Solution 5 to problem e3null
Expressions |
Parameters |
Relevance |
Back to problem e3null
Expressions
The solution is given through the following expressions:
1 2
- ---*b33
4
a22=-------------
c33
1 2
---*b33
4
a33=----------
c33
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
m1=0
m2=0
2*c33*n3
m3=----------
b33
4 2 4 2
- 8*b33 *c33 *n3*q1 + 80*c33 *k1*n1 *n3
r6=------------------------------------------
7
b33
4 2 4 2 4 2
- 4*b33 *c33 *n2*q1 + 40*c33 *k1*n1 *n2 + 40*c33 *k1*n2*n3
r5=--------------------------------------------------------------
7
b33
4 2 4 3 4 2
- 4*b33 *c33 *n1*q1 + 40*c33 *k1*n1 + 40*c33 *k1*n1*n3
r4=-----------------------------------------------------------
7
b33
4 3 2 3 3
- 2*b33 *c33*n3*q1 + 20*c33 *k1*n1 *n3 - 20*c33 *k1*n3
r3=----------------------------------------------------------
6
b33
4 3 2 3 2
- 2*b33 *c33*n2*q1 + 20*c33 *k1*n1 *n2 - 20*c33 *k1*n2*n3
r2=-------------------------------------------------------------
6
b33
4 3 3 3 2
- 2*b33 *c33*n1*q1 + 20*c33 *k1*n1 - 20*c33 *k1*n1*n3
r1=----------------------------------------------------------
6
b33
4
16*c33 *k1*n2*n3
q19=------------------
6
b33
4
16*c33 *k1*n1*n3
q18=------------------
6
b33
4 3 2 3 2
- 4*b33 *c33*q1 + 24*c33 *k1*n1 + 56*c33 *k1*n3
q17=----------------------------------------------------
5
b33
3
56*c33 *k1*n2*n3
q16=------------------
5
b33
3
56*c33 *k1*n1*n3
q15=------------------
5
b33
4 2 4 2 4 2 4 2
4*b33 *c33 *q1 - 40*c33 *k1*n1 - 8*c33 *k1*n2 - 64*c33 *k1*n3
q14=------------------------------------------------------------------
6
b33
4
- 16*c33 *k1*n1*n2
q13=---------------------
6
b33
3
16*c33 *k1*n2*n3
q12=------------------
5
b33
3 2 3 2
- 16*c33 *k1*n1 + 16*c33 *k1*n2
q11=------------------------------------
5
b33
3
16*c33 *k1*n1*n2
q10=------------------
5
b33
4 2 4 2 4 2
4*b33 *c33 *q1 - 48*c33 *k1*n1 - 64*c33 *k1*n3
q9=--------------------------------------------------
6
b33
3
16*c33 *k1*n1*n3
q8=------------------
5
b33
3
16*c33 *k1*n1*n2
q7=------------------
5
b33
2
20*c33 *k1*n2*n3
q5=------------------
4
b33
2
20*c33 *k1*n1*n3
