Solution 19 to problem e3null
Expressions |
Parameters |
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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
n3=0
m1=0
m2=0
m3=0
r6=0
1 2 1 2 1 2
- ---*a33 *b33*n2*q1 + ----*b33*k1*n1 *n2 - ----*b33*k10*n1 *n2
4 16 16
r5=------------------------------------------------------------------
4
a33
1 2 1 3 1 3
- ---*a33 *b33*n1*q1 + ----*b33*k1*n1 - ----*b33*k10*n1
4 16 16
r4=------------------------------------------------------------
4
a33
r3=0
1 2 1 2 1 2
- ---*a33 *n2*q1 + ---*k1*n1 *n2 - ---*k10*n1 *n2
2 8 8
r2=----------------------------------------------------
3
a33
1 2 1 3 1 3
- ---*a33 *n1*q1 + ---*k1*n1 - ---*k10*n1
2 8 8
r1=----------------------------------------------
3
a33
q19=0
q18=0
2 1 2
- a33 *b33*q1 - ---*b33*k10*n1
4
q17=----------------------------------
3
a33
q16=0
q15=0
1 2 2 1 2 2 1 2 2 1 2 2
q14=(---*a33 *b33 *q1 - ----*b33 *k1*n1 + ----*b33 *k1*n2 + ----*b33 *k10*n1
4 16 16 16
1 2 2 4
+ ----*b33 *k10*n2 )/a33
16
1 2 1 2
---*b33 *k1*n1*n2 + ---*b33 *k10*n1*n2
8 8
q13=----------------------------------------
4
a33
q12=0
1 2 1 2
- ---*b33*k1*n1 + ---*b33*k1*n2
4 4
q11=------------------------------------
3
a33
1
---*b33*k1*n1*n2
4
q10=------------------
3
a33
1 2 2 1 2 2
---*a33 *b33 *q1 + ---*b33 *k10*n1
4 8
q9=-------------------------------------
4
a33
q8=0
1
---*b33*k1*n1*n2
4
q7=------------------
3
a33
q5=0
q4=0
2 1 2 1 2 1 2 1 2
a33 *q1 - ---*k1*n1 + ---*k1*n2 + ---*k10*n1 - ---*k10*n2
4 4 4 4
q3=---------------------------------------------------------------
2
a33
1 1
---*k1*n1*n2 - ---*k10*n1*n2
2 2
q2=------------------------------
2
a33
p50=0
4 1 3 1 3
a33 *n2*p16 + ---*b33 *k1*n1*n2 + ----*b33 *k10*n1*n2
8 16
p49=-------------------------------------------------------
4
a33 *n1
4 1 3 1 3
a33 *p16 + ---*b33 *k1*n1 + ----*b33 *k10*n1
8 16
p48=----------------------------------------------
4
a33
p47=0
4 1 3 1 3
2*a33 *n2*p16 + ---*b33 *k1*n1*n2 - ---*b33 *k10*n1*n2
4 8
p46=--------------------------------------------------------
3
a33 *b33*n1
4 1 3 1 3
2*a33 *p16 + ---*b33 *k1*n1 - ---*b33 *k10*n1
4 8
