Solution 28 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
m1=0
m2=0
2 2 4 2
---*b31 *n2 - ---*b32 *n2
3 3
m3=---------------------------
a33*b32
r6=0
r5=0
r4=0
r3=0
52 2 3 52 2 3
----*b31 *k1*n2 + ----*b32 *k1*n2
3 3
r2=-------------------------------------
3 2
a33 *b32
52 3 3 52 2 3
- ----*b31 *k1*n2 - ----*b31*b32 *k1*n2
3 3
r1=--------------------------------------------
3 3
a33 *b32
q20=0
q19=0
q17=0
32 3 2 32 2 2
----*b31 *k1*n2 + ----*b31*b32 *k1*n2
3 3
q16=-----------------------------------------
3 2
a33 *b32
14 4 2 1960 2 2 2 154 4 2
----*b31 *k1*n2 - ------*b31 *b32 *k1*n2 - -----*b32 *k1*n2
39 117 9
q15=----------------------------------------------------------------
4 2
a33 *b32
q14=0
52 2 2 4 2 2
----*b31 *k1*n2 + ---*b32 *k1*n2
3 3
q13=------------------------------------
3
a33 *b32
q12=0
q11=0
14 4 2 1960 2 2 2 154 4 2
----*b31 *k1*n2 - ------*b31 *b32 *k1*n2 - -----*b32 *k1*n2
39 117 9
q10=----------------------------------------------------------------
4 2
a33 *b32
28 3 2 20 2 2
- ----*b31 *k1*n2 + ----*b31*b32 *k1*n2
3 3
q9=--------------------------------------------
3 2
a33 *b32
q8=0
q7=0
q5=0
q4=0
2 2 2 2
- 14*b31 *k1*n2 - 10*b32 *k1*n2
q3=------------------------------------
2 2
a33 *b32
2
- 8*b31*k1*n2
q2=-----------------
2
a33 *b32
14 2 2 26 2 2
- ----*b31 *k1*n2 - ----*b32 *k1*n2
3 3
q1=----------------------------------------
2 2
a33 *b32
p56=0
p55=0
p54=0
p53=0
p52=0
p51=0
p50=0
p49=0
p48=0
p47=0
p46=0
p45=0
p44=0
p43=0
p42=0
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
56 2 8 2
- ----*b31 *k1*n2 - ---*b32 *k1*n2
3 3
p25=-------------------------------------
2
a33 *b32
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
56 2 8 2
- ----*b31 *k1*n2 - ---*b32 *k1*n2
3 3
p14=-------------------------------------
2
a33 *b32
p13=0
p12=0
p11=0
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:
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 2 2
- 6*a33*b31*n2*u1 + 6*a33*b32 *u3*v2 + 6*a33*b32*n2*u2 - 3*b31 *b32*v3
2 3 2 2
+ 4*b31 *n2*v3 - 3*b32 *v3 - 8*b32 *n2*v3)/(6*a33*b32)
4 3 4 4 3 2 2 4 3 4
INT=(k1*(117*a33 *b32 *u1 + 234*a33 *b32 *u1 *u2 + 117*a33 *b32 *u2
3 3 2 3 3
- 468*a33 *b31*b32 *u1 *u3*v1 - 936*a33 *b31*b32 *u1*u2*u3*v2
3 3 2 3 2 3
