Solution 49 to problem e3null
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Parameters |
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Expressions
The solution is given through the following expressions:
1 2 1 2
---*b31 *n3 + ---*b32 *n3
2 2
a22=---------------------------
b31*m1
2 2
b31 *n3 + b32 *n3
a33=-------------------
b31*m1
b22=0
b33=0
c12=0
c13=0
c22=0
c23=0
1
- ---*b31*m1
2
c33=---------------
n3
n1=0
n2=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
2
- 2*b31*m1 *p5
q15=-------------------
2 2
b31 *n3 + b32 *n3
3
- b31*m1 *p5
q14=---------------------
2 2 2 2
b31 *n3 + b32 *n3
q13=0
q12=0
q11=0
q10=0
3
- b31*m1 *p5
q9=---------------------
2 2 2 2
b31 *n3 + b32 *n3
q8=0
q7=0
q5=0
q4=0
b31*m1*p5
q3=-------------
2 2
b31 + b32
q2=0
b31*m1*p5
q1=-------------
2 2
b31 + b32
p50=0
p49=0
2 3
b31 *m1 *p5
p48=---------------------
2 3 2 3
b31 *n3 + b32 *n3
2 2
3*b31 *m1 *p5
p47=---------------------
2 2 2 2
b31 *n3 + b32 *n3
p46=0
p45=0
p44=0
p43=0
p42=0
p41=0
p40=0
p39=0
p38=0
p37=0
p36=0
- 2*b31*b32*m1*p5
p35=--------------------
2 2
b31 *n3 + b32 *n3
2 2
- 3*b31 *m1*p5 - b32 *m1*p5
p34=------------------------------
2 2
b31 *n3 + b32 *n3
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
m1*p5
p18=-------
n3
p17=0
p16=0
p15=0
p14=0
p13=0
p12=0
- m1*p5
p11=----------
n3
p10=0
p9=0
p8=0
p7=p5
p6=0
p4=0
p3=0
p2=0
p1=0
k104=0
k103=0
k102=0
k101=0
k100=0
1 3 3
---*b31 *m1 *p5
2
k99=---------------------
2 4 2 4
b31 *n3 + b32 *n3
k98=0
2 2
b31 *b32*m1 *p5
k97=---------------------
2 3 2 3
b31 *n3 + b32 *n3
k96=0
k95=0
1 3 3
---*b31 *m1 *p5
2
k94=---------------------
2 4 2 4
b31 *n3 + b32 *n3
3 2
b31 *m1 *p5
k93=---------------------
2 3 2 3
b31 *n3 + b32 *n3
k92=0
2*b31*m1*p5
k91=-------------
2
n3
k90=0
k89=0
1
---*b31*m1*p5
2
k88=---------------
2
n3
k87=0
1
---*b31*m1*p5
2
k86=---------------
2
n3
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
- 2*b32*p5
k64=-------------
n3
- b31*p5
k63=-----------
n3
k62=0
k61=0
k60=0
k59=0
k58=0
k57=0
k56=0
1 3 3
---*b31 *m1 *p5
4
k55=---------------------
2 4 2 4
b31 *n3 + b32 *n3
k54=0
k53=0
k52=0
k51=0
1 3 3
---*b31 *m1 *p5
2
k50=---------------------
2 4 2 4
b31 *n3 + b32 *n3
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
- b32*p5
k32=-----------
n3
b31*p5
k31=--------
n3
