Solution 16 to problem e3null
Remaining equations |
Expressions |
Parameters |
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Equations
The following unsolved equations remain:
2 2
0=b31 + b32
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
- b31*n2
n1=-----------
b32
n3=0
- b32*m2
m1=-----------
b31
- 2*b32*n2
m3=-------------
a33
r6=0
2
- 16*k1*m2*n2
r5=-----------------
3
a33
2 2 2 2
- 8*b31 *k1*m2*n2 - 24*b32 *k1*m2*n2
r4=-----------------------------------------
3
a33 *b31*b32
r3=0
2 3 2 3
8*b31 *k1*n2 + 8*b32 *k1*n2
r2=-------------------------------
3 2
a33 *b32
3 3 2 3
- 8*b31 *k1*n2 - 8*b31*b32 *k1*n2
r1=--------------------------------------
3 3
a33 *b32
q19=0
q18=0
- 16*k1*m2*n2
q17=----------------
2
a33
q16=0
q15=0
2 2 2 2
- 8*b31 *k1*m2 - 8*b32 *k1*m2
q14=----------------------------------
2 2
a33 *b31
q13=0
2 2 2 2
8*b31 *k1*n2 - 8*b32 *k1*n2
q12=-------------------------------
3
a33 *b32
q11=0
- 16*b32*k1*m2*n2
q10=--------------------
2
a33 *b31
2 2 2 2
- 8*b31 *k1*m2 - 8*b32 *k1*m2
q9=----------------------------------
2 2
a33 *b31
2
16*b31*k1*n2
q8=---------------
3
a33
16*b32*k1*m2*n2
q7=-----------------
2
a33 *b31
q5=0
q4=0
2 2
- 4*b31 *k1*n2
q3=------------------
2 2
a33 *b32
2
- 8*b31*k1*n2
q2=-----------------
2
a33 *b32
2
- 4*k1*n2
q1=-------------
2
a33
2 3
4*b31 *b32*k1*n2 + 4*b32 *k1*n2
p50=---------------------------------
4
a33
2 2
- 4*b31 *k1*m2 - 4*b32 *k1*m2
p49=--------------------------------
3
a33
4 2 2 4
4*b31 *k1*m2 - 4*b31 *b32 *k1*m2 - 8*b32 *k1*m2
p48=-------------------------------------------------
3
a33 *b31*b32
p47=0
2 2
8*b31 *k1*n2 + 8*b32 *k1*n2
p46=-----------------------------
3
a33
p45=0
p44=0
p43=0
p42=0
p41=0
p40=0
p39=0
p38=0
p37=0
- 16*b32*k1*n2
p36=-----------------
2
a33
8*k1*m2
p35=---------
a33
- 4*b32*k1*m2
p34=----------------
a33*b31
2 2
- 8*b31 *k1*n2 - 8*b32 *k1*n2
p33=--------------------------------
2
a33 *b32
p32=0
p31=0
p30=0
p29=0
p28=0
p27=0
p26=0
p25=0
p24=0
p23=0
p22=0
p21=0
16*b31*k1*n2
p20=--------------
2
a33
4*k1*m2
p19=---------
a33
4*b32*k1*m2
p18=-------------
a33*b31
4*k1*m2
p17=---------
a33
p16=0
p15=0
p14=0
p13=0
- 16*b31*k1*n2
p12=-----------------
2
a33
- 4*b32*k1*m2
