Solution 2 to problem e3c2new
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
c33=0
- b31*n2
n1=-----------
b32
m1=0
m2=0
1 2 5 2
- ---*b31 *n2 - ---*b32 *n2
2 2
m3=------------------------------
a33*b32
2
16*b32*k1*n2*n3
r6=------------------
4
a33
2
16*b32*k1*n2 *n3
r5=------------------
4
a33
2
- 16*b31*k1*n2 *n3
r4=---------------------
4
a33
r3=0
2 3 2 3
8*b31 *k1*n2 + 8*b32 *k1*n2
r2=-------------------------------
3 2
a33 *b32
r1=0
q20=0
q19=0
q17=0
q16=0
2 2 2 2
- 24*b31 *k1*n2 - 24*b32 *k1*n2
q15=------------------------------------
4
a33
5 2 3 2 2 4 2
- 2*b31 *k1*n2 - 12*b31 *b32 *k1*n2 - 10*b31*b32 *k1*n2
q14=-------------------------------------------------------------
4 3
a33 *b32
2
- 16*b32*k1*n2
q13=------------------
3
a33
8*b32*k1*n2*n3
q12=----------------
3
a33
- 8*b31*k1*n2*n3
q11=-------------------
3
a33
q10=0
2
16*b31*k1*n2
q9=---------------
3
a33
8*b31*k1*n2*n3
q8=----------------
3
a33
8*b32*k1*n2*n3
q7=----------------
3
a33
8*k1*n2*n3
q5=------------
2
a33
- 8*b31*k1*n2*n3
q4=-------------------
2
a33 *b32
2 2
4*k1*n2 - 4*k1*n3
q3=---------------------
2
a33
2
- 8*b31*k1*n2
q2=-----------------
2
a33 *b32
2 2
- 4*k1*n2 - 4*k1*n3
q1=------------------------
2
a33
10 4 40 2 2 4
----*b31 *k1*n2 + ----*b31 *b32 *k1*n2 + 10*b32 *k1*n2
3 3
p56=--------------------------------------------------------
4
a33 *b32
p55=0
p54=0
p53=0
p52=0
p51=0
4 2 2 4
2*b31 *k1*n2 + 12*b31 *b32 *k1*n2 + 10*b32 *k1*n2
p50=---------------------------------------------------
4
a33 *b32
p49=0
p48=0
p47=0
p46=0
4 2 2 4
2*b31 *k1*n2 + 12*b31 *b32 *k1*n2 + 10*b32 *k1*n2
p45=---------------------------------------------------
4
a33 *b32
p44=0
p43=0
p42=0
p41=0
8*b32*k1*n3
p40=-------------
2
a33
8*b31*k1*n3
p39=-------------
2
a33
p38=0
p37=0
2 2
- 2*b31 *k1*n2 - 2*b32 *k1*n2
p36=--------------------------------
2
a33 *b32
p35=0
p34=0
p33=0
p32=0
p31=0
p30=0
p29=0
p28=0
p27=0
p26=0
16*b32*k1*n2
p25=--------------
2
a33
16*b31*k1*n2
p24=--------------
2
a33
p23=0
p22=0
p21=0
p20=0
p19=0
p18=0
p17=0
p16=0
- 16*b31*k1*n2
p15=-----------------
2
a33
16*b32*k1*n2
p14=--------------
2
a33
p13=0
p12=0
p11=0
p10=0
p9=0
p8=0
- 4*k1*n3
p7=------------
a33
p6=0
- 4*k1*n3
p5=------------
a33
- 4*k1*n2
p4=------------
a33
4*b31*k1*n2
p3=-------------
