
p37=0
p36=0
9
---*b33*k26*n2
4
p35=----------------
2
a33
9
---*b33*k26*n1
4
p34=----------------
2
a33
3
- ---*k26*n3
2
p33=---------------
a33
3
- ---*k26*n2
2
p32=---------------
a33
3
- ---*k26*n1
2
p31=---------------
a33
9
---*b33*k26*n2
4
p30=----------------
2
a33
p29=0
p28=0
p27=0
p26=0
p25=0
p24=0
p23=0
p22=0
p21=0
9
---*b33*k26*n1
4
p20=----------------
2
a33
3
- ---*k26*n3
2
p19=---------------
a33
3
- ---*k26*n2
2
p18=---------------
a33
3
- ---*k26*n1
2
p17=---------------
a33
p16=0
p15=0
p14=0
p13=0
p12=0
p11=0
p10=0
p9=0
p8=0
p7=0
p6=0
p5=0
p4=0
p3=0
p2=0
p1=0
k125=0
k124=0
k122=0
k121=0
9 2
---*b33 *k26
4
k120=--------------
2
a33
k119=0
k118=0
- 3*b33*k26
k117=--------------
a33
k116=0
9 2
---*b33 *k26
4
k115=--------------
2
a33
k114=0
k113=0
- 3*b33*k26
k112=--------------
a33
k110=0
k109=0
k108=k26
k107=0
k106=k26
k105=0
k104=0
k103=0
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
k95=0
k94=0
k93=0
k92=0
k91=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
9 2
---*b33 *k26
4
k70=--------------
2
a33
k69=0
k68=0
- 3*b33*k26
k67=--------------
a33
k66=0
9 2
---*b33 *k26
2
k65=--------------
2
a33
k64=0
k63=0
- 3*b33*k26
k62=--------------
a33
k61=0
k60=0
k59=0
k58=k26
k57=0
k56=k26
k55=0
k54=0
- 3*b33*k26
k53=--------------
a33
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
k38=0
k37=0
k36=0
9 2
---*b33 *k26
4
k35=--------------
2
a33
k34=0
k33=0
- 3*b33*k26
k32=--------------
a33
k30=0
k29=0
k28=k26
k27=0
k25=0
k24=0
k23=0
k22=0
k21=0
k20=0
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=0
k4=0
k3=0
k2=0
k1=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,k26,n1,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 + 12*a33 *u3 + 12*a33*b33*u3*v3 + 12*a33*n1*u1
2 2
+ 12*a33*n2*u2 + 12*a33*n3*u3 - 9*b33 *v3 + 36*b33*n1*v1 + 36*b33*n2*v2
+ 18*b33*n3*v3)/(12*a33)
2 2 2 2 2 2 2 2 2
INT=(4*a33 *k26*u1 *v1 + 4*a33 *k26*u1 *v2 + 4*a33 *k26*u1 *v3
2 2 2 2 2 2 2 2 2
+ 4*a33 *k26*u2 *v1 + 4*a33 *k26*u2 *v2 + 4*a33 *k26*u2 *v3
2 2 2 2 3
+ 4*a33 *q1*u1 + 4*a33 *q1*u2 - 12*a33*b33*k26*u1*v1
2 2
- 12*a33*b33*k26*u1*v1*v2 - 12*a33*b33*k26*u1*v1*v3
2 3