p11=----------------
a33*b31
p10=0
p9=0
p8=0
p7=0
p6=0
p5=0
- 4*k1*n2
p4=------------
a33
4*b31*k1*n2
p3=-------------
a33*b32
- 4*k1*n2
p2=------------
a33
4*b31*k1*n2
p1=-------------
a33*b32
k104=0
k103=0
k102=0
k101=0
k100=0
4 2 2 4
4*b31 *k1 + 8*b31 *b32 *k1 + 4*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
4*b31 *k1 + 8*b31 *b32 *k1 + 4*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
2*b31 *k1 + 4*b31 *b32 *k1 + 2*b32 *k1
k55=----------------------------------------
4
a33
k54=0
k53=0
k52=0
k51=0
4 2 2 4
4*b31 *k1 + 8*b31 *b32 *k1 + 4*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
2*b31 *k1 + 4*b31 *b32 *k1 + 2*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:
m2,k1,b31,b32,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=(a33 *b31*b32*u1 + a33 *b31*b32*u2 + 2*a33 *b31*b32*u3
2 2 2
+ 2*a33*b31 *b32*u3*v1 - 2*a33*b31 *n2*u1 + 2*a33*b31*b32 *u3*v2
2
+ 2*a33*b31*b32*m2*v2 + 2*a33*b31*b32*n2*u2 - 2*a33*b32 *m2*v1
3 2 3 2 2
- b31 *b32*v3 - b31*b32 *v3 - 4*b31*b32 *n2*v3)/(2*a33*b31*b32)
4 2 3 4 4 2 3 2 2 4 2 3 4
INT=(k1*(a33 *b31 *b32 *u1 + 2*a33 *b31 *b32 *u1 *u2 + a33 *b31 *b32 *u2
3 3 3 3 3 3 2
- 4*a33 *b31 *b32 *u1*u2*u3*v2 + 4*a33 *b31 *b32 *u1*u3 *v3
3 3 3 2 3 3 2 3
+ 4*a33 *b31 *b32 *u2 *u3*v1 + 4*a33 *b31 *b32 *n2*u1
3 3 2 2 3 2 4 2
+ 4*a33 *b31 *b32 *n2*u1*u2 + 4*a33 *b31 *b32 *u1 *u3*v2
3 2 4 2 3 2 4 2
+ 4*a33 *b31 *b32 *u2 *u3*v2 + 8*a33 *b31 *b32 *u2*u3 *v3
3 2 3 2 3 2 3 2
+ 4*a33 *b31 *b32 *m2*u1 *v2 + 4*a33 *b31 *b32 *m2*u2 *v2
3 2 3 3 2 3 2
+ 8*a33 *b31 *b32 *m2*u2*u3*v3 - 4*a33 *b31 *b32 *n2*u1 *u2
3 2 3 3 3 4
- 4*a33 *b31 *b32 *n2*u2 + 4*a33 *b31*b32 *m2*u1*u2*v2
3 4 3 4 2
- 4*a33 *b31*b32 *m2*u1*u3*v3 - 4*a33 *b31*b32 *m2*u2 *v1
2 4 3 2 2 2 4 3 2 2
- 2*a33 *b31 *b32 *u1 *v3 - 2*a33 *b31 *b32 *u2 *v3
2 4 3 2 2 2 4 2 2
- 8*a33 *b31 *b32 *u3 *v3 - 8*a33 *b31 *b32 *n2*u2 *v3
2 4 2 2 2 3 3
- 4*a33 *b31 *b32*n2 *u2 + 16*a33 *b31 *b32 *n2*u1*u3*v2
2 3 3 2 3 2 2
- 16*a33 *b31 *b32 *n2*u2*u3*v1 - 8*a33 *b31 *b32 *n2 *u1*u2
2 2 5 2 2 2 2 5 2 2
- 2*a33 *b31 *b32 *u1 *v3 - 2*a33 *b31 *b32 *u2 *v3
2 2 5 2 2 2 2 4 2
- 8*a33 *b31 *b32 *u3 *v3 - 8*a33 *b31 *b32 *n2*u2 *v3
2 2 4 2 2 2 3 2 2
- 16*a33 *b31 *b32 *n2*u3 *v3 - 8*a33 *b31 *b32 *m2 *v1
