Solution 1 to problem e3null
Remaining equations |
Expressions |
Parameters |
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Equations
The following unsolved equations remain:
2 2
0=n1 + n2
Expressions
The solution is given through the following expressions:
1
a22=---*a33
2
b22=0
- b31*n2
b32=-----------
n1
b33=0
c12=0
c13=0
c22=0
c23=0
c33=0
- m2*n1
m1=----------
n2
2 2
4*b31*n1 + 6*b31*n2
m3=-----------------------
a33*n1
3 4 2 3
r6=(16*a33*k1*m2*n1 *n3 + 640*b31*k1*n1 *n2 + 1336*b31*k1*n1 *n2
2 2 5 3 2 4
- 32*b31*k1*n1 *n2*n3 + 696*b31*k1*n2 - 48*b31*k1*n2 *n3 )/(a33 *n1*n2)
3 2 3
16*a33*k1*m2*n1 + 88*b31*k1*n1 *n2*n3 + 72*b31*k1*n2 *n3
r5=-----------------------------------------------------------
4
a33 *n1
r4
3 2 2 3
8*a33*k1*m2*n1 - 8*a33*k1*m2*n1*n2 - 52*b31*k1*n1 *n2*n3 - 68*b31*k1*n2 *n3
=-------------------------------------------------------------------------------
4
a33 *n2
2 2
- 132*k1*n1 *n3 - 132*k1*n2 *n3
r3=----------------------------------
3
a33
2 3
- 132*k1*n1 *n2 - 132*k1*n2
r2=-------------------------------
3
a33
3 2
- 132*k1*n1 - 132*k1*n1*n2
r1=-------------------------------
3
a33
3 2 2 2
q19=( - 108*a33*b31*k1*m2*n1 - 108*a33*b31*k1*m2*n1*n2 + 120*b31 *k1*n1 *n2*n3
2 3 4 2
+ 120*b31 *k1*n2 *n3)/(a33 *n1 )
3 2 2 2
q18=(52*a33*b31*k1*m2*n1 + 52*a33*b31*k1*m2*n1*n2 + 120*b31 *k1*n1 *n2*n3
2 3 4
+ 120*b31 *k1*n2 *n3)/(a33 *n1*n2)
3 2 3
16*a33*k1*m2*n1 + 88*b31*k1*n1 *n2*n3 + 96*b31*k1*n2 *n3
q17=-----------------------------------------------------------
3
a33 *n1*n2
2 3
8*a33*k1*m2*n1*n3 - 80*b31*k1*n1 *n2 - 80*b31*k1*n2
q16=------------------------------------------------------
3
a33 *n1
2 3
- 8*a33*k1*m2*n1*n3 - 80*b31*k1*n1 *n2 - 80*b31*k1*n2
q15=---------------------------------------------------------
3
a33 *n2
140 3 140 2
q14=( - -----*a33*b31*k1*m2*n1 *n3 - -----*a33*b31*k1*m2*n1*n2 *n3
3 3
160 2 4 208 2 2 3 2 5 4 2
- -----*b31 *k1*n1 *n2 - -----*b31 *k1*n1 *n2 - 16*b31 *k1*n2 )/(a33 *n1
3 3
*n2)
q13=0
2 3
20*b31*k1*n1 *n2 + 36*b31*k1*n2
q12=----------------------------------
3
a33 *n1
q11=0
- 16*a33*k1*m2*n1 - 8*b31*k1*n2*n3
q10=-------------------------------------
3
a33
140 3 140 2
q9=( - -----*a33*b31*k1*m2*n1 *n3 - -----*a33*b31*k1*m2*n1*n2 *n3
