Solution 22 to problem e3c2new
Expressions |
Parameters |
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Back to problem e3c2new
Expressions
The solution is given through the following expressions:
a22=2*a33
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
1 2
- ---*b33
8
c33=-------------
a33
n1=0
n3=0
m1=0
3
- ---*b33*n2
4
m2=---------------
a33
m3=0
r6=0
1
- ---*b33*n2*q1
4
r5=------------------
2
a33
r4=0
r3=0
n2*q1
r2=-------
a33
r1=0
q20=0
q19=0
q17=0
q16=0
1 2 2 7 2 2
----*a33 *b33 *q1 + ----*b33 *k1*n2
16 64
q15=--------------------------------------
4
a33
q14=0
q13=0
1 2 1 2
- ---*a33 *b33*q1 - ---*b33*k1*n2
2 2
q12=-------------------------------------
3
a33
q11=0
1 2
----*b33 *q1
16
q10=--------------
2
a33
q9=0
q8=0
1
- ---*b33*q1
2
q7=---------------
a33
q5=0
q4=0
2 1 2
a33 *q1 + ---*k1*n2
4
q3=----------------------
2
a33
q2=0
p56=0
1 2 1 3
- ---*a33 *b33*k26*n2 - -----*b33 *k1*n2
4 128
p55=-------------------------------------------
4
a33
p54=0
p53=0
2 1 2
a33 *k26*n2 + ---*b33 *k1*n2
8
p52=------------------------------
3
a33
p51=0
p50=0
p49=0
1 11 2
- ---*a33*b33*k23*n2 - ----*b33 *k1*n2
4 32
p48=-----------------------------------------
3
a33
p47=0
p46=0
p45=0
p44=0
p43=0
p42=0
p41=0
5
- a33*k23*n2 + ---*b33*k1*n2
8
p40=-------------------------------
2
a33
p39=0
p38=0
p37=0
p36=0
1 2 13 3
- ---*a33 *b33*k26*n2 - -----*b33 *k1*n2
4 256
p35=-------------------------------------------
4
a33
p34=0
p33=0
2 1 13 2
a33 *k26*n2 - ---*a33*b33*k23*n2 + ----*b33 *k1*n2
4 64
p32=----------------------------------------------------
3
a33
p31=0
1 2 13 3
- ---*a33 *b33*k26*n2 - -----*b33 *k1*n2
4 256
p30=-------------------------------------------
4
a33
p29=0
p28=0
1
- ---*b33*k23*n2
4
p27=-------------------
2
a33
3
- ----*b33*k1*n2
16
p26=-------------------
2
a33
p25=0
p24=0
11
- a33*k23*n2 - ----*b33*k1*n2
16
p23=--------------------------------
2
a33
p22=0
11
- ----*b33*k1*n2
16
p21=-------------------
2
a33
p20=0
p19=0
2 13 2
a33 *k26*n2 + ----*b33 *k1*n2
64
p18=-------------------------------
3
a33
p17=0
p16=0
p15=0
p14=0
p13=0
- k23*n2
p12=-----------
a33
p11=0
p10=0
3
---*k1*n2
4
p9=-----------
a33
p8=0
p7=0
p6=0
p5=0
5
---*k1*n2
4
p4=-----------
a33
p3=0
5
---*k1*n2
4
p2=-----------
a33
p1=0
k125=0
k124=0
k122=0
k121=0
1 2 2 5 4
----*a33 *b33 *k26 + ------*b33 *k1
16 1024
k120=-------------------------------------
4
a33
k119=0
k118=0
1 2 5 3
- ---*a33 *b33*k26 - -----*b33 *k1
2 128
k117=-------------------------------------
3
a33
k116=0
