Solution 17 to problem e3null
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*b32*n2
m3=-------------
a33
2 3 2 3
- 408*b31 *k1*n2 - 408*b32 *k1*n2
r6=--------------------------------------
4
a33 *b32
r5=0
r4=0
r3=0
2 3 2 3
204*b31 *k1*n2 + 204*b32 *k1*n2
r2=-----------------------------------
3 2
a33 *b32
3 3 2 3
796*b31 *k1*n2 + 796*b31*b32 *k1*n2
r1=---------------------------------------
3 3
a33 *b32
q19=0
q18=0
q17=0
2 2 2 2
32*b31 *k1*n2 + 32*b32 *k1*n2
q16=---------------------------------
3
a33 *b32
q15=0
3184 4 2 6400 2 2 2 3216 4 2
- ------*b31 *k1*n2 - ------*b31 *b32 *k1*n2 - ------*b32 *k1*n2
7 7 7
q14=----------------------------------------------------------------------
4 2
a33 *b32
q13=0
4 2 33232 2 2 2 32648 4 2
8*b31 *k1*n2 - -------*b31 *b32 *k1*n2 - -------*b32 *k1*n2
73 73
q12=----------------------------------------------------------------
3 2
a33 *b31 *b32
q11=0
q10=0
3184 4 2 6400 2 2 2 3216 4 2
- ------*b31 *k1*n2 - ------*b31 *b32 *k1*n2 - ------*b32 *k1*n2
7 7 7
q9=----------------------------------------------------------------------
4 2
a33 *b32
33816 2 2 32648 2 2
-------*b31 *k1*n2 + -------*b32 *k1*n2
73 73
q8=-------------------------------------------
3
a33 *b31
q7=0
q5=0
q4=0
2 2 2 2
398*b31 *k1*n2 + 402*b32 *k1*n2
q3=-----------------------------------
2 2
a33 *b32
2
- 8*b31*k1*n2
q2=-----------------
2
a33 *b32
2 2 2 2
- 98*b31 *k1*n2 - 102*b32 *k1*n2
q1=-------------------------------------
2 2
a33 *b32
p50=0
p49=0
p48=0
p47=0
2 2
8*b31 *k1*n2 + 8*b32 *k1*n2
p46=-----------------------------
3
a33
p45=0
p44=0
p43=0
p42=0
p41=0
p40=0
p39=0
p38=0
p37=0
66464 2 65296 3
- -------*b31 *b32*k1*n2 - -------*b32 *k1*n2
73 73
p36=------------------------------------------------
2 2
a33 *b31
p35=0
p34=0
2 2
- 8*b31 *k1*n2 - 8*b32 *k1*n2
p33=--------------------------------
2
a33 *b32
p32=0
p31=0
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
p19=0
p18=0
p17=0
p16=0
p15=0
p14=0
p13=0
- 16*b31*k1*n2
p12=-----------------
2
a33
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
k104=0
k103=0
k102=0
k101=0
k100=0
4 2 2 4
4*b31 *k1 + 8*b31 *b32 *k1 + 4*b32 *k1
k99=----------------------------------------
4
a33
k98=0
2 3
- 4*b31 *b32*k1 - 4*b32 *k1
k97=------------------------------
3
a33
k96=0
k95=0
4 2 2 4
4*b31 *k1 + 8*b31 *b32 *k1 + 4*b32 *k1
k94=----------------------------------------
4
a33
3 2
- 4*b31 *k1 - 4*b31*b32 *k1
k93=------------------------------
3
a33
k92=0
2 2
- 8*b31 *k1 - 8*b32 *k1
k91=--------------------------
2
a33
k90=0
k89=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k88=--------------------------
2
a33
k87=0
2 2
- 2*b31 *k1 - 2*b32 *k1
k86=--------------------------
2
a33
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*b32*k1
k64=----------
a33
4*b31*k1
k63=----------
a33
k62=0
k61=0
k60=0
k59=0
k58=0
k57=0
k56=0
4 2 2 4
2*b31 *k1 + 4*b31 *b32 *k1 + 2*b32 *k1
k55=----------------------------------------