3 3
160 2 4 208 2 2 3 2 5 4 2
- -----*b31 *k1*n1 *n2 - -----*b31 *k1*n1 *n2 - 16*b31 *k1*n2 )/(a33 *n1
3 3
*n2)
2 2
28*b31*k1*n1 + 44*b31*k1*n2
q8=-------------------------------
3
a33
16*a33*k1*m2*n1 + 8*b31*k1*n2*n3
q7=----------------------------------
3
a33
8*k1*n2*n3
q5=------------
2
a33
8*k1*n1*n3
q4=------------
2
a33
2 2 2
66*k1*n1 + 70*k1*n2 - 4*k1*n3
q3=----------------------------------
2
a33
8*k1*n1*n2
q2=------------
2
a33
2 2 2
70*k1*n1 + 66*k1*n2 - 4*k1*n3
q1=----------------------------------
2
a33
p50=0
p49=0
p48=0
p47=0
p46=0
p45=0
p44=0
p43=0
2 2 2 3
- 48*b31 *k1*n1 *n2 - 48*b31 *k1*n2
p42=---------------------------------------
3 2
a33 *n1
p41=0
p40=0
p39=0
2 2 2 2
48*b31 *k1*n1 + 48*b31 *k1*n2
p38=---------------------------------
3
a33 *n1
p37=0
2 2
56*b31*k1*n1 + 72*b31*k1*n2
p36=-------------------------------
2
a33 *n1
8*a33*k1*m2*n1 - 8*b31*k1*n2*n3
p35=---------------------------------
2
a33 *n1
- 4*a33*k1*m2*n1 + 8*b31*k1*n2*n3
p34=------------------------------------
2
a33 *n2
2 2
24*b31*k1*n1 + 24*b31*k1*n2
p33=-------------------------------
2
a33 *n1
p32=0
2 2
16*b31*k1*n1 + 16*b31*k1*n2
p31=-------------------------------
2
a33 *n1
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*k1*m2*n1
p18=------------
a33*n2
4*k1*m2
p17=---------
a33
p16=0
p15=0
p14=0
p13=0
- 16*b31*k1*n2
p12=-----------------
2
a33
- 4*k1*m2*n1
p11=---------------
a33*n2
p10=0
p9=0
p8=0
- 4*k1*n3
p7=------------
a33
p6=0
- 4*k1*n3
p5=------------
a33
- 4*k1*n2
p4=------------
a33
- 4*k1*n1
p3=------------
a33
- 4*k1*n2
p2=------------
a33
- 4*k1*n1
p1=------------
a33
k104=0
k103=0
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
k95=0
k94=0
k93=0
k92=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
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*b31*k1*n2
k64=----------------
a33*n1
4*b31*k1
k63=----------
a33
k62=0
k61=0
k60=0
k59=0
k58=0
k57=0
k56=0
k55=0
k54=0
k53=0
k52=0
k51=0
k50=0
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*b31*k1*n2
k32=----------------
a33*n1
- 4*b31*k1
k31=-------------
a33
- 4*b31*k1*n2
k30=----------------
a33*n1
k29=0
k28=0
k27=0
k26=0
k25=0