p45=-----------------------------------------------
3
a33 *b33
p44=0
p43=0
1 2 1 2
---*b33 *k1*n2 + ---*b33 *k10*n2
2 4
p42=----------------------------------
3
a33
p41=0
p40=0
p39=0
4 1 3 1 3
- 2*a33 *p16 + ---*b33 *k1*n1 + ---*b33 *k10*n1
2 4
p38=--------------------------------------------------
3
a33 *b33
p37=0
p36=0
1
b33*k1*n2 - ---*b33*k10*n2
2
p35=----------------------------
2
a33
3 1
---*b33*k1*n1 - ---*b33*k10*n1
2 4
p34=--------------------------------
2
a33
p33=0
p32=0
p31=0
n2*p16
p30=--------
n1
p29=p16
p28=0
2*a33*n2*p16
p27=--------------
b33*n1
2*a33*p16
p26=-----------
b33
n2*p16
p25=--------
n1
p24=0
- 2*a33*p16
p23=--------------
b33
1
- ---*b33*k10*n2
4
p22=-------------------
2
a33
p21=0
p20=0
1 1
- ---*b33*k1*n2 - ---*b33*k10*n2
2 4
p19=-----------------------------------
2
a33
1 1
---*b33*k1*n1 + ---*b33*k10*n1
2 4
p18=--------------------------------
2
a33
1 1
- ---*b33*k1*n2 - ---*b33*k10*n2
2 4
p17=-----------------------------------
2
a33
p15=0
2*a33*n2*p16
p14=--------------
b33*n1
1
- ---*b33*k10*n1
4
p13=-------------------
2
a33
p12=0
1 1
- ---*b33*k1*n1 - ---*b33*k10*n1
2 4
p11=-----------------------------------
2
a33
p10=0
1
- ---*k10*n2
2
p9=---------------
a33
1
- ---*k10*n1
2
p8=---------------
a33
p7=0
p6=0
p5=0
1
- k1*n2 + ---*k10*n2
2
p4=-----------------------
a33
1
- k1*n1 + ---*k10*n1
2
p3=-----------------------
a33
1
- k1*n2 + ---*k10*n2
2
p2=-----------------------
a33
1
- k1*n1 + ---*k10*n1
2
p1=-----------------------
a33
k104=0
k103=0
4 1 3
4*a33 *p16 + ---*b33 *k1*n1
2
k102=-----------------------------
3
a33 *n1
k101=0
k100=0
4 1 4
- a33 *b33*p16 - ---*b33 *k1*n1
8
k99=----------------------------------
4
a33 *n1
k98=0
k97=0
k96=0
k95=0
4 1 4
- a33 *b33*p16 - ---*b33 *k1*n1
8
k94=----------------------------------
4
a33 *n1
k93=0
k92=0
4 3 1 3
- 4*a33 *p16 + b33 *k1*n1 - ---*b33 *k10*n1
4
k91=----------------------------------------------
2
a33 *b33*n1
k90=0
k89=0
4 1 3
- 4*a33 *p16 - ---*b33 *k1*n1
2
k88=--------------------------------
2
a33 *b33*n1
k87=0
4 1 3
- 4*a33 *p16 - ---*b33 *k1*n1
2
k86=--------------------------------
2
a33 *b33*n1
