Solution 6 to problem e3c2
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
b31*m2
m1=--------
b32
2 2 4 2
---*b31 *n2 - ---*b32 *n2
3 3
m3=---------------------------
a33*b32
4 3 8 2 2 3 32 4 3
8*b31 *k1*n2 - ---*b31 *b32 *k1*n2 - ----*b32 *k1*n2
3 3
r6=---------------------------------------------------------
4 3
a33 *b32
2
- 16*k1*m2*n2
r5=-----------------
3
a33
2
16*b31*k1*m2*n2
r4=------------------
3
a33 *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
32 2 32 2
----*b31 *k1*m2*n2 + ----*b32 *k1*m2*n2
3 3
q20=-----------------------------------------
3
a33 *b32
32 3 32 2
- ----*b31 *k1*m2*n2 - ----*b31*b32 *k1*m2*n2
3 3
q19=------------------------------------------------
3 2
a33 *b32
q17=0
32 3 2 32 2 2
----*b31 *k1*n2 + ----*b31*b32 *k1*n2
3 3
q16=-----------------------------------------
3 2
a33 *b32
68 4 2 40 2 2 2 28 4 2
----*b31 *k1*n2 + ----*b31 *b32 *k1*n2 - ----*b32 *k1*n2
9 9 9
q15=-------------------------------------------------------------
4 2
a33 *b32
q14=0
2 2 2 2
8*b31 *k1*n2 - 8*b32 *k1*n2
q13=-------------------------------
3
a33 *b32
16*k1*m2*n2
q12=-------------
2
a33
16*b31*k1*m2*n2
q11=-----------------
2
a33 *b32
68 4 2 40 2 2 2 28 4 2
----*b31 *k1*n2 + ----*b31 *b32 *k1*n2 - ----*b32 *k1*n2
9 9 9
q10=-------------------------------------------------------------
4 2
a33 *b32
2
16*b31*k1*n2
q9=---------------
3
a33
- 16*b31*k1*m2*n2
q8=--------------------
2
a33 *b32
16*k1*m2*n2
q7=-------------
2
a33
q5=0
q4=0
2
4*k1*n2
q3=----------
2
a33
2
- 8*b31*k1*n2
q2=-----------------
2
a33 *b32
2
- 4*k1*n2
q1=-------------
2
a33
p56=0
2 2
- 4*b31 *k1*m2 - 4*b32 *k1*m2
p55=--------------------------------
3
a33
3 2
- 4*b31 *k1*m2 - 4*b31*b32 *k1*m2
p54=------------------------------------
3
a33 *b32
16 2 16 2
- ----*b31 *k1*m2 - ----*b32 *k1*m2
3 3
p53=--------------------------------------
2
a33 *b32
p52=0
p51=0
p50=0
p49=0
p48=0
16 2 16 2
- ----*b31 *k1*m2 - ----*b32 *k1*m2
3 3
p47=--------------------------------------
2
a33 *b32
p46=0
p45=0
p44=0
p43=0
16 2 16 2
- ----*b31 *k1*m2 - ----*b32 *k1*m2
3 3
p42=--------------------------------------
2
a33 *b32
p41=0
p40=0
p39=0
16 2 16 2
- ----*b31 *k1*n2 - ----*b32 *k1*n2
3 3
p38=--------------------------------------
2
a33 *b32
p37=0
8 2 8 2
---*b31 *k1*n2 + ---*b32 *k1*n2
3 3
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
- 4*k1*m2
p23=------------
a33
- 8*b31*k1*m2
p22=----------------
a33*b32
4*k1*m2
p21=---------
a33
p20=0
p19=0
p18=0
p17=0
p16=0
- 16*b31*k1*n2
p15=-----------------
2
a33
16*b32*k1*n2
p14=--------------
2