1 2 2 5 4
----*a33 *b33 *k26 + ------*b33 *k1
16 1024
k115=-------------------------------------
4
a33
k114=0
k113=0
1 2 5 3
- ---*a33 *b33*k26 - -----*b33 *k1
2 128
k112=-------------------------------------
3
a33
k110=0
k109=0
2 5 2
a33 *k26 + ----*b33 *k1
64
k108=-------------------------
2
a33
k107=0
2 5 2
a33 *k26 + ----*b33 *k1
64
k106=-------------------------
2
a33
k105=0
k104=0
1 3
----*b33 *k1
16
k103=--------------
3
a33
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
13 2
- ----*b33 *k1
32
k95=-----------------
2
a33
k94=0
k93=0
k92=0
k91=0
k90=0
1 3
----*b33 *k1
16
k89=--------------
3
a33
k88=0
k87=0
k86=0
k85=0
13 2
- ----*b33 *k1
32
k84=-----------------
2
a33
k83=0
k82=0
k81=0
3
a33*k23 + ---*b33*k1
8
k80=----------------------
a33
k79=0
k78=0
b33*k1
k77=--------
a33
k76=0
b33*k1
k75=--------
a33
k74=0
k73=0
k72=0
k71=0
1 2 2 9 4
----*a33 *b33 *k26 + ------*b33 *k1
16 1024
k70=-------------------------------------
4
a33
k69=0
k68=0
1 2 5 3
- ---*a33 *b33*k26 - -----*b33 *k1
2 128
k67=-------------------------------------
3
a33
k66=0
1 2 2 9 4
---*a33 *b33 *k26 + -----*b33 *k1
8 512
k65=-----------------------------------
4
a33
k64=0
k63=0
1 2 5 3
- ---*a33 *b33*k26 - -----*b33 *k1
2 128
k62=-------------------------------------
3
a33
k61=0
k60=0
k59=0
k58=k26
k57=0
2 13 2
a33 *k26 + ----*b33 *k1
64
k56=-------------------------
2
a33
k55=0
k54=0
1 2 5 3
- ---*a33 *b33*k26 - -----*b33 *k1
2 128
k53=-------------------------------------
3
a33
k52=0
k51=0
k50=0
k49=0
k48=0
13 2
- ----*b33 *k1
32
k47=-----------------
2
a33
k46=0
k45=0
k44=k23
k43=0
k42=0
k41=0
k40=0
k38=0
k37=0
k36=0
1 2 2 9 4
----*a33 *b33 *k26 + ------*b33 *k1
16 1024
k35=-------------------------------------
4
a33
k34=0
k33=0
1 2 5 3
- ---*a33 *b33*k26 - -----*b33 *k1
2 128
k32=-------------------------------------
3
a33
k30=0
k29=0
2 13 2
a33 *k26 + ----*b33 *k1
64
k28=-------------------------
2
a33
k27=0
k25=0
k24=0
k22=0
k21=0
k20=0
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
3
k12=---*k1
4
k11=0
3
k10=---*k1
4
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:
q1,k26,k23,k1,b33,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=(16*a33 *u1 + 16*a33 *u2 + 8*a33 *u3 + 8*a33*b33*u3*v3 + 8*a33*n2*u2
2 2
- b33 *v3 - 6*b33*n2*v2)/(8*a33)
4 4 4 2 2 4 2 2
INT=(1024*a33 *k1*u1 + 2048*a33 *k1*u1 *u2 + 768*a33 *k1*u1 *u3
4 4 4 2 2 4 2
+ 1024*a33 *k1*u2 + 768*a33 *k1*u2 *u3 + 1024*a33 *k23*u1*u3 *v1
4 2 4 3 4 2 2
+ 1024*a33 *k23*u2*u3 *v2 + 1024*a33 *k23*u3 *v3 + 1024*a33 *k26*u1 *v1
