Solution 52 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
2
b22=0
b33=0
c12=0
c13=0
c22=0
c23=0
1 2 1 2
- ---*b31 - ---*b32
2 2
c33=------------------------
a33
n1=0
n2=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
q17=0
q16=0
q15=0
q14=0
q13=0
q12=0
q11=0
q10=0
q9=0
q8=0
q7=0
q5=0
q4=0
q3=0
q2=0
q1=0
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
p10=0
p9=0
p8=0
p7=0
p6=0
p5=0
p4=0
p3=0
p2=0
p1=0
k104=0
k103=0
k102=0
k101=0
k100=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k99=-------------------------------------------
4
a33
k98=0
2 3
- 4*b31 *b32*k1 - 4*b32 *k1
k97=------------------------------
3
a33
k96=0
k95=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k94=-------------------------------------------
4
a33
3 2
- 4*b31 *k1 - 4*b31*b32 *k1
k93=------------------------------
3
a33
k92=0
2 2
- 8*b31 *k1 - 8*b32 *k1
k91=--------------------------
2
a33
k90=0
k89=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k88=--------------------------
2
a33
k87=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k86=--------------------------
2
a33
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
8*b32*k1
k64=----------
a33
4*b31*k1
k63=----------
a33
k62=0
k61=0
k60=0
k59=0
k58=0
k57=0
k56=0
4 2 2 4
- b31 *k1 - 2*b31 *b32 *k1 - b32 *k1
k55=---------------------------------------
4
a33
k54=0
k53=0
k52=0
k51=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k50=-------------------------------------------
4
a33
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
4*b32*k1
k32=----------
a33
- 4*b31*k1
k31=-------------
a33
4*b32*k1
k30=----------
a33
k29=0
k28=0
k27=0
k26=0
4 2 2 4
- b31 *k1 - 2*b31 *b32 *k1 - b32 *k1
k25=---------------------------------------
4
a33
k24=0
k23=0
k21=0
k20=0
k19=0
k18=0
4*b31*k1
k17=----------
a33
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:
k1,b31,b32,a33
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2 2 2 2
HAM=(a33 *u1 + a33 *u2 + 2*a33 *u3 + 2*a33*b31*u3*v1 + 2*a33*b32*u3*v2
2 2 2 2
- b31 *v3 - b32 *v3 )/(2*a33)
4 4 4 2 2 4 4 3
INT=(k1*(a33 *u1 + 2*a33 *u1 *u2 + a33 *u2 - 4*a33 *b31*u1*u2*u3*v2
3 2 3 2 3 2
+ 4*a33 *b31*u1*u3 *v3 + 4*a33 *b31*u2 *u3*v1 + 4*a33 *b32*u1 *u3*v2
3 2 3 2 2 2 2 2
+ 4*a33 *b32*u2 *u3*v2 + 8*a33 *b32*u2*u3 *v3 - 2*a33 *b31 *u1 *v3
2 2 2 2 2 2 2 2 2 2 2 2
- 2*a33 *b31 *u2 *v3 - 8*a33 *b31 *u3 *v3 - 2*a33 *b32 *u1 *v3
2 2 2 2 2 2 2 2 3 2
- 2*a33 *b32 *u2 *v3 - 8*a33 *b32 *u3 *v3 - 4*a33*b31 *u3*v1*v3
2 2 2 2
- 4*a33*b31 *b32*u3*v2*v3 - 4*a33*b31*b32 *u3*v1*v3
3 2 4 4 4 2 2 4 2 2
- 4*a33*b32 *u3*v2*v3 - b31 *v1 - 2*b31 *v1 *v2 - 2*b31 *v1 *v3
4 4 4 2 2 2 2 4 2 2 2 2
- b31 *v2 - 2*b31 *v2 *v3 - 2*b31 *b32 *v1 - 4*b31 *b32 *v1 *v2
2 2 2 2 2 2 4 2 2 2 2
- 4*b31 *b32 *v1 *v3 - 2*b31 *b32 *v2 - 4*b31 *b32 *v2 *v3
4 4 4 2 2 4 2 2 4 4
- b32 *v1 - 2*b32 *v1 *v2 - 2*b32 *v1 *v3 - b32 *v2
4 2 2 4
- 2*b32 *v2 *v3 ))/a33
And again in machine readable form:
HAM=(a33**2*u1**2 + a33**2*u2**2 + 2*a33**2*u3**2 + 2*a33*b31*u3*v1 + 2*a33*b32*
u3*v2 - b31**2*v3**2 - b32**2*v3**2)/(2*a33)$
INT=(k1*(a33**4*u1**4 + 2*a33**4*u1**2*u2**2 + a33**4*u2**4 - 4*a33**3*b31*u1*u2
*u3*v2 + 4*a33**3*b31*u1*u3**2*v3 + 4*a33**3*b31*u2**2*u3*v1 + 4*a33**3*b32*u1**
2*u3*v2 + 4*a33**3*b32*u2**2*u3*v2 + 8*a33**3*b32*u2*u3**2*v3 - 2*a33**2*b31**2*
u1**2*v3**2 - 2*a33**2*b31**2*u2**2*v3**2 - 8*a33**2*b31**2*u3**2*v3**2 - 2*a33
**2*b32**2*u1**2*v3**2 - 2*a33**2*b32**2*u2**2*v3**2 - 8*a33**2*b32**2*u3**2*v3
**2 - 4*a33*b31**3*u3*v1*v3**2 - 4*a33*b31**2*b32*u3*v2*v3**2 - 4*a33*b31*b32**2
*u3*v1*v3**2 - 4*a33*b32**3*u3*v2*v3**2 - b31**4*v1**4 - 2*b31**4*v1**2*v2**2 -
2*b31**4*v1**2*v3**2 - b31**4*v2**4 - 2*b31**4*v2**2*v3**2 - 2*b31**2*b32**2*v1
**4 - 4*b31**2*b32**2*v1**2*v2**2 - 4*b31**2*b32**2*v1**2*v3**2 - 2*b31**2*b32**
2*v2**4 - 4*b31**2*b32**2*v2**2*v3**2 - b32**4*v1**4 - 2*b32**4*v1**2*v2**2 - 2*
b32**4*v1**2*v3**2 - b32**4*v2**4 - 2*b32**4*v2**2*v3**2))/a33**4$