a33
4*b31*k1*m2
p13=-------------
a33*b32
- 8*k1*m2
p12=------------
a33
- 4*b31*k1*m2
p11=----------------
a33*b32
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
k125=0
k124=0
k122=0
k121=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k120=-------------------------------------------
4
a33
k119=0
2 3
- 4*b31 *b32*k1 - 4*b32 *k1
k118=------------------------------
3
a33
k117=0
k116=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k115=-------------------------------------------
4
a33
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
4 2 2 4
- b31 *k1 - 2*b31 *b32 *k1 - b32 *k1
k70=---------------------------------------
4
a33
k69=0
k68=0
k67=0
k66=0
4 2 2 4
- 2*b31 *k1 - 4*b31 *b32 *k1 - 2*b32 *k1
k65=-------------------------------------------
4
a33
k64=0
k63=0
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
- 4*b32*k1
k42=-------------
a33
- 8*b31*k1
k41=-------------
a33
4*b32*k1
k40=----------
a33
k38=0
k37=0
k36=0
4 2 2 4
- b31 *k1 - 2*b31 *b32 *k1 - b32 *k1
k35=---------------------------------------
4
a33
k34=0
k33=0
k32=0
k30=0
k29=0
k28=0
k27=0
k26=0
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:
m2,k1,b32,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=(3*a33 *b32*u1 + 3*a33 *b32*u2 + 6*a33 *b32*u3 + 6*a33*b31*b32*u3*v1
2
+ 6*a33*b31*m2*v1 - 6*a33*b31*n2*u1 + 6*a33*b32 *u3*v2 + 6*a33*b32*m2*v2
2 2 2 3 2
+ 6*a33*b32*n2*u2 - 3*b31 *b32*v3 + 4*b31 *n2*v3 - 3*b32 *v3
2
- 8*b32 *n2*v3)/(6*a33*b32)
4 3 4 4 3 2 2 4 3 4
INT=(k1*(9*a33 *b32 *u1 + 18*a33 *b32 *u1 *u2 + 9*a33 *b32 *u2
3 3 2 3 3
- 36*a33 *b31*b32 *u1 *u3*v1 - 72*a33 *b31*b32 *u1*u2*u3*v2
3 3 2 3 2 2
+ 36*a33 *b31*b32 *u2 *u3*v1 - 36*a33 *b31*b32 *m2*u1 *v1
3 2 3 2 2
- 72*a33 *b31*b32 *m2*u1*u2*v2 + 36*a33 *b31*b32 *m2*u2 *v1
3 2 3 3 2 2
+ 36*a33 *b31*b32 *n2*u1 + 36*a33 *b31*b32 *n2*u1*u2
3 4 2 3 4
+ 36*a33 *b32 *u1 *u3*v2 - 72*a33 *b32 *u1*u2*u3*v1
3 4 2 3 3 2
- 36*a33 *b32 *u2 *u3*v2 + 36*a33 *b32 *m2*u1 *v2
3 3 3 3 2
- 72*a33 *b32 *m2*u1*u2*v1 - 36*a33 *b32 *m2*u2 *v2
3 3 2 3 3 3
- 36*a33 *b32 *n2*u1 *u2 - 36*a33 *b32 *n2*u2
2 2 3 2 2 2 2 3
- 18*a33 *b31 *b32 *u1 *v3 + 72*a33 *b31 *b32 *u1*u3*v1*v3
2 2 3 2 2 2 2 3
- 18*a33 *b31 *b32 *u2 *v3 + 72*a33 *b31 *b32 *u2*u3*v2*v3
2 2 2 2 2 2
- 48*a33 *b31 *b32 *m2*u1*v1*v3 - 48*a33 *b31 *b32 *m2*u2*v2*v3
2 2 2 2 2 2 2 2
- 48*a33 *b31 *b32 *m2*u3*v3 + 24*a33 *b31 *b32 *n2*u1 *v3
2 2 2 2 2 3
- 48*a33 *b31 *b32 *n2*u2 *v3 + 144*a33 *b31*b32 *n2*u1*u3*v2
2 3 2 2
- 144*a33 *b31*b32 *n2*u2*u3*v1 + 144*a33 *b31*b32 *m2*n2*u1*v2
2 2 2 2 2
- 144*a33 *b31*b32 *m2*n2*u2*v1 - 72*a33 *b31*b32 *n2 *u1*u2