2 2 3 2 2 2 2 3
- 8*a33 *b31 *b32 *m2 *v2 - 16*a33 *b31 *b32 *m2*n2*u3*v3
2 2 3 2 2 2 4
- 4*a33 *b31 *b32 *n2 *u1 - 16*a33 *b31*b32 *m2*n2*u1*v2
2 4 2 5 2 2
+ 16*a33 *b31*b32 *m2*n2*u2*v1 - 8*a33 *b32 *m2 *v1
2 5 2 2 5 3 2
- 8*a33 *b32 *m2 *v2 - 4*a33*b31 *b32 *u3*v1*v3
5 2 2 5 3
+ 4*a33*b31 *b32 *m2*v1*v3 - 8*a33*b31 *n2 *u1
4 4 2 4 3 2
- 4*a33*b31 *b32 *u3*v2*v3 - 4*a33*b31 *b32 *m2*v2*v3
4 3 2 4 2 2
+ 8*a33*b31 *b32 *n2*u2*v3 + 8*a33*b31 *b32 *n2 *u3*v2
4 3 3 5 2
+ 8*a33*b31 *b32*n2 *u2 - 4*a33*b31 *b32 *u3*v1*v3
3 4 2 3 3 2
- 4*a33*b31 *b32 *m2*v1*v3 + 16*a33*b31 *b32 *n2 *u3*v1
3 2 2 3 2 3
- 8*a33*b31 *b32 *m2*n2 *v1 - 8*a33*b31 *b32 *n2 *u1
2 6 2 2 5 2
- 4*a33*b31 *b32 *u3*v2*v3 - 4*a33*b31 *b32 *m2*v2*v3
2 5 2 2 4 2
+ 8*a33*b31 *b32 *n2*u2*v3 - 8*a33*b31 *b32 *n2 *u3*v2
2 3 2 2 3 3
- 16*a33*b31 *b32 *m2*n2 *v2 + 8*a33*b31 *b32 *n2 *u2
6 2 4 2
- 8*a33*b31*b32 *m2*v1*v3 - 24*a33*b31*b32 *m2*n2 *v1
6 3 4 6 3 2 2 6 3 2 2
+ 2*b31 *b32 *v1 + 4*b31 *b32 *v1 *v2 + 4*b31 *b32 *v1 *v3
6 3 4 6 3 2 2 4 5 4
+ 2*b31 *b32 *v2 + 4*b31 *b32 *v2 *v3 + 4*b31 *b32 *v1
4 5 2 2 4 5 2 2 4 5 4
+ 8*b31 *b32 *v1 *v2 + 8*b31 *b32 *v1 *v3 + 4*b31 *b32 *v2
4 5 2 2 4 4 3 2 7 4
+ 8*b31 *b32 *v2 *v3 + 4*b31 *b32 *n2*v3 + 2*b31 *b32 *v1
2 7 2 2 2 7 2 2 2 7 4
+ 4*b31 *b32 *v1 *v2 + 4*b31 *b32 *v1 *v3 + 2*b31 *b32 *v2
2 7 2 2 2 6 3 4 2 3
+ 4*b31 *b32 *v2 *v3 + 4*b31 *b32 *n2*v3 ))/(a33 *b31 *b32 )
And again in machine readable form:
HAM=(a33**2*b31*b32*u1**2 + a33**2*b31*b32*u2**2 + 2*a33**2*b31*b32*u3**2 + 2*
a33*b31**2*b32*u3*v1 - 2*a33*b31**2*n2*u1 + 2*a33*b31*b32**2*u3*v2 + 2*a33*b31*
b32*m2*v2 + 2*a33*b31*b32*n2*u2 - 2*a33*b32**2*m2*v1 - b31**3*b32*v3**2 - b31*
b32**3*v3**2 - 4*b31*b32**2*n2*v3)/(2*a33*b31*b32)$
INT=(k1*(a33**4*b31**2*b32**3*u1**4 + 2*a33**4*b31**2*b32**3*u1**2*u2**2 + a33**
4*b31**2*b32**3*u2**4 - 4*a33**3*b31**3*b32**3*u1*u2*u3*v2 + 4*a33**3*b31**3*b32
**3*u1*u3**2*v3 + 4*a33**3*b31**3*b32**3*u2**2*u3*v1 + 4*a33**3*b31**3*b32**2*n2
*u1**3 + 4*a33**3*b31**3*b32**2*n2*u1*u2**2 + 4*a33**3*b31**2*b32**4*u1**2*u3*v2
+ 4*a33**3*b31**2*b32**4*u2**2*u3*v2 + 8*a33**3*b31**2*b32**4*u2*u3**2*v3 + 4*
a33**3*b31**2*b32**3*m2*u1**2*v2 + 4*a33**3*b31**2*b32**3*m2*u2**2*v2 + 8*a33**3