q4=------------------
4
b33
4 2 2 2 2
b33 *q1 - 10*c33 *k1*n1 + 10*c33 *k1*n2
q3=-------------------------------------------
4
b33
2
20*c33 *k1*n1*n2
q2=------------------
4
b33
4
64*c33 *k1*n3
p50=---------------
5
b33
4
8*c33 *k1*n2
p49=--------------
5
b33
4
8*c33 *k1*n1
p48=--------------
5
b33
3
92*c33 *k1*n3
p47=---------------
4
b33
3
28*c33 *k1*n2
p46=---------------
4
b33
3
28*c33 *k1*n1
p45=---------------
4
b33
p44=0
p43=0
3
8*c33 *k1*n2
p42=--------------
4
b33
p41=0
p40=0
p39=0
3
8*c33 *k1*n1
p38=--------------
4
b33
p37=0
2
40*c33 *k1*n3
p36=---------------
3
b33
2
28*c33 *k1*n2
p35=---------------
3
b33
2
30*c33 *k1*n1
p34=---------------
3
b33
2
- 16*c33 *k1*n3
p33=------------------
3
b33
p32=0
2
- 16*c33 *k1*n3
p31=------------------
3
b33
p30=0
p29=0
p28=0
p27=0
p26=0
p25=0
p24=0
p23=0
2
6*c33 *k1*n2
p22=--------------
3
b33
p21=0
p20=0
2
- 2*c33 *k1*n2
p19=-----------------
3
b33
2
2*c33 *k1*n1
p18=--------------
3
b33
2
- 2*c33 *k1*n2
p17=-----------------
3
b33
p16=0
p15=0
p14=0
2
6*c33 *k1*n1
p13=--------------
3
b33
p12=0
2
- 2*c33 *k1*n1
p11=-----------------
3
b33
3*c33*k1*n3
p10=-------------
2
b33
3*c33*k1*n2
p9=-------------
2
b33
3*c33*k1*n1
p8=-------------
2
b33
- 7*c33*k1*n3
p7=----------------
2
b33
p6=0
- 7*c33*k1*n3
p5=----------------
2
b33
- 7*c33*k1*n2
p4=----------------
2
b33
- 7*c33*k1*n1
p3=----------------
2
b33
- 7*c33*k1*n2
p2=----------------
2
b33
- 7*c33*k1*n1
p1=----------------
2
b33
k104=0
k103=0
3
32*c33 *k1
k102=------------
3
b33
k101=0
k100=0
4
- 32*c33 *k1
k99=---------------
4
b33
k98=0
k97=0
k96=0
k95=0
4
- 32*c33 *k1
k94=---------------
4
b33
k93=0
k92=0
2
22*c33 *k1
k91=------------
2
b33
k90=0
k89=0
2
- 8*c33 *k1
k88=--------------
2
b33
k87=0
2
- 8*c33 *k1
k86=--------------
2
b33
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
6*c33*k1
k65=----------
b33
k64=0
k63=0
- 8*c33*k1
k62=-------------
b33
k61=0
- 8*c33*k1
k60=-------------
b33
k59=0
k58=0
k57=0
k56=0
4
- 16*c33 *k1
k55=---------------
4
b33
k54=0
k53=0
k52=0
k51=0
4
- 32*c33 *k1
k50=---------------
4
b33
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