2 5 2 2 2 5
- 18*a33 *b32 *u1 *v3 + 72*a33 *b32 *u1*u3*v1*v3
2 5 2 2 2 5
- 18*a33 *b32 *u2 *v3 + 72*a33 *b32 *u2*u3*v2*v3
2 4 2 4
- 48*a33 *b32 *m2*u1*v1*v3 - 48*a33 *b32 *m2*u2*v2*v3
2 4 2 2 4 2
- 48*a33 *b32 *m2*u3*v3 + 24*a33 *b32 *n2*u1 *v3
2 4 2 4 2
+ 144*a33 *b32 *n2*u1*u3*v1 - 48*a33 *b32 *n2*u2 *v3
2 4 2 3
+ 144*a33 *b32 *n2*u2*u3*v2 + 144*a33 *b32 *m2*n2*u1*v1
2 3 2 3 2 2
+ 144*a33 *b32 *m2*n2*u2*v2 - 36*a33 *b32 *n2 *u1
2 3 2 2 3 3 2
+ 36*a33 *b32 *n2 *u2 - 36*a33*b31 *b32 *u3*v1*v3
3 2 2 3
- 36*a33*b31 *b32 *m2*v1*v3 - 96*a33*b31 *b32*m2*n2*v1*v3
3 2 3 3
+ 96*a33*b31 *b32*n2 *u1*v3 - 72*a33*b31 *n2 *u1
2 4 2 2 3 2
- 36*a33*b31 *b32 *u3*v2*v3 - 36*a33*b31 *b32 *m2*v2*v3
2 2 2 2 2
+ 96*a33*b31 *b32 *m2*n2*v2*v3 + 72*a33*b31 *b32 *n2 *u3*v2
2 3 5 2
+ 72*a33*b31 *b32*n2 *u2 - 36*a33*b31*b32 *u3*v1*v3
4 2 3
- 36*a33*b31*b32 *m2*v1*v3 - 96*a33*b31*b32 *m2*n2*v1*v3
3 2 3 2
+ 96*a33*b31*b32 *n2 *u1*v3 + 144*a33*b31*b32 *n2 *u3*v1
2 2 2 3
+ 144*a33*b31*b32 *m2*n2 *v1 - 72*a33*b31*b32 *n2 *u1
6 2 5 2
- 36*a33*b32 *u3*v2*v3 - 36*a33*b32 *m2*v2*v3
4 4 2
+ 96*a33*b32 *m2*n2*v2*v3 - 72*a33*b32 *n2 *u3*v2
3 2 3 3 4 3 4
- 144*a33*b32 *m2*n2 *v2 + 72*a33*b32 *n2 *u2 - 9*b31 *b32 *v1
4 3 2 2 4 3 2 2 4 3 4
- 18*b31 *b32 *v1 *v2 - 18*b31 *b32 *v1 *v3 - 9*b31 *b32 *v2
4 3 2 2 4 2 2 4 2 2
- 18*b31 *b32 *v2 *v3 + 68*b31 *b32*n2 *v1 + 68*b31 *b32*n2 *v2
4 3 2 5 4 2 5 2 2
+ 72*b31 *n2 *v3 - 18*b31 *b32 *v1 - 36*b31 *b32 *v1 *v2
2 5 2 2 2 5 4 2 5 2 2
- 36*b31 *b32 *v1 *v3 - 18*b31 *b32 *v2 - 36*b31 *b32 *v2 *v3
2 3 2 2 2 3 2 2 2 2 3
+ 40*b31 *b32 *n2 *v1 + 40*b31 *b32 *n2 *v2 - 24*b31 *b32 *n2 *v3
7 4 7 2 2 7 2 2 7 4
- 9*b32 *v1 - 18*b32 *v1 *v2 - 18*b32 *v1 *v3 - 9*b32 *v2
7 2 2 5 2 2 5 2 2 4 3
- 18*b32 *v2 *v3 - 28*b32 *n2 *v1 - 28*b32 *n2 *v2 - 96*b32 *n2 *v3
4 3
))/(9*a33 *b32 )
And again in machine readable form:
HAM=(3*a33**2*b32*u1**2 + 3*a33**2*b32*u2**2 + 6*a33**2*b32*u3**2 + 6*a33*b31*
b32*u3*v1 + 6*a33*b31*m2*v1 - 6*a33*b31*n2*u1 + 6*a33*b32**2*u3*v2 + 6*a33*b32*
m2*v2 + 6*a33*b32*n2*u2 - 3*b31**2*b32*v3**2 + 4*b31**2*n2*v3 - 3*b32**3*v3**2 -
8*b32**2*n2*v3)/(6*a33*b32)$
INT=(k1*(9*a33**4*b32**3*u1**4 + 18*a33**4*b32**3*u1**2*u2**2 + 9*a33**4*b32**3*
u2**4 - 36*a33**3*b31*b32**3*u1**2*u3*v1 - 72*a33**3*b31*b32**3*u1*u2*u3*v2 + 36
*a33**3*b31*b32**3*u2**2*u3*v1 - 36*a33**3*b31*b32**2*m2*u1**2*v1 - 72*a33**3*
b31*b32**2*m2*u1*u2*v2 + 36*a33**3*b31*b32**2*m2*u2**2*v1 + 36*a33**3*b31*b32**2
*n2*u1**3 + 36*a33**3*b31*b32**2*n2*u1*u2**2 + 36*a33**3*b32**4*u1**2*u3*v2 - 72