+ 468*a33 *b31*b32 *u2 *u3*v1 + 468*a33 *b31*b32 *n2*u1
3 2 2 3 4 2
+ 468*a33 *b31*b32 *n2*u1*u2 + 468*a33 *b32 *u1 *u3*v2
3 4 3 4 2
- 936*a33 *b32 *u1*u2*u3*v1 - 468*a33 *b32 *u2 *u3*v2
3 3 2 3 3 3
- 468*a33 *b32 *n2*u1 *u2 - 468*a33 *b32 *n2*u2
2 2 3 2 2 2 2 3
- 234*a33 *b31 *b32 *u1 *v3 + 936*a33 *b31 *b32 *u1*u3*v1*v3
2 2 3 2 2 2 2 3
- 234*a33 *b31 *b32 *u2 *v3 + 936*a33 *b31 *b32 *u2*u3*v2*v3
2 2 2 2 2 2 2
+ 312*a33 *b31 *b32 *n2*u1 *v3 - 2184*a33 *b31 *b32 *n2*u1*u3*v1
2 2 2 2 2 2 2
- 624*a33 *b31 *b32 *n2*u2 *v3 - 2184*a33 *b31 *b32 *n2*u2*u3*v2
2 2 2 2 2 2 2 2
- 546*a33 *b31 *b32*n2 *u1 - 1638*a33 *b31 *b32*n2 *u2
2 3 2 3
+ 1872*a33 *b31*b32 *n2*u1*u3*v2 - 1872*a33 *b31*b32 *n2*u2*u3*v1
2 2 2 2 5 2 2
- 936*a33 *b31*b32 *n2 *u1*u2 - 234*a33 *b32 *u1 *v3
2 5 2 5 2 2
+ 936*a33 *b32 *u1*u3*v1*v3 - 234*a33 *b32 *u2 *v3
2 5 2 4 2
+ 936*a33 *b32 *u2*u3*v2*v3 + 312*a33 *b32 *n2*u1 *v3
2 4 2 4 2
- 312*a33 *b32 *n2*u1*u3*v1 - 624*a33 *b32 *n2*u2 *v3
2 4 2 3 2 2
- 312*a33 *b32 *n2*u2*u3*v2 - 1014*a33 *b32 *n2 *u1
2 3 2 2 3 3 2
- 1170*a33 *b32 *n2 *u2 - 468*a33*b31 *b32 *u3*v1*v3
3 2 3 2
+ 1248*a33*b31 *b32*n2 *u1*v3 - 1092*a33*b31 *b32*n2 *u3*v1
3 3 2 4 2
- 2028*a33*b31 *n2 *u1 - 468*a33*b31 *b32 *u3*v2*v3
2 2 2 2 3
+ 2028*a33*b31 *b32 *n2 *u3*v2 + 2028*a33*b31 *b32*n2 *u2
5 2 3 2
- 468*a33*b31*b32 *u3*v1*v3 + 1248*a33*b31*b32 *n2 *u1*v3
3 2 2 3
+ 780*a33*b31*b32 *n2 *u3*v1 - 2028*a33*b31*b32 *n2 *u1
6 2 4 2
- 468*a33*b32 *u3*v2*v3 + 156*a33*b32 *n2 *u3*v2
3 3 4 3 4 4 3 2 2
+ 2028*a33*b32 *n2 *u2 - 117*b31 *b32 *v1 - 234*b31 *b32 *v1 *v2
4 3 2 2 4 3 4 4 3 2 2
- 234*b31 *b32 *v1 *v3 - 117*b31 *b32 *v2 - 234*b31 *b32 *v2 *v3
4 2 2 4 2 2 2 5 4
+ 42*b31 *b32*n2 *v1 + 42*b31 *b32*n2 *v2 - 234*b31 *b32 *v1
2 5 2 2 2 5 2 2 2 5 4
- 468*b31 *b32 *v1 *v2 - 468*b31 *b32 *v1 *v3 - 234*b31 *b32 *v2
2 5 2 2 2 3 2 2
- 468*b31 *b32 *v2 *v3 - 1960*b31 *b32 *n2 *v1
2 3 2 2 7 4 7 2 2
- 1960*b31 *b32 *n2 *v2 - 117*b32 *v1 - 234*b32 *v1 *v2
7 2 2 7 4 7 2 2
- 234*b32 *v1 *v3 - 117*b32 *v2 - 234*b32 *v2 *v3
5 2 2 5 2 2 4 3
- 2002*b32 *n2 *v1 - 2002*b32 *n2 *v2 ))/(117*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*n2*u1 + 6*a33*b32**2*u3*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*(117*a33**4*b32**3*u1**4 + 234*a33**4*b32**3*u1**2*u2**2 + 117*a33**4*