k85=0
k84=0
4*a33*p16
k83=-----------
n1
k82=0
k81=0
k80=0
k79=0
k78=0
k77=0
2
- 8*a33 *p16
k76=---------------
b33*n1
k75=0
k74=0
k73=0
k72=0
k71=0
4*a33*p16
k70=-----------
n1
k69=0
k68=0
k67=0
k66=0
- b33*k10
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
4 1 4
- a33 *b33*p16 - ----*b33 *k1*n1
16
k55=-----------------------------------
4
a33 *n1
k54=0
k53=0
k52=0
k51=0
4 1 4
- 2*a33 *b33*p16 - ---*b33 *k1*n1
8
k50=------------------------------------
4
a33 *n1
k49=0
k48=0
k47=0
k46=0
k45=0
2
- 8*a33 *p16
k44=---------------
b33*n1
k43=0
2
- 4*a33 *p16
k42=---------------
b33*n1
k41=0
k40=0
k39=0
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=0
k31=0
k30=0
k29=0
k28=0
k27=0
k26=0
4 1 4
- a33 *b33*p16 - ----*b33 *k1*n1
16
k25=-----------------------------------
4
a33 *n1
k24=0
k23=0
k21=0
2
- 4*a33 *p16
k20=---------------
b33*n1
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
k12=k10
k11=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:
q1,k10,k1,n1,p16,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*n1*u1
2 2
+ 4*a33*n2*u2 + b33 *v3 )/(4*a33)
6 2 2 6 2 2 6 2 2
INT=( - 64*a33 *p16*u1 *v2 - 64*a33 *p16*u1 *v3 - 64*a33 *p16*u2 *v1
6 2 2 6 2 2 6
- 128*a33 *p16*u2 *v2 - 64*a33 *p16*u2 *v3 - 128*a33 *p16*u2*u3*v2*v3
6 2 2 5 2
- 64*a33 *p16*u3 *v3 + 64*a33 *b33*p16*u3*v1 *v3
5 2 5 3
+ 64*a33 *b33*p16*u3*v2 *v3 + 64*a33 *b33*p16*u3*v3
5 2 5 2 5
+ 32*a33 *n1*p16*u1*v2 + 32*a33 *n1*p16*u1*v3 - 32*a33 *n1*p16*u2*v1*v2
5 5 2 5 2
- 32*a33 *n1*p16*u3*v1*v3 + 32*a33 *n2*p16*u2*v1 + 32*a33 *n2*p16*u2*v2
5 2 4 2 4 4 2 2 2
+ 32*a33 *n2*p16*u2*v3 - 16*a33 *b33 *p16*v1 - 32*a33 *b33 *p16*v1 *v2
4 2 2 2 4 2 4
- 16*a33 *b33 *p16*v1 *v3 - 16*a33 *b33 *p16*v2
4 2 2 2 4 4
- 16*a33 *b33 *p16*v2 *v3 + 16*a33 *b33*k1*n1*u1
4 2 2 4 4
+ 32*a33 *b33*k1*n1*u1 *u2 + 16*a33 *b33*k1*n1*u2
4 2 2 4 2 2
+ 16*a33 *b33*k10*n1*u1 *u3 + 16*a33 *b33*k10*n1*u2 *u3
4 3 4 2
+ 16*a33 *b33*n1*p16*v1 + 16*a33 *b33*n1*p16*v1*v2
4 2 4 2
+ 16*a33 *b33*n1*p16*v1*v3 + 16*a33 *b33*n1*q1*u1
4 2 4 2
+ 16*a33 *b33*n1*q1*u2 + 16*a33 *b33*n2*p16*v1 *v2
4 3 4 2
+ 16*a33 *b33*n2*p16*v2 + 16*a33 *b33*n2*p16*v2*v3