4 2 2 4 2 2 4 2 2
+ 1024*a33 *k26*u1 *v2 + 1024*a33 *k26*u1 *v3 + 1024*a33 *k26*u2 *v1
4 2 2 4 2 2 4 2
+ 1024*a33 *k26*u2 *v2 + 1024*a33 *k26*u2 *v3 + 1024*a33 *q1*u1
4 2 3 2
+ 1024*a33 *q1*u2 + 1024*a33 *b33*k1*u1 *u3*v3
3 2 3 3
+ 1024*a33 *b33*k1*u2 *u3*v3 + 384*a33 *b33*k1*u3 *v3
3 3 3 2
- 512*a33 *b33*k26*u1*v1 - 512*a33 *b33*k26*u1*v1*v2
3 2 3 2
- 512*a33 *b33*k26*u1*v1*v3 - 512*a33 *b33*k26*u2*v1 *v2
3 3 3 2
- 512*a33 *b33*k26*u2*v2 - 512*a33 *b33*k26*u2*v2*v3
3 3 3 2
- 512*a33 *b33*q1*u1*v1 - 512*a33 *b33*q1*u2*v2 + 1280*a33 *k1*n2*u1 *u2
3 3 3 2 3
+ 1280*a33 *k1*n2*u2 + 768*a33 *k1*n2*u2*u3 - 1024*a33 *k23*n2*u1*u2*v1
3 2 3
- 1024*a33 *k23*n2*u2 *v2 - 1024*a33 *k23*n2*u2*u3*v3
3 2 3 2
+ 1024*a33 *k26*n2*u2*v1 + 1024*a33 *k26*n2*u2*v2
3 2 3 2 2 2 2
+ 1024*a33 *k26*n2*u2*v3 + 1024*a33 *n2*q1*u2 + 208*a33 *b33 *k1*u1 *v2
2 2 2 2 2 2
+ 80*a33 *b33 *k1*u1 *v3 - 416*a33 *b33 *k1*u1*u2*v1*v2
2 2 2 2 2 2
- 416*a33 *b33 *k1*u1*u3*v1*v3 + 208*a33 *b33 *k1*u2 *v1
2 2 2 2 2 2
+ 80*a33 *b33 *k1*u2 *v3 - 416*a33 *b33 *k1*u2*u3*v2*v3
2 2 4 2 2 2 2
+ 64*a33 *b33 *k26*v1 + 128*a33 *b33 *k26*v1 *v2
2 2 2 2 2 2 4
+ 64*a33 *b33 *k26*v1 *v3 + 64*a33 *b33 *k26*v2
2 2 2 2 2 2 2 2 2 2
+ 64*a33 *b33 *k26*v2 *v3 + 64*a33 *b33 *q1*v1 + 64*a33 *b33 *q1*v2
2 2 2 2
- 704*a33 *b33*k1*n2*u1 *v2 - 704*a33 *b33*k1*n2*u2 *v2
2 2 2
+ 640*a33 *b33*k1*n2*u2*u3*v3 - 192*a33 *b33*k1*n2*u3 *v2
2 2 2
- 256*a33 *b33*k23*n2*u1*v1*v2 - 256*a33 *b33*k23*n2*u2*v2
2 2 2
- 256*a33 *b33*k23*n2*u3*v2*v3 - 256*a33 *b33*k26*n2*v1 *v2
2 3 2 2
- 256*a33 *b33*k26*n2*v2 - 256*a33 *b33*k26*n2*v2*v3
2 2 2 2 3 3
- 256*a33 *b33*n2*q1*v2 + 256*a33 *k1*n2 *u2 - 40*a33*b33 *k1*u1*v1
3 2 3 2
- 40*a33*b33 *k1*u1*v1*v2 - 40*a33*b33 *k1*u1*v1*v3
3 2 3 3
- 40*a33*b33 *k1*u2*v1 *v2 - 40*a33*b33 *k1*u2*v2
3 2 3 2
- 40*a33*b33 *k1*u2*v2*v3 + 64*a33*b33 *k1*u3*v1 *v3
3 2 2 2
+ 64*a33*b33 *k1*u3*v2 *v3 + 208*a33*b33 *k1*n2*u2*v1
2 2 2 2
+ 208*a33*b33 *k1*n2*u2*v2 + 128*a33*b33 *k1*n2*u2*v3
2 2 4 4
- 352*a33*b33 *k1*n2*u3*v2*v3 - 512*a33*b33*k1*n2 *u2*v2 + 9*b33 *k1*v1
4 2 2 4 2 2 4 4
+ 18*b33 *k1*v1 *v2 + 5*b33 *k1*v1 *v3 + 9*b33 *k1*v2
4 2 2 3 2 3 3
+ 5*b33 *k1*v2 *v3 - 52*b33 *k1*n2*v1 *v2 - 52*b33 *k1*n2*v2
3 2 2 2 2 4
- 8*b33 *k1*n2*v2*v3 + 112*b33 *k1*n2 *v2 )/(1024*a33 )
And again in machine readable form:
HAM=(16*a33**2*u1**2 + 16*a33**2*u2**2 + 8*a33**2*u3**2 + 8*a33*b33*u3*v3 + 8*
a33*n2*u2 - b33**2*v3**2 - 6*b33*n2*v2)/(8*a33)$
INT=(1024*a33**4*k1*u1**4 + 2048*a33**4*k1*u1**2*u2**2 + 768*a33**4*k1*u1**2*u3