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,n1,n3,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 *n1*n2*u1 + a33 *n1*n2*u2 + 2*a33 *n1*n2*u3 + 2*a33*b31*n1*n2*u3*v1
2 2
- 2*a33*b31*n2 *u3*v2 - 2*a33*m2*n1 *v1 + 2*a33*m2*n1*n2*v2
2 2 2
+ 2*a33*n1 *n2*u1 + 2*a33*n1*n2 *u2 + 2*a33*n1*n2*n3*u3 + 8*b31*n1 *n2*v3
3
+ 12*b31*n2 *v3)/(2*a33*n1*n2)
4 2 4 4 2 2 2 4 2 4
INT=(k1*(3*a33 *n1 *n2*u1 + 6*a33 *n1 *n2*u1 *u2 + 3*a33 *n1 *n2*u2
3 2 3 2 2
- 12*a33 *b31*n1 *n2*u1*u2*u3*v2 + 12*a33 *b31*n1 *n2*u1*u3 *v3
3 2 2 3 2 2
+ 12*a33 *b31*n1 *n2*u2 *u3*v1 - 12*a33 *b31*n1*n2 *u1 *u3*v2
3 2 2 3 2 2
- 12*a33 *b31*n1*n2 *u2 *u3*v2 - 24*a33 *b31*n1*n2 *u2*u3 *v3
3 3 3 3
+ 12*a33 *m2*n1 *u1*u2*v2 - 12*a33 *m2*n1 *u1*u3*v3
3 3 2 3 2 2
- 12*a33 *m2*n1 *u2 *v1 + 12*a33 *m2*n1 *n2*u1 *v2
3 2 2 3 2
+ 12*a33 *m2*n1 *n2*u2 *v2 + 24*a33 *m2*n1 *n2*u2*u3*v3
3 3 3 3 3 2 3 2 2 2
- 12*a33 *n1 *n2*u1 - 12*a33 *n1 *n2*u1*u2 - 12*a33 *n1 *n2 *u1 *u2
3 2 2 3 3 2 2
- 12*a33 *n1 *n2 *u2 - 12*a33 *n1 *n2*n3*u1 *u3
3 2 2 2 3 2
- 12*a33 *n1 *n2*n3*u2 *u3 + 48*a33 *b31*n1 *n2*u1 *v3
2 3 2 2 3 2
+ 72*a33 *b31*n1 *n2*u2 *v3 + 168*a33 *b31*n1 *n2*u3 *v3
2 2 2 2 2 2
+ 48*a33 *b31*n1 *n2 *u1*u3*v2 - 48*a33 *b31*n1 *n2 *u2*u3*v1
2 2 2 3 2
+ 24*a33 *b31*n1 *n2*n3*u1*u3*v3 + 48*a33 *b31*n1*n2 *u1 *v3
2 3 2 2 3 2
+ 72*a33 *b31*n1*n2 *u2 *v3 + 216*a33 *b31*n1*n2 *u3 *v3
2 2 2 4
- 24*a33 *b31*n1*n2 *n3*u2*u3*v3 + 48*a33 *m2*n1 *u3*v3
2 3 2 3
- 48*a33 *m2*n1 *n2*u1*v2 + 48*a33 *m2*n1 *n2*u2*v1
2 3 2 2
- 24*a33 *m2*n1 *n3*u1*v3 + 24*a33 *m2*n1 *n2*n3*u2*v3
2 4 2 2 4 2 2 3 2
+ 210*a33 *n1 *n2*u1 + 198*a33 *n1 *n2*u2 + 24*a33 *n1 *n2 *u1*u2
2 3 2 2 3 2
+ 24*a33 *n1 *n2*n3*u1*u3 + 198*a33 *n1 *n2 *u1
2 2 3 2 2 2 2
+ 210*a33 *n1 *n2 *u2 + 24*a33 *n1 *n2 *n3*u2*u3
2 2 2 2 2 2 2 2
- 12*a33 *n1 *n2*n3 *u1 - 12*a33 *n1 *n2*n3 *u2
2 3 2 2 2
+ 144*a33*b31 *n1 *n2*u3*v1*v3 - 144*a33*b31 *n1 *n2 *u3*v2*v3
2 3 2 4
+ 144*a33*b31 *n1*n2 *u3*v1*v3 - 144*a33*b31 *n2 *u3*v2*v3
4 3
+ 156*a33*b31*m2*n1 *v1*v3 - 324*a33*b31*m2*n1 *n2*v2*v3
3 2 3 2
- 140*a33*b31*m2*n1 *n3*v1 - 140*a33*b31*m2*n1 *n3*v2
2 2 3
+ 156*a33*b31*m2*n1 *n2 *v1*v3 - 324*a33*b31*m2*n1*n2 *v2*v3
2 2 2 2
- 140*a33*b31*m2*n1*n2 *n3*v1 - 140*a33*b31*m2*n1*n2 *n3*v2
4 4
- 240*a33*b31*n1 *n2*u1*v3 + 84*a33*b31*n1 *n2*u3*v1
3 2 3 2