4
a33
k54=0
k53=0
k52=0
k51=0
4 2 2 4
4*b31 *k1 + 8*b31 *b32 *k1 + 4*b32 *k1
k50=----------------------------------------
4
a33
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*b32*k1
k32=----------
a33
- 4*b31*k1
k31=-------------
a33
4*b32*k1
k30=----------
a33
k29=0
k28=0
k27=0
k26=0
4 2 2 4
2*b31 *k1 + 4*b31 *b32 *k1 + 2*b32 *k1
k25=----------------------------------------
4
a33
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:
k1,b31,b32,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 2
- 2*a33*b31*n2*u1 + 2*a33*b32 *u3*v2 + 2*a33*b32*n2*u2 - b31 *b32*v3
3 2 2
- b32 *v3 - 4*b32 *n2*v3)/(2*a33*b32)
4 2 3 4 4 2 3 2 2
INT=(k1*(511*a33 *b31 *b32 *u1 + 1022*a33 *b31 *b32 *u1 *u2
4 2 3 4 3 3 3
+ 511*a33 *b31 *b32 *u2 - 2044*a33 *b31 *b32 *u1*u2*u3*v2
3 3 3 2 3 3 3 2
+ 2044*a33 *b31 *b32 *u1*u3 *v3 + 2044*a33 *b31 *b32 *u2 *u3*v1
3 3 2 3 3 3 2 2
+ 2044*a33 *b31 *b32 *n2*u1 + 2044*a33 *b31 *b32 *n2*u1*u2
3 2 4 2 3 2 4 2
+ 2044*a33 *b31 *b32 *u1 *u3*v2 + 2044*a33 *b31 *b32 *u2 *u3*v2
3 2 4 2 3 2 3 2
+ 4088*a33 *b31 *b32 *u2*u3 *v3 - 2044*a33 *b31 *b32 *n2*u1 *u2
3 2 3 3 2 4 3 2 2
- 2044*a33 *b31 *b32 *n2*u2 - 1022*a33 *b31 *b32 *u1 *v3
2 4 3 2 2 2 4 3 2 2
- 1022*a33 *b31 *b32 *u2 *v3 - 4088*a33 *b31 *b32 *u3 *v3
2 4 2 2 2 4 2 2
- 4088*a33 *b31 *b32 *n2*u2 *v3 - 50078*a33 *b31 *b32*n2 *u1
2 4 2 2 2 3 3
+ 203378*a33 *b31 *b32*n2 *u2 + 8176*a33 *b31 *b32 *n2*u1*u3*v2
2 3 3 2 3 2 2
- 8176*a33 *b31 *b32 *n2*u2*u3*v1 - 4088*a33 *b31 *b32 *n2 *u1*u2
2 2 5 2 2 2 2 5 2 2
- 1022*a33 *b31 *b32 *u1 *v3 - 1022*a33 *b31 *b32 *u2 *v3
2 2 5 2 2 2 2 4 2
- 4088*a33 *b31 *b32 *u3 *v3 - 4088*a33 *b31 *b32 *n2*u2 *v3
2 2 4 2 2 2 3 2 2
- 465248*a33 *b31 *b32 *n2*u3 *v3 - 52122*a33 *b31 *b32 *n2 *u1
2 2 3 2 2 2 6 2
+ 205422*a33 *b31 *b32 *n2 *u2 - 457072*a33 *b32 *n2*u3 *v3
5 3 2 5 3
- 2044*a33*b31 *b32 *u3*v1*v3 + 406756*a33*b31 *n2 *u1
4 4 2 4 3 2
- 2044*a33*b31 *b32 *u3*v2*v3 + 4088*a33*b31 *b32 *n2*u2*v3
4 2 2 4 2 2
+ 16352*a33*b31 *b32 *n2 *u2*v3 + 4088*a33*b31 *b32 *n2 *u3*v2
4 3 3 5 2
+ 104244*a33*b31 *b32*n2 *u2 - 2044*a33*b31 *b32 *u3*v1*v3
3 3 2 3 2 3
+ 236712*a33*b31 *b32 *n2 *u3*v1 + 406756*a33*b31 *b32 *n2 *u1
2 6 2 2 5 2
- 2044*a33*b31 *b32 *u3*v2*v3 + 4088*a33*b31 *b32 *n2*u2*v3
2 4 2 2 4 2
+ 16352*a33*b31 *b32 *n2 *u2*v3 - 232624*a33*b31 *b32 *n2 *u3*v2
2 3 3 5 2
+ 104244*a33*b31 *b32 *n2 *u2 + 228536*a33*b31*b32 *n2 *u3*v1
6 2 6 3 4
- 228536*a33*b32 *n2 *u3*v2 + 1022*b31 *b32 *v1
6 3 2 2 6 3 2 2 6 3 4
+ 2044*b31 *b32 *v1 *v2 + 2044*b31 *b32 *v1 *v3 + 1022*b31 *b32 *v2
6 3 2 2 6 2 2
+ 2044*b31 *b32 *v2 *v3 - 232432*b31 *b32*n2 *v1
6 2 2 4 5 4
- 232432*b31 *b32*n2 *v2 + 2044*b31 *b32 *v1
4 5 2 2 4 5 2 2 4 5 4
+ 4088*b31 *b32 *v1 *v2 + 4088*b31 *b32 *v1 *v3 + 2044*b31 *b32 *v2
4 5 2 2 4 3 2 2
+ 4088*b31 *b32 *v2 *v3 - 467200*b31 *b32 *n2 *v1
4 3 2 2 4 2 3
- 467200*b31 *b32 *n2 *v2 - 208488*b31 *b32 *n2 *v3
2 7 4 2 7 2 2 2 7 2 2