*a33**3*b32**4*u1*u2*u3*v1 - 36*a33**3*b32**4*u2**2*u3*v2 + 36*a33**3*b32**3*m2*
u1**2*v2 - 72*a33**3*b32**3*m2*u1*u2*v1 - 36*a33**3*b32**3*m2*u2**2*v2 - 36*a33
**3*b32**3*n2*u1**2*u2 - 36*a33**3*b32**3*n2*u2**3 - 18*a33**2*b31**2*b32**3*u1
**2*v3**2 + 72*a33**2*b31**2*b32**3*u1*u3*v1*v3 - 18*a33**2*b31**2*b32**3*u2**2*
v3**2 + 72*a33**2*b31**2*b32**3*u2*u3*v2*v3 - 48*a33**2*b31**2*b32**2*m2*u1*v1*
v3 - 48*a33**2*b31**2*b32**2*m2*u2*v2*v3 - 48*a33**2*b31**2*b32**2*m2*u3*v3**2 +
24*a33**2*b31**2*b32**2*n2*u1**2*v3 - 48*a33**2*b31**2*b32**2*n2*u2**2*v3 + 144
*a33**2*b31*b32**3*n2*u1*u3*v2 - 144*a33**2*b31*b32**3*n2*u2*u3*v1 + 144*a33**2*
b31*b32**2*m2*n2*u1*v2 - 144*a33**2*b31*b32**2*m2*n2*u2*v1 - 72*a33**2*b31*b32**
2*n2**2*u1*u2 - 18*a33**2*b32**5*u1**2*v3**2 + 72*a33**2*b32**5*u1*u3*v1*v3 - 18
*a33**2*b32**5*u2**2*v3**2 + 72*a33**2*b32**5*u2*u3*v2*v3 - 48*a33**2*b32**4*m2*
u1*v1*v3 - 48*a33**2*b32**4*m2*u2*v2*v3 - 48*a33**2*b32**4*m2*u3*v3**2 + 24*a33
**2*b32**4*n2*u1**2*v3 + 144*a33**2*b32**4*n2*u1*u3*v1 - 48*a33**2*b32**4*n2*u2
**2*v3 + 144*a33**2*b32**4*n2*u2*u3*v2 + 144*a33**2*b32**3*m2*n2*u1*v1 + 144*a33
**2*b32**3*m2*n2*u2*v2 - 36*a33**2*b32**3*n2**2*u1**2 + 36*a33**2*b32**3*n2**2*
u2**2 - 36*a33*b31**3*b32**3*u3*v1*v3**2 - 36*a33*b31**3*b32**2*m2*v1*v3**2 - 96
*a33*b31**3*b32*m2*n2*v1*v3 + 96*a33*b31**3*b32*n2**2*u1*v3 - 72*a33*b31**3*n2**
3*u1 - 36*a33*b31**2*b32**4*u3*v2*v3**2 - 36*a33*b31**2*b32**3*m2*v2*v3**2 + 96*
a33*b31**2*b32**2*m2*n2*v2*v3 + 72*a33*b31**2*b32**2*n2**2*u3*v2 + 72*a33*b31**2
*b32*n2**3*u2 - 36*a33*b31*b32**5*u3*v1*v3**2 - 36*a33*b31*b32**4*m2*v1*v3**2 -
96*a33*b31*b32**3*m2*n2*v1*v3 + 96*a33*b31*b32**3*n2**2*u1*v3 + 144*a33*b31*b32
**3*n2**2*u3*v1 + 144*a33*b31*b32**2*m2*n2**2*v1 - 72*a33*b31*b32**2*n2**3*u1 -
36*a33*b32**6*u3*v2*v3**2 - 36*a33*b32**5*m2*v2*v3**2 + 96*a33*b32**4*m2*n2*v2*
v3 - 72*a33*b32**4*n2**2*u3*v2 - 144*a33*b32**3*m2*n2**2*v2 + 72*a33*b32**3*n2**
3*u2 - 9*b31**4*b32**3*v1**4 - 18*b31**4*b32**3*v1**2*v2**2 - 18*b31**4*b32**3*
v1**2*v3**2 - 9*b31**4*b32**3*v2**4 - 18*b31**4*b32**3*v2**2*v3**2 + 68*b31**4*
b32*n2**2*v1**2 + 68*b31**4*b32*n2**2*v2**2 + 72*b31**4*n2**3*v3 - 18*b31**2*b32
**5*v1**4 - 36*b31**2*b32**5*v1**2*v2**2 - 36*b31**2*b32**5*v1**2*v3**2 - 18*b31
**2*b32**5*v2**4 - 36*b31**2*b32**5*v2**2*v3**2 + 40*b31**2*b32**3*n2**2*v1**2 +
40*b31**2*b32**3*n2**2*v2**2 - 24*b31**2*b32**2*n2**3*v3 - 9*b32**7*v1**4 - 18*
b32**7*v1**2*v2**2 - 18*b32**7*v1**2*v3**2 - 9*b32**7*v2**4 - 18*b32**7*v2**2*v3
**2 - 28*b32**5*n2**2*v1**2 - 28*b32**5*n2**2*v2**2 - 96*b32**4*n2**3*v3))/(9*
a33**4*b32**3)$