k32=0
k31=0
k30=0
k29=0
k28=0
k27=0
k26=0
4
- 16*c33 *k1
k25=---------------
4
b33
k24=0
k23=0
k21=0
k20=0
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
3
k12= - ---*k1
2
k11=0
3
k10= - ---*k1
2
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,k1,c33,b33,n1,n3,n2
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
3 2 3 2 3 2 2 2 2
HAM=( - b33 *u1 - b33 *u2 + b33 *u3 + 4*b33 *c33*u3*v3 + 4*b33*c33 *v3
2
+ 4*b33*c33*n1*u1 + 4*b33*c33*n2*u2 + 4*b33*c33*n3*u3 + 8*c33 *n3*v3)/(4
*b33*c33)
7 4 7 2 2 7 2 2 7 4
INT=(2*b33 *k1*u1 + 4*b33 *k1*u1 *u2 - 3*b33 *k1*u1 *u3 + 2*b33 *k1*u2
7 2 2 7 2 7 2
- 3*b33 *k1*u2 *u3 + 2*b33 *q1*u1 + 2*b33 *q1*u2
6 2 6 2
- 16*b33 *c33*k1*u1 *u3*v3 - 16*b33 *c33*k1*u2 *u3*v3
6 3 6 5 2 2 2
+ 12*b33 *c33*k1*u3 *v3 - 8*b33 *c33*q1*u3*v3 - 16*b33 *c33 *k1*u1 *v3
5 2 2 2 5 2 2 2 5 2 2
- 16*b33 *c33 *k1*u2 *v3 + 44*b33 *c33 *k1*u3 *v3 + 8*b33 *c33 *q1*v1
5 2 2 5 3 5 2
+ 8*b33 *c33 *q1*v2 - 14*b33 *c33*k1*n1*u1 - 14*b33 *c33*k1*n1*u1*u2
5 2 5 2
+ 6*b33 *c33*k1*n1*u1*u3 - 14*b33 *c33*k1*n2*u1 *u2
5 3 5 2
- 14*b33 *c33*k1*n2*u2 + 6*b33 *c33*k1*n2*u2*u3
5 2 5 2
- 14*b33 *c33*k1*n3*u1 *u3 - 14*b33 *c33*k1*n3*u2 *u3
5 3 5 5
+ 6*b33 *c33*k1*n3*u3 - 4*b33 *c33*n1*q1*u1 - 4*b33 *c33*n2*q1*u2
5 4 3 3
- 4*b33 *c33*n3*q1*u3 + 64*b33 *c33 *k1*u3*v3
4 2 4 2
+ 4*b33 *c33 *k1*n1*u1*u2*v2 + 60*b33 *c33 *k1*n1*u1*u3*v3
4 2 2 4 2 2
- 4*b33 *c33 *k1*n1*u2 *v1 + 12*b33 *c33 *k1*n1*u3 *v1
4 2 2 4 2 2
- 4*b33 *c33 *k1*n2*u1 *v2 - 4*b33 *c33 *k1*n2*u2 *v2
4 2 4 2 2
+ 56*b33 *c33 *k1*n2*u2*u3*v3 + 12*b33 *c33 *k1*n2*u3 *v2
4 2 2 4 2 2
- 32*b33 *c33 *k1*n3*u1 *v3 - 32*b33 *c33 *k1*n3*u2 *v3
4 2 2 4 2 4 2
+ 80*b33 *c33 *k1*n3*u3 *v3 - 8*b33 *c33 *n1*q1*v1 - 8*b33 *c33 *n2*q1*v2
4 2 3 4 4 3 4 2 2
- 16*b33 *c33 *n3*q1*v3 - 32*b33 *c33 *k1*v1 - 64*b33 *c33 *k1*v1 *v2
3 4 2 2 3 4 4 3 4 2 2
- 64*b33 *c33 *k1*v1 *v3 - 32*b33 *c33 *k1*v2 - 64*b33 *c33 *k1*v2 *v3
3 3 2 3 3
+ 56*b33 *c33 *k1*n1*u1*v3 + 16*b33 *c33 *k1*n1*u3*v1*v3
3 3 2 3 3
+ 56*b33 *c33 *k1*n2*u2*v3 + 16*b33 *c33 *k1*n2*u3*v2*v3
3 3 2 3 2 2 2
+ 184*b33 *c33 *k1*n3*u3*v3 - 20*b33 *c33 *k1*n1 *u2