b32**3*u2**4 - 468*a33**3*b31*b32**3*u1**2*u3*v1 - 936*a33**3*b31*b32**3*u1*u2*
u3*v2 + 468*a33**3*b31*b32**3*u2**2*u3*v1 + 468*a33**3*b31*b32**2*n2*u1**3 + 468
*a33**3*b31*b32**2*n2*u1*u2**2 + 468*a33**3*b32**4*u1**2*u3*v2 - 936*a33**3*b32
**4*u1*u2*u3*v1 - 468*a33**3*b32**4*u2**2*u3*v2 - 468*a33**3*b32**3*n2*u1**2*u2
- 468*a33**3*b32**3*n2*u2**3 - 234*a33**2*b31**2*b32**3*u1**2*v3**2 + 936*a33**2
*b31**2*b32**3*u1*u3*v1*v3 - 234*a33**2*b31**2*b32**3*u2**2*v3**2 + 936*a33**2*
b31**2*b32**3*u2*u3*v2*v3 + 312*a33**2*b31**2*b32**2*n2*u1**2*v3 - 2184*a33**2*
b31**2*b32**2*n2*u1*u3*v1 - 624*a33**2*b31**2*b32**2*n2*u2**2*v3 - 2184*a33**2*
b31**2*b32**2*n2*u2*u3*v2 - 546*a33**2*b31**2*b32*n2**2*u1**2 - 1638*a33**2*b31
**2*b32*n2**2*u2**2 + 1872*a33**2*b31*b32**3*n2*u1*u3*v2 - 1872*a33**2*b31*b32**
3*n2*u2*u3*v1 - 936*a33**2*b31*b32**2*n2**2*u1*u2 - 234*a33**2*b32**5*u1**2*v3**
2 + 936*a33**2*b32**5*u1*u3*v1*v3 - 234*a33**2*b32**5*u2**2*v3**2 + 936*a33**2*
b32**5*u2*u3*v2*v3 + 312*a33**2*b32**4*n2*u1**2*v3 - 312*a33**2*b32**4*n2*u1*u3*
v1 - 624*a33**2*b32**4*n2*u2**2*v3 - 312*a33**2*b32**4*n2*u2*u3*v2 - 1014*a33**2
*b32**3*n2**2*u1**2 - 1170*a33**2*b32**3*n2**2*u2**2 - 468*a33*b31**3*b32**3*u3*
v1*v3**2 + 1248*a33*b31**3*b32*n2**2*u1*v3 - 1092*a33*b31**3*b32*n2**2*u3*v1 -
2028*a33*b31**3*n2**3*u1 - 468*a33*b31**2*b32**4*u3*v2*v3**2 + 2028*a33*b31**2*
b32**2*n2**2*u3*v2 + 2028*a33*b31**2*b32*n2**3*u2 - 468*a33*b31*b32**5*u3*v1*v3
**2 + 1248*a33*b31*b32**3*n2**2*u1*v3 + 780*a33*b31*b32**3*n2**2*u3*v1 - 2028*
a33*b31*b32**2*n2**3*u1 - 468*a33*b32**6*u3*v2*v3**2 + 156*a33*b32**4*n2**2*u3*
v2 + 2028*a33*b32**3*n2**3*u2 - 117*b31**4*b32**3*v1**4 - 234*b31**4*b32**3*v1**
2*v2**2 - 234*b31**4*b32**3*v1**2*v3**2 - 117*b31**4*b32**3*v2**4 - 234*b31**4*
b32**3*v2**2*v3**2 + 42*b31**4*b32*n2**2*v1**2 + 42*b31**4*b32*n2**2*v2**2 - 234
*b31**2*b32**5*v1**4 - 468*b31**2*b32**5*v1**2*v2**2 - 468*b31**2*b32**5*v1**2*
v3**2 - 234*b31**2*b32**5*v2**4 - 468*b31**2*b32**5*v2**2*v3**2 - 1960*b31**2*
b32**3*n2**2*v1**2 - 1960*b31**2*b32**3*n2**2*v2**2 - 117*b32**7*v1**4 - 234*b32
**7*v1**2*v2**2 - 234*b32**7*v1**2*v3**2 - 117*b32**7*v2**4 - 234*b32**7*v2**2*
v3**2 - 2002*b32**5*n2**2*v1**2 - 2002*b32**5*n2**2*v2**2))/(117*a33**4*b32**3)$