- b32*p5
k30=-----------
n3
k29=0
k28=0
k27=0
k26=0
1 3 3
---*b31 *m1 *p5
4
k25=---------------------
2 4 2 4
b31 *n3 + b32 *n3
k24=0
k23=0
k21=0
k20=0
k19=0
k18=0
- b31*p5
k17=-----------
n3
k16=0
k14=0
k13=0
k12=0
k11=0
k10=0
k9=0
k8=0
k7=0
k6=0
1 2 1 2
- ---*b31 *p5 - ---*b32 *p5
4 4
k5=------------------------------
b31*m1
k4=0
1 2 1 2
- ---*b31 *p5 - ---*b32 *p5
2 2
k3=------------------------------
b31*m1
k2=0
1 2 1 2
- ---*b31 *p5 - ---*b32 *p5
4 4
k1=------------------------------
b31*m1
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:
b31,p5,m1,n3,b32
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2 2 2 2 2 2 2 2
HAM=( - b31 *m1 *v3 + 2*b31 *m1*n3*u3*v1 + b31 *n3 *u1 + b31 *n3 *u2
2 2 2 2
+ 2*b31 *n3 *u3 + 2*b31*b32*m1*n3*u3*v2 + 2*b31*m1 *n3*v1
2 2 2 2 2 2 2 2 2 2
+ 2*b31*m1*n3 *u3 + b32 *n3 *u1 + b32 *n3 *u2 + 2*b32 *n3 *u3 )/(2*b31
*m1*n3)
4 4 4 4 4 2 2 4 4 2 2 4 4 4
INT=(p5*(b31 *m1 *v1 + 2*b31 *m1 *v1 *v2 + 2*b31 *m1 *v1 *v3 + b31 *m1 *v2
4 4 2 2 4 3 2
+ 2*b31 *m1 *v2 *v3 + 4*b31 *m1 *n3*u3*v1*v3
4 2 2 2 2 4 2 2 2 2
+ 2*b31 *m1 *n3 *u1 *v3 + 2*b31 *m1 *n3 *u2 *v3
4 2 2 2 2 4 3
+ 8*b31 *m1 *n3 *u3 *v3 + 4*b31 *m1*n3 *u1*u2*u3*v2
4 3 2 4 3 2 4 4 4
- 4*b31 *m1*n3 *u1*u3 *v3 - 4*b31 *m1*n3 *u2 *u3*v1 - b31 *n3 *u1
4 4 2 2 4 4 4 3 3 2
- 2*b31 *n3 *u1 *u2 - b31 *n3 *u2 + 4*b31 *b32*m1 *n3*u3*v2*v3
3 3 2 3 3 2
- 4*b31 *b32*m1*n3 *u1 *u3*v2 - 4*b31 *b32*m1*n3 *u2 *u3*v2
3 3 2 3 4 2
- 8*b31 *b32*m1*n3 *u2*u3 *v3 + 4*b31 *m1 *n3*v1*v3
3 3 2 2 3 2 3
+ 12*b31 *m1 *n3 *u3*v3 + 4*b31 *m1 *n3 *u1*u2*v2
3 2 3 3 2 3 2
- 12*b31 *m1 *n3 *u1*u3*v3 - 4*b31 *m1 *n3 *u2 *v1
3 4 2 3 4 2
+ 4*b31 *m1*n3 *u1 *u3 + 4*b31 *m1*n3 *u2 *u3
2 2 2 2 2 2 2 2 2 2 2 2
+ 2*b31 *b32 *m1 *n3 *u1 *v3 + 2*b31 *b32 *m1 *n3 *u2 *v3
2 2 2 2 2 2 2 2 3
+ 8*b31 *b32 *m1 *n3 *u3 *v3 + 4*b31 *b32 *m1*n3 *u1*u2*u3*v2
2 2 3 2 2 2 3 2
- 4*b31 *b32 *m1*n3 *u1*u3 *v3 - 4*b31 *b32 *m1*n3 *u2 *u3*v1
2 2 4 4 2 2 4 2 2 2 2 4 4
- 2*b31 *b32 *n3 *u1 - 4*b31 *b32 *n3 *u1 *u2 - 2*b31 *b32 *n3 *u2
2 2 3 2 4 2 2
- 8*b31 *b32*m1 *n3 *u2*u3*v3 - 4*b31 *m1 *n3 *v1
2 4 2 2 2 3 3 2 2 4 2
- 4*b31 *m1 *n3 *v2 - 8*b31 *m1 *n3 *u1*v3 + 4*b31 *m1 *n3 *u1
2 2 4 2 3 3 2
+ 4*b31 *m1 *n3 *u2 - 4*b31*b32 *m1*n3 *u1 *u3*v2
3 3 2 3 3 2
- 4*b31*b32 *m1*n3 *u2 *u3*v2 - 8*b31*b32 *m1*n3 *u2*u3 *v3
2 2 3 2 2 3