*b31**2*b32**3*m2*u2*u3*v3 - 4*a33**3*b31**2*b32**3*n2*u1**2*u2 - 4*a33**3*b31**
2*b32**3*n2*u2**3 + 4*a33**3*b31*b32**4*m2*u1*u2*v2 - 4*a33**3*b31*b32**4*m2*u1*
u3*v3 - 4*a33**3*b31*b32**4*m2*u2**2*v1 - 2*a33**2*b31**4*b32**3*u1**2*v3**2 - 2
*a33**2*b31**4*b32**3*u2**2*v3**2 - 8*a33**2*b31**4*b32**3*u3**2*v3**2 - 8*a33**
2*b31**4*b32**2*n2*u2**2*v3 - 4*a33**2*b31**4*b32*n2**2*u2**2 + 16*a33**2*b31**3
*b32**3*n2*u1*u3*v2 - 16*a33**2*b31**3*b32**3*n2*u2*u3*v1 - 8*a33**2*b31**3*b32
**2*n2**2*u1*u2 - 2*a33**2*b31**2*b32**5*u1**2*v3**2 - 2*a33**2*b31**2*b32**5*u2
**2*v3**2 - 8*a33**2*b31**2*b32**5*u3**2*v3**2 - 8*a33**2*b31**2*b32**4*n2*u2**2
*v3 - 16*a33**2*b31**2*b32**4*n2*u3**2*v3 - 8*a33**2*b31**2*b32**3*m2**2*v1**2 -
8*a33**2*b31**2*b32**3*m2**2*v2**2 - 16*a33**2*b31**2*b32**3*m2*n2*u3*v3 - 4*
a33**2*b31**2*b32**3*n2**2*u1**2 - 16*a33**2*b31*b32**4*m2*n2*u1*v2 + 16*a33**2*
b31*b32**4*m2*n2*u2*v1 - 8*a33**2*b32**5*m2**2*v1**2 - 8*a33**2*b32**5*m2**2*v2
**2 - 4*a33*b31**5*b32**3*u3*v1*v3**2 + 4*a33*b31**5*b32**2*m2*v1*v3**2 - 8*a33*
b31**5*n2**3*u1 - 4*a33*b31**4*b32**4*u3*v2*v3**2 - 4*a33*b31**4*b32**3*m2*v2*v3
**2 + 8*a33*b31**4*b32**3*n2*u2*v3**2 + 8*a33*b31**4*b32**2*n2**2*u3*v2 + 8*a33*
b31**4*b32*n2**3*u2 - 4*a33*b31**3*b32**5*u3*v1*v3**2 - 4*a33*b31**3*b32**4*m2*
v1*v3**2 + 16*a33*b31**3*b32**3*n2**2*u3*v1 - 8*a33*b31**3*b32**2*m2*n2**2*v1 -
8*a33*b31**3*b32**2*n2**3*u1 - 4*a33*b31**2*b32**6*u3*v2*v3**2 - 4*a33*b31**2*
b32**5*m2*v2*v3**2 + 8*a33*b31**2*b32**5*n2*u2*v3**2 - 8*a33*b31**2*b32**4*n2**2
*u3*v2 - 16*a33*b31**2*b32**3*m2*n2**2*v2 + 8*a33*b31**2*b32**3*n2**3*u2 - 8*a33
*b31*b32**6*m2*v1*v3**2 - 24*a33*b31*b32**4*m2*n2**2*v1 + 2*b31**6*b32**3*v1**4
+ 4*b31**6*b32**3*v1**2*v2**2 + 4*b31**6*b32**3*v1**2*v3**2 + 2*b31**6*b32**3*v2
**4 + 4*b31**6*b32**3*v2**2*v3**2 + 4*b31**4*b32**5*v1**4 + 8*b31**4*b32**5*v1**
2*v2**2 + 8*b31**4*b32**5*v1**2*v3**2 + 4*b31**4*b32**5*v2**4 + 8*b31**4*b32**5*
v2**2*v3**2 + 4*b31**4*b32**4*n2*v3**3 + 2*b31**2*b32**7*v1**4 + 4*b31**2*b32**7
*v1**2*v2**2 + 4*b31**2*b32**7*v1**2*v3**2 + 2*b31**2*b32**7*v2**4 + 4*b31**2*
b32**7*v2**2*v3**2 + 4*b31**2*b32**6*n2*v3**3))/(a33**4*b31**2*b32**3)$