- 240*a33*b31*n1 *n2 *u2*v3 + 60*a33*b31*n1 *n2 *u3*v2
3 2 3
+ 264*a33*b31*n1 *n2*n3*u3*v3 - 240*a33*b31*n1 *n2 *u1*v3
2 3 2 2
+ 132*a33*b31*n1 *n2 *u3*v1 - 24*a33*b31*n1 *n2 *n3*u1*v2
2 2 4
+ 24*a33*b31*n1 *n2 *n3*u2*v1 - 240*a33*b31*n1*n2 *u2*v3
4 3
+ 108*a33*b31*n1*n2 *u3*v2 + 288*a33*b31*n1*n2 *n3*u3*v3
5 4 4
+ 24*a33*m2*n1 *v1 + 48*a33*m2*n1 *n2*v2 + 48*a33*m2*n1 *n3*v3
3 2 5 4 2
- 24*a33*m2*n1 *n2 *v1 - 396*a33*n1 *n2*u1 - 396*a33*n1 *n2 *u2
4 3 3 2 4
- 396*a33*n1 *n2*n3*u3 - 396*a33*n1 *n2 *u1 - 396*a33*n1 *n2 *u2
2 3 2 4 2 2 4 2
- 396*a33*n1 *n2 *n3*u3 - 160*b31 *n1 *n2*v1 - 160*b31 *n1 *n2*v2
2 3 2 2 3 2
+ 360*b31 *n1 *n2*n3*v1*v3 - 208*b31 *n1 *n2 *v1
2 2 3 2 2 2 2
- 208*b31 *n1 *n2 *v2 + 360*b31 *n1 *n2 *n3*v2*v3
2 3 2 5 2 2 5 2
+ 360*b31 *n1*n2 *n3*v1*v3 - 48*b31 *n2 *v1 - 48*b31 *n2 *v2
2 4 5 4
+ 360*b31 *n2 *n3*v2*v3 + 1920*b31*n1 *n2*v3 - 156*b31*n1 *n2*n3*v1
3 3 3 2 3 2
+ 4008*b31*n1 *n2 *v3 + 264*b31*n1 *n2 *n3*v2 - 96*b31*n1 *n2*n3 *v3
2 3 5 4
- 204*b31*n1 *n2 *n3*v1 + 2088*b31*n1*n2 *v3 + 216*b31*n1*n2 *n3*v2
3 2 4 2
- 144*b31*n1*n2 *n3 *v3))/(3*a33 *n1 *n2)
And again in machine readable form:
HAM=(a33**2*n1*n2*u1**2 + a33**2*n1*n2*u2**2 + 2*a33**2*n1*n2*u3**2 + 2*a33*b31*
n1*n2*u3*v1 - 2*a33*b31*n2**2*u3*v2 - 2*a33*m2*n1**2*v1 + 2*a33*m2*n1*n2*v2 + 2*
a33*n1**2*n2*u1 + 2*a33*n1*n2**2*u2 + 2*a33*n1*n2*n3*u3 + 8*b31*n1**2*n2*v3 + 12
*b31*n2**3*v3)/(2*a33*n1*n2)$
INT=(k1*(3*a33**4*n1**2*n2*u1**4 + 6*a33**4*n1**2*n2*u1**2*u2**2 + 3*a33**4*n1**
2*n2*u2**4 - 12*a33**3*b31*n1**2*n2*u1*u2*u3*v2 + 12*a33**3*b31*n1**2*n2*u1*u3**
2*v3 + 12*a33**3*b31*n1**2*n2*u2**2*u3*v1 - 12*a33**3*b31*n1*n2**2*u1**2*u3*v2 -
12*a33**3*b31*n1*n2**2*u2**2*u3*v2 - 24*a33**3*b31*n1*n2**2*u2*u3**2*v3 + 12*
a33**3*m2*n1**3*u1*u2*v2 - 12*a33**3*m2*n1**3*u1*u3*v3 - 12*a33**3*m2*n1**3*u2**
2*v1 + 12*a33**3*m2*n1**2*n2*u1**2*v2 + 12*a33**3*m2*n1**2*n2*u2**2*v2 + 24*a33
**3*m2*n1**2*n2*u2*u3*v3 - 12*a33**3*n1**3*n2*u1**3 - 12*a33**3*n1**3*n2*u1*u2**
2 - 12*a33**3*n1**2*n2**2*u1**2*u2 - 12*a33**3*n1**2*n2**2*u2**3 - 12*a33**3*n1
**2*n2*n3*u1**2*u3 - 12*a33**3*n1**2*n2*n3*u2**2*u3 + 48*a33**2*b31*n1**3*n2*u1
**2*v3 + 72*a33**2*b31*n1**3*n2*u2**2*v3 + 168*a33**2*b31*n1**3*n2*u3**2*v3 + 48
*a33**2*b31*n1**2*n2**2*u1*u3*v2 - 48*a33**2*b31*n1**2*n2**2*u2*u3*v1 + 24*a33**