a33*b32
- 4*k1*n2
p2=------------
a33
4*b31*k1*n2
p1=-------------
a33*b32
k125=0
k124=0
k122=0
k121=0
k120=0
k119=0
2 3
- 4*b31 *b32*k1 - 4*b32 *k1
k118=------------------------------
3
a33
k117=0
k116=0
k115=0
3 2
- 4*b31 *k1 - 4*b31*b32 *k1
k114=------------------------------
3
a33
k113=0
k112=0
k110=0
k109=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k108=--------------------------
2
a33
k107=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k106=--------------------------
2
a33
k105=0
k104=0
k103=0
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
2 2
8*b31 *k1 + 8*b32 *k1
k95=-----------------------
2
a33
k94=0
k93=0
k92=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
k85=0
2 2
8*b31 *k1 + 8*b32 *k1
k84=-----------------------
2
a33
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
k64=0
k63=0
k62=0
k61=0
k60=0
k59=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k58=--------------------------
2
a33
k57=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k56=--------------------------
2
a33
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
- 4*b32*k1
k42=-------------
a33
- 8*b31*k1
k41=-------------
a33
4*b32*k1
k40=----------
a33
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=0
k30=0
k29=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k28=--------------------------
2
a33
k27=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k26=--------------------------
2
a33
k25=0
k24=0
k23=0
4*b31*k1
k22=----------
a33
- 8*b32*k1
k21=-------------
a33
- 4*b31*k1
k20=-------------
a33
k19=0
k18=0
k17=0
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:
b32,n3,k1,b31,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 *b32*u1 + a33 *b32*u2 + 2*a33 *b32*u3 + 2*a33*b31*b32*u3*v1
2
- 2*a33*b31*n2*u1 + 2*a33*b32 *u3*v2 + 2*a33*b32*n2*u2 + 2*a33*b32*n3*u3
2 2
- b31 *n2*v3 - 5*b32 *n2*v3)/(2*a33*b32)
4 3 4 4 3 2 2 4 3 4
INT=(k1*(3*a33 *b32 *u1 + 6*a33 *b32 *u1 *u2 + 3*a33 *b32 *u2
3 3 2 3 3
- 12*a33 *b31*b32 *u1 *u3*v1 - 24*a33 *b31*b32 *u1*u2*u3*v2
3 3 2 3 2 3
+ 12*a33 *b31*b32 *u2 *u3*v1 + 12*a33 *b31*b32 *n2*u1
3 2 2 3 4 2
+ 12*a33 *b31*b32 *n2*u1*u2 + 12*a33 *b32 *u1 *u3*v2
3 4 3 4 2
- 24*a33 *b32 *u1*u2*u3*v1 - 12*a33 *b32 *u2 *u3*v2
3 3 2 3 3 3
- 12*a33 *b32 *n2*u1 *u2 - 12*a33 *b32 *n2*u2