3 2 2 3 2 2
- 32*a33 *b33 *k1*n1*u1 *u3*v3 - 32*a33 *b33 *k1*n1*u2 *u3*v3
3 2 3 3 2
- 16*a33 *b33 *k10*n1*u3 *v3 - 16*a33 *b33 *n1*q1*u3*v3
3 2 3 3 2 2
- 16*a33 *b33*k1*n1 *u1 - 16*a33 *b33*k1*n1 *u1*u2
3 2 3 3
- 16*a33 *b33*k1*n1*n2*u1 *u2 - 16*a33 *b33*k1*n1*n2*u2
3 2 3 3 2 2
+ 8*a33 *b33*k10*n1 *u1 + 8*a33 *b33*k10*n1 *u1*u2
3 2 2 3 2
- 8*a33 *b33*k10*n1 *u1*u3 + 8*a33 *b33*k10*n1*n2*u1 *u2
3 3 3 2
+ 8*a33 *b33*k10*n1*n2*u2 - 8*a33 *b33*k10*n1*n2*u2*u3
3 2 3
- 8*a33 *b33*n1 *q1*u1 - 8*a33 *b33*n1*n2*q1*u2
2 3 2 2 2 3 2 2
- 8*a33 *b33 *k1*n1*u1 *v3 - 8*a33 *b33 *k1*n1*u2 *v3
2 3 2 2 2 3 2 2
+ 16*a33 *b33 *k1*n1*u3 *v3 - 4*a33 *b33 *k10*n1*u3 *v3
2 3 2 2 3 2
+ 4*a33 *b33 *n1*q1*v1 + 4*a33 *b33 *n1*q1*v2
2 2 2 2 2 2
+ 8*a33 *b33 *k1*n1 *u1*u2*v2 + 24*a33 *b33 *k1*n1 *u1*u3*v3
2 2 2 2 2 2 2
- 8*a33 *b33 *k1*n1 *u2 *v1 - 8*a33 *b33 *k1*n1*n2*u1 *v2
2 2 2 2 2
- 8*a33 *b33 *k1*n1*n2*u2 *v2 + 16*a33 *b33 *k1*n1*n2*u2*u3*v3
2 2 2 2 2 2
+ 4*a33 *b33 *k10*n1 *u1*u2*v2 - 4*a33 *b33 *k10*n1 *u1*u3*v3
2 2 2 2 2 2 2 2
- 4*a33 *b33 *k10*n1 *u2 *v1 - 4*a33 *b33 *k10*n1 *u3 *v1
2 2 2 2 2 2
- 4*a33 *b33 *k10*n1*n2*u1 *v2 - 4*a33 *b33 *k10*n1*n2*u2 *v2
2 2 2 2 2
- 8*a33 *b33 *k10*n1*n2*u2*u3*v3 - 4*a33 *b33 *k10*n1*n2*u3 *v2
2 2 2 2 2 2 3 2
- 4*a33 *b33 *n1 *q1*v1 - 4*a33 *b33 *n1*n2*q1*v2 - 4*a33 *b33*k1*n1 *u2
2 2 2 2 2
+ 8*a33 *b33*k1*n1 *n2*u1*u2 + 4*a33 *b33*k1*n1*n2 *u2
2 3 2 2 2
+ 4*a33 *b33*k10*n1 *u2 - 8*a33 *b33*k10*n1 *n2*u1*u2
2 2 2 4 3
- 4*a33 *b33*k10*n1*n2 *u2 + 8*a33*b33 *k1*n1*u3*v3
3 2 2 3 2
+ 4*a33*b33 *k1*n1 *u1*v3 + 8*a33*b33 *k1*n1 *u3*v1*v3
3 2 3
+ 4*a33*b33 *k1*n1*n2*u2*v3 + 8*a33*b33 *k1*n1*n2*u3*v2*v3
3 2 2 3 2
- 2*a33*b33 *k10*n1 *u1*v3 + 4*a33*b33 *k10*n1 *u3*v1*v3
3 2 3
- 2*a33*b33 *k10*n1*n2*u2*v3 + 4*a33*b33 *k10*n1*n2*u3*v2*v3
2 3 2 2
- 4*a33*b33 *k1*n1 *u2*v2 + 4*a33*b33 *k1*n1 *n2*u1*v2
2 2 2 2
+ 4*a33*b33 *k1*n1 *n2*u2*v1 + 4*a33*b33 *k1*n1*n2 *u2*v2
2 3 4 3
- 4*a33*b33 *k10*n1 *u3*v3 + 2*a33*b33*k1*n1 *u1 + 2*a33*b33*k1*n1 *n2*u2
4 3 5 4
- 2*a33*b33*k10*n1 *u1 - 2*a33*b33*k10*n1 *n2*u2 - b33 *k1*n1*v1
5 2 2 5 2 2 5 4
- 2*b33 *k1*n1*v1 *v2 - 2*b33 *k1*n1*v1 *v3 - b33 *k1*n1*v2