3 2 3 2
+ 40*b33 *c33 *k1*n1*n2*u1*u2 + 40*b33 *c33 *k1*n1*n3*u1*u3
3 2 2 2 3 2
+ 20*b33 *c33 *k1*n2 *u2 + 40*b33 *c33 *k1*n2*n3*u2*u3
2 4 2 2 4 2
+ 16*b33 *c33 *k1*n1*v1*v3 + 16*b33 *c33 *k1*n2*v2*v3
2 4 3 2 3 2
+ 128*b33 *c33 *k1*n3*v3 - 32*b33 *c33 *k1*n1 *u2*v2
2 3 2 2 3
+ 48*b33 *c33 *k1*n1 *u3*v3 + 32*b33 *c33 *k1*n1*n2*u1*v2
2 3 2 3
+ 32*b33 *c33 *k1*n1*n2*u2*v1 + 112*b33 *c33 *k1*n1*n3*u1*v3
2 3 2 3 2
+ 32*b33 *c33 *k1*n1*n3*u3*v1 + 32*b33 *c33 *k1*n2 *u2*v2
2 3 2 3
+ 112*b33 *c33 *k1*n2*n3*u2*v3 + 32*b33 *c33 *k1*n2*n3*u3*v2
2 3 2 4 2 2
+ 112*b33 *c33 *k1*n3 *u3*v3 - 96*b33*c33 *k1*n1 *v1
4 2 2 4
- 80*b33*c33 *k1*n1 *v2 - 32*b33*c33 *k1*n1*n2*v1*v2
4 4 2 2
+ 32*b33*c33 *k1*n1*n3*v1*v3 - 16*b33*c33 *k1*n2 *v2
4 4 2 2
+ 32*b33*c33 *k1*n2*n3*v2*v3 - 128*b33*c33 *k1*n3 *v1
4 2 2 3 3
- 128*b33*c33 *k1*n3 *v2 + 40*b33*c33 *k1*n1 *u1
3 2 3 2
+ 40*b33*c33 *k1*n1 *n2*u2 + 40*b33*c33 *k1*n1 *n3*u3
3 2 3 2
- 40*b33*c33 *k1*n1*n3 *u1 - 40*b33*c33 *k1*n2*n3 *u2
3 3 4 3 4 2
- 40*b33*c33 *k1*n3 *u3 + 80*c33 *k1*n1 *v1 + 80*c33 *k1*n1 *n2*v2
4 2 4 2 4 2
+ 160*c33 *k1*n1 *n3*v3 + 80*c33 *k1*n1*n3 *v1 + 80*c33 *k1*n2*n3 *v2)/(2
7
*b33 )
And again in machine readable form:
HAM=( - b33**3*u1**2 - b33**3*u2**2 + b33**3*u3**2 + 4*b33**2*c33*u3*v3 + 4*b33*
c33**2*v3**2 + 4*b33*c33*n1*u1 + 4*b33*c33*n2*u2 + 4*b33*c33*n3*u3 + 8*c33**2*n3
*v3)/(4*b33*c33)$
INT=(2*b33**7*k1*u1**4 + 4*b33**7*k1*u1**2*u2**2 - 3*b33**7*k1*u1**2*u3**2 + 2*
b33**7*k1*u2**4 - 3*b33**7*k1*u2**2*u3**2 + 2*b33**7*q1*u1**2 + 2*b33**7*q1*u2**
2 - 16*b33**6*c33*k1*u1**2*u3*v3 - 16*b33**6*c33*k1*u2**2*u3*v3 + 12*b33**6*c33*
k1*u3**3*v3 - 8*b33**6*c33*q1*u3*v3 - 16*b33**5*c33**2*k1*u1**2*v3**2 - 16*b33**
5*c33**2*k1*u2**2*v3**2 + 44*b33**5*c33**2*k1*u3**2*v3**2 + 8*b33**5*c33**2*q1*
v1**2 + 8*b33**5*c33**2*q1*v2**2 - 14*b33**5*c33*k1*n1*u1**3 - 14*b33**5*c33*k1*
n1*u1*u2**2 + 6*b33**5*c33*k1*n1*u1*u3**2 - 14*b33**5*c33*k1*n2*u1**2*u2 - 14*
b33**5*c33*k1*n2*u2**3 + 6*b33**5*c33*k1*n2*u2*u3**2 - 14*b33**5*c33*k1*n3*u1**2
*u3 - 14*b33**5*c33*k1*n3*u2**2*u3 + 6*b33**5*c33*k1*n3*u3**3 - 4*b33**5*c33*n1*