+ 4*b31*b32 *m1 *n3 *u1*u2*v2 - 4*b31*b32 *m1 *n3 *u1*u3*v3
2 2 3 2 2 4 2
- 4*b31*b32 *m1 *n3 *u2 *v1 + 4*b31*b32 *m1*n3 *u1 *u3
2 4 2 4 4 4 4 4 2 2
+ 4*b31*b32 *m1*n3 *u2 *u3 - b32 *n3 *u1 - 2*b32 *n3 *u1 *u2
4 4 4 4 2 2
- b32 *n3 *u2 ))/(4*b31*m1*n3 *(b31 + b32 ))
And again in machine readable form:
HAM=( - b31**2*m1**2*v3**2 + 2*b31**2*m1*n3*u3*v1 + b31**2*n3**2*u1**2 + b31**2*
n3**2*u2**2 + 2*b31**2*n3**2*u3**2 + 2*b31*b32*m1*n3*u3*v2 + 2*b31*m1**2*n3*v1 +
2*b31*m1*n3**2*u3 + b32**2*n3**2*u1**2 + b32**2*n3**2*u2**2 + 2*b32**2*n3**2*u3
**2)/(2*b31*m1*n3)$
INT=(p5*(b31**4*m1**4*v1**4 + 2*b31**4*m1**4*v1**2*v2**2 + 2*b31**4*m1**4*v1**2*
v3**2 + b31**4*m1**4*v2**4 + 2*b31**4*m1**4*v2**2*v3**2 + 4*b31**4*m1**3*n3*u3*
v1*v3**2 + 2*b31**4*m1**2*n3**2*u1**2*v3**2 + 2*b31**4*m1**2*n3**2*u2**2*v3**2 +
8*b31**4*m1**2*n3**2*u3**2*v3**2 + 4*b31**4*m1*n3**3*u1*u2*u3*v2 - 4*b31**4*m1*
n3**3*u1*u3**2*v3 - 4*b31**4*m1*n3**3*u2**2*u3*v1 - b31**4*n3**4*u1**4 - 2*b31**
4*n3**4*u1**2*u2**2 - b31**4*n3**4*u2**4 + 4*b31**3*b32*m1**3*n3*u3*v2*v3**2 - 4
*b31**3*b32*m1*n3**3*u1**2*u3*v2 - 4*b31**3*b32*m1*n3**3*u2**2*u3*v2 - 8*b31**3*
b32*m1*n3**3*u2*u3**2*v3 + 4*b31**3*m1**4*n3*v1*v3**2 + 12*b31**3*m1**3*n3**2*u3
*v3**2 + 4*b31**3*m1**2*n3**3*u1*u2*v2 - 12*b31**3*m1**2*n3**3*u1*u3*v3 - 4*b31
**3*m1**2*n3**3*u2**2*v1 + 4*b31**3*m1*n3**4*u1**2*u3 + 4*b31**3*m1*n3**4*u2**2*
u3 + 2*b31**2*b32**2*m1**2*n3**2*u1**2*v3**2 + 2*b31**2*b32**2*m1**2*n3**2*u2**2
*v3**2 + 8*b31**2*b32**2*m1**2*n3**2*u3**2*v3**2 + 4*b31**2*b32**2*m1*n3**3*u1*
u2*u3*v2 - 4*b31**2*b32**2*m1*n3**3*u1*u3**2*v3 - 4*b31**2*b32**2*m1*n3**3*u2**2
*u3*v1 - 2*b31**2*b32**2*n3**4*u1**4 - 4*b31**2*b32**2*n3**4*u1**2*u2**2 - 2*b31
**2*b32**2*n3**4*u2**4 - 8*b31**2*b32*m1**2*n3**3*u2*u3*v3 - 4*b31**2*m1**4*n3**
2*v1**2 - 4*b31**2*m1**4*n3**2*v2**2 - 8*b31**2*m1**3*n3**3*u1*v3 + 4*b31**2*m1
**2*n3**4*u1**2 + 4*b31**2*m1**2*n3**4*u2**2 - 4*b31*b32**3*m1*n3**3*u1**2*u3*v2
- 4*b31*b32**3*m1*n3**3*u2**2*u3*v2 - 8*b31*b32**3*m1*n3**3*u2*u3**2*v3 + 4*b31
*b32**2*m1**2*n3**3*u1*u2*v2 - 4*b31*b32**2*m1**2*n3**3*u1*u3*v3 - 4*b31*b32**2*
m1**2*n3**3*u2**2*v1 + 4*b31*b32**2*m1*n3**4*u1**2*u3 + 4*b31*b32**2*m1*n3**4*u2
**2*u3 - b32**4*n3**4*u1**4 - 2*b32**4*n3**4*u1**2*u2**2 - b32**4*n3**4*u2**4))/
(4*b31*m1*n3**4*(b31**2 + b32**2))$