5 2 2 4 2 2 4 2
- 2*b33 *k1*n1*v2 *v3 + 2*b33 *k1*n1 *v1*v3 + 2*b33 *k1*n1*n2*v2*v3
4 2 2 4 2 3 3 2
+ b33 *k10*n1 *v1*v3 + b33 *k10*n1*n2*v2*v3 - b33 *k1*n1 *v2
3 2 3 2 2 3 3 2
+ 2*b33 *k1*n1 *n2*v1*v2 + b33 *k1*n1*n2 *v2 + 2*b33 *k10*n1 *v1
3 3 2 3 2 3 2 2
+ b33 *k10*n1 *v2 + 2*b33 *k10*n1 *n2*v1*v2 + b33 *k10*n1*n2 *v2
2 4 2 3 2 4
+ b33 *k1*n1 *v1 + b33 *k1*n1 *n2*v2 - b33 *k10*n1 *v1
2 3 4
- b33 *k10*n1 *n2*v2)/(16*a33 *b33*n1)
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*n1*u1 + 4*a33*n2*u2 + b33**2*v3**2)/(4*a33)$
INT=( - 64*a33**6*p16*u1**2*v2**2 - 64*a33**6*p16*u1**2*v3**2 - 64*a33**6*p16*u2
**2*v1**2 - 128*a33**6*p16*u2**2*v2**2 - 64*a33**6*p16*u2**2*v3**2 - 128*a33**6*
p16*u2*u3*v2*v3 - 64*a33**6*p16*u3**2*v3**2 + 64*a33**5*b33*p16*u3*v1**2*v3 + 64
*a33**5*b33*p16*u3*v2**2*v3 + 64*a33**5*b33*p16*u3*v3**3 + 32*a33**5*n1*p16*u1*
v2**2 + 32*a33**5*n1*p16*u1*v3**2 - 32*a33**5*n1*p16*u2*v1*v2 - 32*a33**5*n1*p16
*u3*v1*v3 + 32*a33**5*n2*p16*u2*v1**2 + 32*a33**5*n2*p16*u2*v2**2 + 32*a33**5*n2
*p16*u2*v3**2 - 16*a33**4*b33**2*p16*v1**4 - 32*a33**4*b33**2*p16*v1**2*v2**2 -
16*a33**4*b33**2*p16*v1**2*v3**2 - 16*a33**4*b33**2*p16*v2**4 - 16*a33**4*b33**2
*p16*v2**2*v3**2 + 16*a33**4*b33*k1*n1*u1**4 + 32*a33**4*b33*k1*n1*u1**2*u2**2 +
16*a33**4*b33*k1*n1*u2**4 + 16*a33**4*b33*k10*n1*u1**2*u3**2 + 16*a33**4*b33*
k10*n1*u2**2*u3**2 + 16*a33**4*b33*n1*p16*v1**3 + 16*a33**4*b33*n1*p16*v1*v2**2
+ 16*a33**4*b33*n1*p16*v1*v3**2 + 16*a33**4*b33*n1*q1*u1**2 + 16*a33**4*b33*n1*
q1*u2**2 + 16*a33**4*b33*n2*p16*v1**2*v2 + 16*a33**4*b33*n2*p16*v2**3 + 16*a33**
4*b33*n2*p16*v2*v3**2 - 32*a33**3*b33**2*k1*n1*u1**2*u3*v3 - 32*a33**3*b33**2*k1
*n1*u2**2*u3*v3 - 16*a33**3*b33**2*k10*n1*u3**3*v3 - 16*a33**3*b33**2*n1*q1*u3*
v3 - 16*a33**3*b33*k1*n1**2*u1**3 - 16*a33**3*b33*k1*n1**2*u1*u2**2 - 16*a33**3*
b33*k1*n1*n2*u1**2*u2 - 16*a33**3*b33*k1*n1*n2*u2**3 + 8*a33**3*b33*k10*n1**2*u1
**3 + 8*a33**3*b33*k10*n1**2*u1*u2**2 - 8*a33**3*b33*k10*n1**2*u1*u3**2 + 8*a33
**3*b33*k10*n1*n2*u1**2*u2 + 8*a33**3*b33*k10*n1*n2*u2**3 - 8*a33**3*b33*k10*n1*
n2*u2*u3**2 - 8*a33**3*b33*n1**2*q1*u1 - 8*a33**3*b33*n1*n2*q1*u2 - 8*a33**2*b33