3 3 2 3 3 2
- 12*a33 *b32 *n3*u1 *u3 - 12*a33 *b32 *n3*u2 *u3
2 2 3 2 2 2 2 3 2 2
- 6*a33 *b31 *b32 *u1 *v1 - 6*a33 *b31 *b32 *u1 *v2
2 2 3 2 2 2 2 3
- 6*a33 *b31 *b32 *u1 *v3 + 24*a33 *b31 *b32 *u1*u3*v1*v3
2 2 3 2 2 2 2 3 2 2
- 6*a33 *b31 *b32 *u2 *v1 - 6*a33 *b31 *b32 *u2 *v2
2 2 3 2 2 2 2 3
- 6*a33 *b31 *b32 *u2 *v3 + 24*a33 *b31 *b32 *u2*u3*v2*v3
2 2 2 2 2 3
- 6*a33 *b31 *b32 *n2*u1 *v3 + 48*a33 *b31*b32 *n2*u1*u3*v2
2 3 2 3
- 48*a33 *b31*b32 *n2*u2*u3*v1 + 24*a33 *b31*b32 *n3*u1*u3*v3
2 2 2 2 2
- 24*a33 *b31*b32 *n2 *u1*u2 - 24*a33 *b31*b32 *n2*n3*u1*u3
2 5 2 2 2 5 2 2 2 5 2 2
- 6*a33 *b32 *u1 *v1 - 6*a33 *b32 *u1 *v2 - 6*a33 *b32 *u1 *v3
2 5 2 5 2 2 2 5 2 2
+ 24*a33 *b32 *u1*u3*v1*v3 - 6*a33 *b32 *u2 *v1 - 6*a33 *b32 *u2 *v2
2 5 2 2 2 5
- 6*a33 *b32 *u2 *v3 + 24*a33 *b32 *u2*u3*v2*v3
2 4 2 2 4
- 6*a33 *b32 *n2*u1 *v3 + 48*a33 *b32 *n2*u1*u3*v1
2 4 2 4
+ 48*a33 *b32 *n2*u2*u3*v2 + 24*a33 *b32 *n3*u2*u3*v3
2 3 2 2 2 3 2 2
- 12*a33 *b32 *n2 *u1 + 12*a33 *b32 *n2 *u2
2 3 2 3 2 2
+ 24*a33 *b32 *n2*n3*u2*u3 - 12*a33 *b32 *n3 *u1
2 3 2 2 3 3 2
- 12*a33 *b32 *n3 *u2 - 12*a33*b31 *b32 *u3*v1*v3
2 4 2 2 3
- 12*a33*b31 *b32 *u3*v2*v3 + 24*a33*b31 *b32*n2 *u2
5 2 3 2
- 12*a33*b31*b32 *u3*v1*v3 + 48*a33*b31*b32 *n2 *u3*v1
3 3
- 24*a33*b31*b32 *n2*n3*u1*v2 + 24*a33*b31*b32 *n2*n3*u2*v1
6 2 4 2
- 12*a33*b32 *u3*v2*v3 - 48*a33*b32 *n2 *u3*v2
4 4
+ 24*a33*b32 *n2*n3*u1*v1 + 24*a33*b32 *n2*n3*u2*v2
3 3 5 2 4 2 2
+ 24*a33*b32 *n2 *u2 - 6*b31 *n2 *v1*v2 + 6*b31 *b32 *n2*v1 *v3
4 2 2 4 2 3 3 2 2
+ 6*b31 *b32 *n2*v2 *v3 + 10*b31 *b32 *n2*v3 - 36*b31 *b32 *n2 *v1*v2
2 4 2 2 4 2
+ 36*b31 *b32 *n2*v1 *v3 + 36*b31 *b32 *n2*v2 *v3
2 4 3 2 3 2 2 4 2
+ 40*b31 *b32 *n2*v3 - 72*b31 *b32 *n2 *v2 - 30*b31*b32 *n2 *v1*v2
3 2 6 2 6 2
- 48*b31*b32 *n2 *n3*v1 + 30*b32 *n2*v1 *v3 + 30*b32 *n2*v2 *v3
6 3 5 2 2 4 2
+ 30*b32 *n2*v3 - 72*b32 *n2 *v2 + 48*b32 *n2 *n3*v2
4 2 4 3
+ 48*b32 *n2*n3 *v3))/(3*a33 *b32 )
And again in machine readable form:
HAM=(a33**2*b32*u1**2 + a33**2*b32*u2**2 + 2*a33**2*b32*u3**2 + 2*a33*b31*b32*u3
*v1 - 2*a33*b31*n2*u1 + 2*a33*b32**2*u3*v2 + 2*a33*b32*n2*u2 + 2*a33*b32*n3*u3 -
b31**2*n2*v3 - 5*b32**2*n2*v3)/(2*a33*b32)$
INT=(k1*(3*a33**4*b32**3*u1**4 + 6*a33**4*b32**3*u1**2*u2**2 + 3*a33**4*b32**3*