2*b31*n1**2*n2*n3*u1*u3*v3 + 48*a33**2*b31*n1*n2**3*u1**2*v3 + 72*a33**2*b31*n1*
n2**3*u2**2*v3 + 216*a33**2*b31*n1*n2**3*u3**2*v3 - 24*a33**2*b31*n1*n2**2*n3*u2
*u3*v3 + 48*a33**2*m2*n1**4*u3*v3 - 48*a33**2*m2*n1**3*n2*u1*v2 + 48*a33**2*m2*
n1**3*n2*u2*v1 - 24*a33**2*m2*n1**3*n3*u1*v3 + 24*a33**2*m2*n1**2*n2*n3*u2*v3 +
210*a33**2*n1**4*n2*u1**2 + 198*a33**2*n1**4*n2*u2**2 + 24*a33**2*n1**3*n2**2*u1
*u2 + 24*a33**2*n1**3*n2*n3*u1*u3 + 198*a33**2*n1**2*n2**3*u1**2 + 210*a33**2*n1
**2*n2**3*u2**2 + 24*a33**2*n1**2*n2**2*n3*u2*u3 - 12*a33**2*n1**2*n2*n3**2*u1**
2 - 12*a33**2*n1**2*n2*n3**2*u2**2 + 144*a33*b31**2*n1**3*n2*u3*v1*v3 - 144*a33*
b31**2*n1**2*n2**2*u3*v2*v3 + 144*a33*b31**2*n1*n2**3*u3*v1*v3 - 144*a33*b31**2*
n2**4*u3*v2*v3 + 156*a33*b31*m2*n1**4*v1*v3 - 324*a33*b31*m2*n1**3*n2*v2*v3 -
140*a33*b31*m2*n1**3*n3*v1**2 - 140*a33*b31*m2*n1**3*n3*v2**2 + 156*a33*b31*m2*
n1**2*n2**2*v1*v3 - 324*a33*b31*m2*n1*n2**3*v2*v3 - 140*a33*b31*m2*n1*n2**2*n3*
v1**2 - 140*a33*b31*m2*n1*n2**2*n3*v2**2 - 240*a33*b31*n1**4*n2*u1*v3 + 84*a33*
b31*n1**4*n2*u3*v1 - 240*a33*b31*n1**3*n2**2*u2*v3 + 60*a33*b31*n1**3*n2**2*u3*
v2 + 264*a33*b31*n1**3*n2*n3*u3*v3 - 240*a33*b31*n1**2*n2**3*u1*v3 + 132*a33*b31
*n1**2*n2**3*u3*v1 - 24*a33*b31*n1**2*n2**2*n3*u1*v2 + 24*a33*b31*n1**2*n2**2*n3
*u2*v1 - 240*a33*b31*n1*n2**4*u2*v3 + 108*a33*b31*n1*n2**4*u3*v2 + 288*a33*b31*
n1*n2**3*n3*u3*v3 + 24*a33*m2*n1**5*v1 + 48*a33*m2*n1**4*n2*v2 + 48*a33*m2*n1**4
*n3*v3 - 24*a33*m2*n1**3*n2**2*v1 - 396*a33*n1**5*n2*u1 - 396*a33*n1**4*n2**2*u2
- 396*a33*n1**4*n2*n3*u3 - 396*a33*n1**3*n2**3*u1 - 396*a33*n1**2*n2**4*u2 -
396*a33*n1**2*n2**3*n3*u3 - 160*b31**2*n1**4*n2*v1**2 - 160*b31**2*n1**4*n2*v2**
2 + 360*b31**2*n1**3*n2*n3*v1*v3 - 208*b31**2*n1**2*n2**3*v1**2 - 208*b31**2*n1
**2*n2**3*v2**2 + 360*b31**2*n1**2*n2**2*n3*v2*v3 + 360*b31**2*n1*n2**3*n3*v1*v3
- 48*b31**2*n2**5*v1**2 - 48*b31**2*n2**5*v2**2 + 360*b31**2*n2**4*n3*v2*v3 +
1920*b31*n1**5*n2*v3 - 156*b31*n1**4*n2*n3*v1 + 4008*b31*n1**3*n2**3*v3 + 264*
b31*n1**3*n2**2*n3*v2 - 96*b31*n1**3*n2*n3**2*v3 - 204*b31*n1**2*n2**3*n3*v1 +
2088*b31*n1*n2**5*v3 + 216*b31*n1*n2**4*n3*v2 - 144*b31*n1*n2**3*n3**2*v3))/(3*
a33**4*n1**2*n2)$