**3*k1*n1*u1**2*v3**2 - 8*a33**2*b33**3*k1*n1*u2**2*v3**2 + 16*a33**2*b33**3*k1*
n1*u3**2*v3**2 - 4*a33**2*b33**3*k10*n1*u3**2*v3**2 + 4*a33**2*b33**3*n1*q1*v1**
2 + 4*a33**2*b33**3*n1*q1*v2**2 + 8*a33**2*b33**2*k1*n1**2*u1*u2*v2 + 24*a33**2*
b33**2*k1*n1**2*u1*u3*v3 - 8*a33**2*b33**2*k1*n1**2*u2**2*v1 - 8*a33**2*b33**2*
k1*n1*n2*u1**2*v2 - 8*a33**2*b33**2*k1*n1*n2*u2**2*v2 + 16*a33**2*b33**2*k1*n1*
n2*u2*u3*v3 + 4*a33**2*b33**2*k10*n1**2*u1*u2*v2 - 4*a33**2*b33**2*k10*n1**2*u1*
u3*v3 - 4*a33**2*b33**2*k10*n1**2*u2**2*v1 - 4*a33**2*b33**2*k10*n1**2*u3**2*v1
- 4*a33**2*b33**2*k10*n1*n2*u1**2*v2 - 4*a33**2*b33**2*k10*n1*n2*u2**2*v2 - 8*
a33**2*b33**2*k10*n1*n2*u2*u3*v3 - 4*a33**2*b33**2*k10*n1*n2*u3**2*v2 - 4*a33**2
*b33**2*n1**2*q1*v1 - 4*a33**2*b33**2*n1*n2*q1*v2 - 4*a33**2*b33*k1*n1**3*u2**2
+ 8*a33**2*b33*k1*n1**2*n2*u1*u2 + 4*a33**2*b33*k1*n1*n2**2*u2**2 + 4*a33**2*b33
*k10*n1**3*u2**2 - 8*a33**2*b33*k10*n1**2*n2*u1*u2 - 4*a33**2*b33*k10*n1*n2**2*
u2**2 + 8*a33*b33**4*k1*n1*u3*v3**3 + 4*a33*b33**3*k1*n1**2*u1*v3**2 + 8*a33*b33
**3*k1*n1**2*u3*v1*v3 + 4*a33*b33**3*k1*n1*n2*u2*v3**2 + 8*a33*b33**3*k1*n1*n2*
u3*v2*v3 - 2*a33*b33**3*k10*n1**2*u1*v3**2 + 4*a33*b33**3*k10*n1**2*u3*v1*v3 - 2
*a33*b33**3*k10*n1*n2*u2*v3**2 + 4*a33*b33**3*k10*n1*n2*u3*v2*v3 - 4*a33*b33**2*
k1*n1**3*u2*v2 + 4*a33*b33**2*k1*n1**2*n2*u1*v2 + 4*a33*b33**2*k1*n1**2*n2*u2*v1
+ 4*a33*b33**2*k1*n1*n2**2*u2*v2 - 4*a33*b33**2*k10*n1**3*u3*v3 + 2*a33*b33*k1*
n1**4*u1 + 2*a33*b33*k1*n1**3*n2*u2 - 2*a33*b33*k10*n1**4*u1 - 2*a33*b33*k10*n1
**3*n2*u2 - b33**5*k1*n1*v1**4 - 2*b33**5*k1*n1*v1**2*v2**2 - 2*b33**5*k1*n1*v1
**2*v3**2 - b33**5*k1*n1*v2**4 - 2*b33**5*k1*n1*v2**2*v3**2 + 2*b33**4*k1*n1**2*
v1*v3**2 + 2*b33**4*k1*n1*n2*v2*v3**2 + b33**4*k10*n1**2*v1*v3**2 + b33**4*k10*
n1*n2*v2*v3**2 - b33**3*k1*n1**3*v2**2 + 2*b33**3*k1*n1**2*n2*v1*v2 + b33**3*k1*
n1*n2**2*v2**2 + 2*b33**3*k10*n1**3*v1**2 + b33**3*k10*n1**3*v2**2 + 2*b33**3*
k10*n1**2*n2*v1*v2 + b33**3*k10*n1*n2**2*v2**2 + b33**2*k1*n1**4*v1 + b33**2*k1*
n1**3*n2*v2 - b33**2*k10*n1**4*v1 - b33**2*k10*n1**3*n2*v2)/(16*a33**4*b33*n1)$