u2**4 - 12*a33**3*b31*b32**3*u1**2*u3*v1 - 24*a33**3*b31*b32**3*u1*u2*u3*v2 + 12
*a33**3*b31*b32**3*u2**2*u3*v1 + 12*a33**3*b31*b32**2*n2*u1**3 + 12*a33**3*b31*
b32**2*n2*u1*u2**2 + 12*a33**3*b32**4*u1**2*u3*v2 - 24*a33**3*b32**4*u1*u2*u3*v1
- 12*a33**3*b32**4*u2**2*u3*v2 - 12*a33**3*b32**3*n2*u1**2*u2 - 12*a33**3*b32**
3*n2*u2**3 - 12*a33**3*b32**3*n3*u1**2*u3 - 12*a33**3*b32**3*n3*u2**2*u3 - 6*a33
**2*b31**2*b32**3*u1**2*v1**2 - 6*a33**2*b31**2*b32**3*u1**2*v2**2 - 6*a33**2*
b31**2*b32**3*u1**2*v3**2 + 24*a33**2*b31**2*b32**3*u1*u3*v1*v3 - 6*a33**2*b31**
2*b32**3*u2**2*v1**2 - 6*a33**2*b31**2*b32**3*u2**2*v2**2 - 6*a33**2*b31**2*b32
**3*u2**2*v3**2 + 24*a33**2*b31**2*b32**3*u2*u3*v2*v3 - 6*a33**2*b31**2*b32**2*
n2*u1**2*v3 + 48*a33**2*b31*b32**3*n2*u1*u3*v2 - 48*a33**2*b31*b32**3*n2*u2*u3*
v1 + 24*a33**2*b31*b32**3*n3*u1*u3*v3 - 24*a33**2*b31*b32**2*n2**2*u1*u2 - 24*
a33**2*b31*b32**2*n2*n3*u1*u3 - 6*a33**2*b32**5*u1**2*v1**2 - 6*a33**2*b32**5*u1
**2*v2**2 - 6*a33**2*b32**5*u1**2*v3**2 + 24*a33**2*b32**5*u1*u3*v1*v3 - 6*a33**
2*b32**5*u2**2*v1**2 - 6*a33**2*b32**5*u2**2*v2**2 - 6*a33**2*b32**5*u2**2*v3**2
+ 24*a33**2*b32**5*u2*u3*v2*v3 - 6*a33**2*b32**4*n2*u1**2*v3 + 48*a33**2*b32**4
*n2*u1*u3*v1 + 48*a33**2*b32**4*n2*u2*u3*v2 + 24*a33**2*b32**4*n3*u2*u3*v3 - 12*
a33**2*b32**3*n2**2*u1**2 + 12*a33**2*b32**3*n2**2*u2**2 + 24*a33**2*b32**3*n2*
n3*u2*u3 - 12*a33**2*b32**3*n3**2*u1**2 - 12*a33**2*b32**3*n3**2*u2**2 - 12*a33*
b31**3*b32**3*u3*v1*v3**2 - 12*a33*b31**2*b32**4*u3*v2*v3**2 + 24*a33*b31**2*b32
*n2**3*u2 - 12*a33*b31*b32**5*u3*v1*v3**2 + 48*a33*b31*b32**3*n2**2*u3*v1 - 24*
a33*b31*b32**3*n2*n3*u1*v2 + 24*a33*b31*b32**3*n2*n3*u2*v1 - 12*a33*b32**6*u3*v2
*v3**2 - 48*a33*b32**4*n2**2*u3*v2 + 24*a33*b32**4*n2*n3*u1*v1 + 24*a33*b32**4*
n2*n3*u2*v2 + 24*a33*b32**3*n2**3*u2 - 6*b31**5*n2**2*v1*v2 + 6*b31**4*b32**2*n2
*v1**2*v3 + 6*b31**4*b32**2*n2*v2**2*v3 + 10*b31**4*b32**2*n2*v3**3 - 36*b31**3*
b32**2*n2**2*v1*v2 + 36*b31**2*b32**4*n2*v1**2*v3 + 36*b31**2*b32**4*n2*v2**2*v3
+ 40*b31**2*b32**4*n2*v3**3 - 72*b31**2*b32**3*n2**2*v2**2 - 30*b31*b32**4*n2**
2*v1*v2 - 48*b31*b32**3*n2**2*n3*v1 + 30*b32**6*n2*v1**2*v3 + 30*b32**6*n2*v2**2
*v3 + 30*b32**6*n2*v3**3 - 72*b32**5*n2**2*v2**2 + 48*b32**4*n2**2*n3*v2 + 48*
b32**4*n2*n3**2*v3))/(3*a33**4*b32**3)$