+ 1022*b31 *b32 *v1 + 2044*b31 *b32 *v1 *v2 + 2044*b31 *b32 *v1 *v3
2 7 4 2 7 2 2
+ 1022*b31 *b32 *v2 + 2044*b31 *b32 *v2 *v3
2 5 2 2 2 5 2 2
- 234768*b31 *b32 *n2 *v1 - 234768*b31 *b32 *n2 *v2
2 4 3 4 2 3
- 208488*b31 *b32 *n2 *v3))/(511*a33 *b31 *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 - b31**2*b32*v3**2
- b32**3*v3**2 - 4*b32**2*n2*v3)/(2*a33*b32)$
INT=(k1*(511*a33**4*b31**2*b32**3*u1**4 + 1022*a33**4*b31**2*b32**3*u1**2*u2**2
+ 511*a33**4*b31**2*b32**3*u2**4 - 2044*a33**3*b31**3*b32**3*u1*u2*u3*v2 + 2044*
a33**3*b31**3*b32**3*u1*u3**2*v3 + 2044*a33**3*b31**3*b32**3*u2**2*u3*v1 + 2044*
a33**3*b31**3*b32**2*n2*u1**3 + 2044*a33**3*b31**3*b32**2*n2*u1*u2**2 + 2044*a33
**3*b31**2*b32**4*u1**2*u3*v2 + 2044*a33**3*b31**2*b32**4*u2**2*u3*v2 + 4088*a33
**3*b31**2*b32**4*u2*u3**2*v3 - 2044*a33**3*b31**2*b32**3*n2*u1**2*u2 - 2044*a33
**3*b31**2*b32**3*n2*u2**3 - 1022*a33**2*b31**4*b32**3*u1**2*v3**2 - 1022*a33**2
*b31**4*b32**3*u2**2*v3**2 - 4088*a33**2*b31**4*b32**3*u3**2*v3**2 - 4088*a33**2
*b31**4*b32**2*n2*u2**2*v3 - 50078*a33**2*b31**4*b32*n2**2*u1**2 + 203378*a33**2
*b31**4*b32*n2**2*u2**2 + 8176*a33**2*b31**3*b32**3*n2*u1*u3*v2 - 8176*a33**2*
b31**3*b32**3*n2*u2*u3*v1 - 4088*a33**2*b31**3*b32**2*n2**2*u1*u2 - 1022*a33**2*
b31**2*b32**5*u1**2*v3**2 - 1022*a33**2*b31**2*b32**5*u2**2*v3**2 - 4088*a33**2*
b31**2*b32**5*u3**2*v3**2 - 4088*a33**2*b31**2*b32**4*n2*u2**2*v3 - 465248*a33**
2*b31**2*b32**4*n2*u3**2*v3 - 52122*a33**2*b31**2*b32**3*n2**2*u1**2 + 205422*
a33**2*b31**2*b32**3*n2**2*u2**2 - 457072*a33**2*b32**6*n2*u3**2*v3 - 2044*a33*
b31**5*b32**3*u3*v1*v3**2 + 406756*a33*b31**5*n2**3*u1 - 2044*a33*b31**4*b32**4*
u3*v2*v3**2 + 4088*a33*b31**4*b32**3*n2*u2*v3**2 + 16352*a33*b31**4*b32**2*n2**2
*u2*v3 + 4088*a33*b31**4*b32**2*n2**2*u3*v2 + 104244*a33*b31**4*b32*n2**3*u2 -
2044*a33*b31**3*b32**5*u3*v1*v3**2 + 236712*a33*b31**3*b32**3*n2**2*u3*v1 +
406756*a33*b31**3*b32**2*n2**3*u1 - 2044*a33*b31**2*b32**6*u3*v2*v3**2 + 4088*
a33*b31**2*b32**5*n2*u2*v3**2 + 16352*a33*b31**2*b32**4*n2**2*u2*v3 - 232624*a33
*b31**2*b32**4*n2**2*u3*v2 + 104244*a33*b31**2*b32**3*n2**3*u2 + 228536*a33*b31*
b32**5*n2**2*u3*v1 - 228536*a33*b32**6*n2**2*u3*v2 + 1022*b31**6*b32**3*v1**4 +
2044*b31**6*b32**3*v1**2*v2**2 + 2044*b31**6*b32**3*v1**2*v3**2 + 1022*b31**6*
b32**3*v2**4 + 2044*b31**6*b32**3*v2**2*v3**2 - 232432*b31**6*b32*n2**2*v1**2 -
232432*b31**6*b32*n2**2*v2**2 + 2044*b31**4*b32**5*v1**4 + 4088*b31**4*b32**5*v1
**2*v2**2 + 4088*b31**4*b32**5*v1**2*v3**2 + 2044*b31**4*b32**5*v2**4 + 4088*b31
**4*b32**5*v2**2*v3**2 - 467200*b31**4*b32**3*n2**2*v1**2 - 467200*b31**4*b32**3
*n2**2*v2**2 - 208488*b31**4*b32**2*n2**3*v3 + 1022*b31**2*b32**7*v1**4 + 2044*
b31**2*b32**7*v1**2*v2**2 + 2044*b31**2*b32**7*v1**2*v3**2 + 1022*b31**2*b32**7*
v2**4 + 2044*b31**2*b32**7*v2**2*v3**2 - 234768*b31**2*b32**5*n2**2*v1**2 -
234768*b31**2*b32**5*n2**2*v2**2 - 208488*b31**2*b32**4*n2**3*v3))/(511*a33**4*
b31**2*b32**3)$