**2 + 1024*a33**4*k1*u2**4 + 768*a33**4*k1*u2**2*u3**2 + 1024*a33**4*k23*u1*u3**
2*v1 + 1024*a33**4*k23*u2*u3**2*v2 + 1024*a33**4*k23*u3**3*v3 + 1024*a33**4*k26*
u1**2*v1**2 + 1024*a33**4*k26*u1**2*v2**2 + 1024*a33**4*k26*u1**2*v3**2 + 1024*
a33**4*k26*u2**2*v1**2 + 1024*a33**4*k26*u2**2*v2**2 + 1024*a33**4*k26*u2**2*v3
**2 + 1024*a33**4*q1*u1**2 + 1024*a33**4*q1*u2**2 + 1024*a33**3*b33*k1*u1**2*u3*
v3 + 1024*a33**3*b33*k1*u2**2*u3*v3 + 384*a33**3*b33*k1*u3**3*v3 - 512*a33**3*
b33*k26*u1*v1**3 - 512*a33**3*b33*k26*u1*v1*v2**2 - 512*a33**3*b33*k26*u1*v1*v3
**2 - 512*a33**3*b33*k26*u2*v1**2*v2 - 512*a33**3*b33*k26*u2*v2**3 - 512*a33**3*
b33*k26*u2*v2*v3**2 - 512*a33**3*b33*q1*u1*v1 - 512*a33**3*b33*q1*u2*v2 + 1280*
a33**3*k1*n2*u1**2*u2 + 1280*a33**3*k1*n2*u2**3 + 768*a33**3*k1*n2*u2*u3**2 -
1024*a33**3*k23*n2*u1*u2*v1 - 1024*a33**3*k23*n2*u2**2*v2 - 1024*a33**3*k23*n2*
u2*u3*v3 + 1024*a33**3*k26*n2*u2*v1**2 + 1024*a33**3*k26*n2*u2*v2**2 + 1024*a33
**3*k26*n2*u2*v3**2 + 1024*a33**3*n2*q1*u2 + 208*a33**2*b33**2*k1*u1**2*v2**2 +
80*a33**2*b33**2*k1*u1**2*v3**2 - 416*a33**2*b33**2*k1*u1*u2*v1*v2 - 416*a33**2*
b33**2*k1*u1*u3*v1*v3 + 208*a33**2*b33**2*k1*u2**2*v1**2 + 80*a33**2*b33**2*k1*
u2**2*v3**2 - 416*a33**2*b33**2*k1*u2*u3*v2*v3 + 64*a33**2*b33**2*k26*v1**4 +
128*a33**2*b33**2*k26*v1**2*v2**2 + 64*a33**2*b33**2*k26*v1**2*v3**2 + 64*a33**2
*b33**2*k26*v2**4 + 64*a33**2*b33**2*k26*v2**2*v3**2 + 64*a33**2*b33**2*q1*v1**2
+ 64*a33**2*b33**2*q1*v2**2 - 704*a33**2*b33*k1*n2*u1**2*v2 - 704*a33**2*b33*k1
*n2*u2**2*v2 + 640*a33**2*b33*k1*n2*u2*u3*v3 - 192*a33**2*b33*k1*n2*u3**2*v2 -
256*a33**2*b33*k23*n2*u1*v1*v2 - 256*a33**2*b33*k23*n2*u2*v2**2 - 256*a33**2*b33
*k23*n2*u3*v2*v3 - 256*a33**2*b33*k26*n2*v1**2*v2 - 256*a33**2*b33*k26*n2*v2**3
- 256*a33**2*b33*k26*n2*v2*v3**2 - 256*a33**2*b33*n2*q1*v2 + 256*a33**2*k1*n2**2
*u2**2 - 40*a33*b33**3*k1*u1*v1**3 - 40*a33*b33**3*k1*u1*v1*v2**2 - 40*a33*b33**
3*k1*u1*v1*v3**2 - 40*a33*b33**3*k1*u2*v1**2*v2 - 40*a33*b33**3*k1*u2*v2**3 - 40
*a33*b33**3*k1*u2*v2*v3**2 + 64*a33*b33**3*k1*u3*v1**2*v3 + 64*a33*b33**3*k1*u3*
v2**2*v3 + 208*a33*b33**2*k1*n2*u2*v1**2 + 208*a33*b33**2*k1*n2*u2*v2**2 + 128*
a33*b33**2*k1*n2*u2*v3**2 - 352*a33*b33**2*k1*n2*u3*v2*v3 - 512*a33*b33*k1*n2**2
*u2*v2 + 9*b33**4*k1*v1**4 + 18*b33**4*k1*v1**2*v2**2 + 5*b33**4*k1*v1**2*v3**2
+ 9*b33**4*k1*v2**4 + 5*b33**4*k1*v2**2*v3**2 - 52*b33**3*k1*n2*v1**2*v2 - 52*
b33**3*k1*n2*v2**3 - 8*b33**3*k1*n2*v2*v3**2 + 112*b33**2*k1*n2**2*v2**2)/(1024*
a33**4)$