- 12*a33*b33*k26*u2*v1 *v2 - 12*a33*b33*k26*u2*v2
2
- 12*a33*b33*k26*u2*v2*v3 - 12*a33*b33*q1*u1*v1 - 12*a33*b33*q1*u2*v2
2 2 2
- 6*a33*k26*n1*u1*v1 - 6*a33*k26*n1*u1*v2 - 6*a33*k26*n1*u1*v3
2 2 2
- 6*a33*k26*n2*u2*v1 - 6*a33*k26*n2*u2*v2 - 6*a33*k26*n2*u2*v3
2 2 2
- 6*a33*k26*n3*u3*v1 - 6*a33*k26*n3*u3*v2 - 6*a33*k26*n3*u3*v3
2 4
- 6*a33*n1*q1*u1 - 6*a33*n2*q1*u2 - 6*a33*n3*q1*u3 + 9*b33 *k26*v1
2 2 2 2 2 2 2 4
+ 18*b33 *k26*v1 *v2 + 9*b33 *k26*v1 *v3 + 9*b33 *k26*v2
2 2 2 2 2 2 2 3
+ 9*b33 *k26*v2 *v3 + 9*b33 *q1*v1 + 9*b33 *q1*v2 + 9*b33*k26*n1*v1
2 2 2
+ 9*b33*k26*n1*v1*v2 + 9*b33*k26*n1*v1*v3 + 9*b33*k26*n2*v1 *v2
3 2 2
+ 9*b33*k26*n2*v2 + 9*b33*k26*n2*v2*v3 + 18*b33*k26*n3*v1 *v3
2 3
+ 18*b33*k26*n3*v2 *v3 + 18*b33*k26*n3*v3 + 9*b33*n1*q1*v1
2
+ 9*b33*n2*q1*v2 + 18*b33*n3*q1*v3)/(4*a33 )
And again in machine readable form:
HAM=(4*a33**2*u1**2 + 4*a33**2*u2**2 + 12*a33**2*u3**2 + 12*a33*b33*u3*v3 + 12*
a33*n1*u1 + 12*a33*n2*u2 + 12*a33*n3*u3 - 9*b33**2*v3**2 + 36*b33*n1*v1 + 36*b33
*n2*v2 + 18*b33*n3*v3)/(12*a33)$
INT=(4*a33**2*k26*u1**2*v1**2 + 4*a33**2*k26*u1**2*v2**2 + 4*a33**2*k26*u1**2*v3
**2 + 4*a33**2*k26*u2**2*v1**2 + 4*a33**2*k26*u2**2*v2**2 + 4*a33**2*k26*u2**2*
v3**2 + 4*a33**2*q1*u1**2 + 4*a33**2*q1*u2**2 - 12*a33*b33*k26*u1*v1**3 - 12*a33
*b33*k26*u1*v1*v2**2 - 12*a33*b33*k26*u1*v1*v3**2 - 12*a33*b33*k26*u2*v1**2*v2 -
12*a33*b33*k26*u2*v2**3 - 12*a33*b33*k26*u2*v2*v3**2 - 12*a33*b33*q1*u1*v1 - 12
*a33*b33*q1*u2*v2 - 6*a33*k26*n1*u1*v1**2 - 6*a33*k26*n1*u1*v2**2 - 6*a33*k26*n1
*u1*v3**2 - 6*a33*k26*n2*u2*v1**2 - 6*a33*k26*n2*u2*v2**2 - 6*a33*k26*n2*u2*v3**
2 - 6*a33*k26*n3*u3*v1**2 - 6*a33*k26*n3*u3*v2**2 - 6*a33*k26*n3*u3*v3**2 - 6*
a33*n1*q1*u1 - 6*a33*n2*q1*u2 - 6*a33*n3*q1*u3 + 9*b33**2*k26*v1**4 + 18*b33**2*
k26*v1**2*v2**2 + 9*b33**2*k26*v1**2*v3**2 + 9*b33**2*k26*v2**4 + 9*b33**2*k26*
v2**2*v3**2 + 9*b33**2*q1*v1**2 + 9*b33**2*q1*v2**2 + 9*b33*k26*n1*v1**3 + 9*b33
*k26*n1*v1*v2**2 + 9*b33*k26*n1*v1*v3**2 + 9*b33*k26*n2*v1**2*v2 + 9*b33*k26*n2*
v2**3 + 9*b33*k26*n2*v2*v3**2 + 18*b33*k26*n3*v1**2*v3 + 18*b33*k26*n3*v2**2*v3
+ 18*b33*k26*n3*v3**3 + 9*b33*n1*q1*v1 + 9*b33*n2*q1*v2 + 18*b33*n3*q1*v3)/(4*
a33**2)$