q1*u1 - 4*b33**5*c33*n2*q1*u2 - 4*b33**5*c33*n3*q1*u3 + 64*b33**4*c33**3*k1*u3*
v3**3 + 4*b33**4*c33**2*k1*n1*u1*u2*v2 + 60*b33**4*c33**2*k1*n1*u1*u3*v3 - 4*b33
**4*c33**2*k1*n1*u2**2*v1 + 12*b33**4*c33**2*k1*n1*u3**2*v1 - 4*b33**4*c33**2*k1
*n2*u1**2*v2 - 4*b33**4*c33**2*k1*n2*u2**2*v2 + 56*b33**4*c33**2*k1*n2*u2*u3*v3
+ 12*b33**4*c33**2*k1*n2*u3**2*v2 - 32*b33**4*c33**2*k1*n3*u1**2*v3 - 32*b33**4*
c33**2*k1*n3*u2**2*v3 + 80*b33**4*c33**2*k1*n3*u3**2*v3 - 8*b33**4*c33**2*n1*q1*
v1 - 8*b33**4*c33**2*n2*q1*v2 - 16*b33**4*c33**2*n3*q1*v3 - 32*b33**3*c33**4*k1*
v1**4 - 64*b33**3*c33**4*k1*v1**2*v2**2 - 64*b33**3*c33**4*k1*v1**2*v3**2 - 32*
b33**3*c33**4*k1*v2**4 - 64*b33**3*c33**4*k1*v2**2*v3**2 + 56*b33**3*c33**3*k1*
n1*u1*v3**2 + 16*b33**3*c33**3*k1*n1*u3*v1*v3 + 56*b33**3*c33**3*k1*n2*u2*v3**2
+ 16*b33**3*c33**3*k1*n2*u3*v2*v3 + 184*b33**3*c33**3*k1*n3*u3*v3**2 - 20*b33**3
*c33**2*k1*n1**2*u2**2 + 40*b33**3*c33**2*k1*n1*n2*u1*u2 + 40*b33**3*c33**2*k1*
n1*n3*u1*u3 + 20*b33**3*c33**2*k1*n2**2*u2**2 + 40*b33**3*c33**2*k1*n2*n3*u2*u3
+ 16*b33**2*c33**4*k1*n1*v1*v3**2 + 16*b33**2*c33**4*k1*n2*v2*v3**2 + 128*b33**2
*c33**4*k1*n3*v3**3 - 32*b33**2*c33**3*k1*n1**2*u2*v2 + 48*b33**2*c33**3*k1*n1**
2*u3*v3 + 32*b33**2*c33**3*k1*n1*n2*u1*v2 + 32*b33**2*c33**3*k1*n1*n2*u2*v1 +
112*b33**2*c33**3*k1*n1*n3*u1*v3 + 32*b33**2*c33**3*k1*n1*n3*u3*v1 + 32*b33**2*
c33**3*k1*n2**2*u2*v2 + 112*b33**2*c33**3*k1*n2*n3*u2*v3 + 32*b33**2*c33**3*k1*
n2*n3*u3*v2 + 112*b33**2*c33**3*k1*n3**2*u3*v3 - 96*b33*c33**4*k1*n1**2*v1**2 -
80*b33*c33**4*k1*n1**2*v2**2 - 32*b33*c33**4*k1*n1*n2*v1*v2 + 32*b33*c33**4*k1*
n1*n3*v1*v3 - 16*b33*c33**4*k1*n2**2*v2**2 + 32*b33*c33**4*k1*n2*n3*v2*v3 - 128*
b33*c33**4*k1*n3**2*v1**2 - 128*b33*c33**4*k1*n3**2*v2**2 + 40*b33*c33**3*k1*n1
**3*u1 + 40*b33*c33**3*k1*n1**2*n2*u2 + 40*b33*c33**3*k1*n1**2*n3*u3 - 40*b33*
c33**3*k1*n1*n3**2*u1 - 40*b33*c33**3*k1*n2*n3**2*u2 - 40*b33*c33**3*k1*n3**3*u3
+ 80*c33**4*k1*n1**3*v1 + 80*c33**4*k1*n1**2*n2*v2 + 160*c33**4*k1*n1**2*n3*v3
+ 80*c33**4*k1*n1*n3**2*v1 + 80*c33**4*k1*n2*n3**2*v2)/(2*b33**7)$