Solution 17 to problem e3quant
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
a22=4*a33
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
1 2
- ----*b33
16
c33=--------------
a33
n2=0
n3=0
5
- ----*b33*n1
24
m1=----------------
a33
m2=0
m3=0
r6=0
r5=0
39 2 1 2 5 3
-----*a33 *b33*k1*n1 - ----*a33 *b33*n1*q1 + ------*b33*k1*n1
176 24 1584
r4=----------------------------------------------------------------
4
a33
r3=0
r2=0
5 2 1 2 5 3
----*a33 *k1*n1 + ---*a33 *n1*q1 - -----*k1*n1
22 3 198
r1=-------------------------------------------------
3
a33
7 2
------*b33 *k1*n1
2112
q20=-------------------
3
a33
1 2 2 1 2 2 5 2 2
- ---*a33 *b33 *k1 + ----*a33 *b33 *q1 - ------*b33 *k1*n1
8 64 4224
q19=--------------------------------------------------------------
4
a33
q18=0
q17=0
1 2 2 1 2 2 7 2 2
- ---*a33 *b33 *k1 + ----*a33 *b33 *q1 + ------*b33 *k1*n1
8 64 6336
q16=--------------------------------------------------------------
4
a33
1
----*b33*k1*n1
12
q14=----------------
2
a33
q13=0
7
- -----*b33*k1*n1
264
q11=--------------------
2
a33
3 3 2 1 2 5 2
- a33 *k38 + ---*a33 *b33*k1 - ---*a33 *b33*q1 + -----*b33*k1*n1
2 4 264
q10=--------------------------------------------------------------------
3
a33
q9=0
5
- ----*k1*n1
11
q8=---------------
a33
2 2 5 2
- 4*a33 *k1 + a33 *q1 - ----*k1*n1
66
q7=--------------------------------------
2
a33
q6=0
q5=0
3 3 2 1 2 7 2
- a33 *k38 + ---*a33 *b33*k1 - ---*a33 *b33*q1 - -----*b33*k1*n1
2 4 792
q4=--------------------------------------------------------------------
3
a33
q3=0
q2=0
p56=0
p55=0
p54=0
p53=0
1 2 5 3
- ----*a33 *b33*k16*n1 - ------*b33 *k1*n1
24 5632
p52=---------------------------------------------
4
a33
p51=0
1 2 27 3
- ----*a33 *b33*k16*n1 - -------*b33 *k1*n1
24 11264
p50=----------------------------------------------
4
a33
p49=0
p48=0
1 2 27 3
- ----*a33 *b33*k16*n1 - -------*b33 *k1*n1
24 11264
p47=----------------------------------------------
4
a33
p46=0
p45=0
p44=0
1 17 2
- ----*a33*b33*k38*n1 - -----*b33 *k1*n1
24 704
p43=-------------------------------------------
3
a33
81 2
------*b33 *k1
1408
p42=----------------
2
a33
p41=0
p40=0
p39=0
7
- -----*b33*k1*n1
528
p38=--------------------
2
a33
p37=0
p36=0
p35=0
p34=0
7 2
------*b33 *k1
1408
p33=----------------
2
a33
1
- ----*b33*k38*n1
24
p32=--------------------
2
a33
p31=0
p30=0
p29=0
1
2*a33*k38 - ---*b33*k1
2
p28=------------------------
a33
p27=0
p26=0
p25=0
17
- -----*b33*k1*n1
176
p24=--------------------
2
a33
p23=0
p22=0
1 2 1 2
---*a33 *k16*n1 + ----*b33 *k1*n1
3 96
p21=-----------------------------------
3
a33
7 2
- ------*b33 *k1
1408
p20=-------------------
2
a33
1 2 27 2
---*a33 *k16*n1 + ------*b33 *k1*n1
3 1408
p19=-------------------------------------
3
a33
p18=0
p17=0
1 2 1 27 2
---*a33 *k16*n1 - ----*a33*b33*k38*n1 + ------*b33 *k1*n1
3 24 1408
p16=-----------------------------------------------------------
3
a33
1 37
- ---*a33*k38*n1 + -----*b33*k1*n1
3 264
p15=-------------------------------------
2
a33
1
- 2*a33*k38 + ---*b33*k1
2
p14=---------------------------
a33
p13=0
7
----*k1*n1
66
p12=------------
a33
p11=0
1
- ---*k38*n1
3
p10=---------------
a33
p9=0
p8= - 4*k1
37
----*k1*n1
66
p7=------------
a33
p6=0
p5=0
1 17
- ---*a33*k38*n1 - -----*b33*k1*n1
3 176
p4=-------------------------------------
2
a33
p3=0
p2=0
37
----*k1*n1
66
p1=------------
a33
k125=0
1 2 2 37 4
----*a33 *b33 *k16 + -------*b33 *k1
64 90112
k124=--------------------------------------
4
a33
k123=0
1 2 2 59 4
----*a33 *b33 *k16 + -------*b33 *k1
64 90112
k122=--------------------------------------
4
a33
k121=0
k120=0
k119=0
k118=0
1 2 2 37 4
----*a33 *b33 *k16 + -------*b33 *k1
64 90112
k117=--------------------------------------
4
a33
k116=0
1 2 2 59 4
----*a33 *b33 *k16 + -------*b33 *k1
32 45056
k115=--------------------------------------
4
a33
k114=0
k113=0
1 2 2 59 4
----*a33 *b33 *k16 + -------*b33 *k1
64 90112
k112=--------------------------------------
4
a33
k110=0
1 3
-----*b33 *k1
128
k109=---------------
3
a33
k108=0
k107=0
k106=0
k105=0
1 3
-----*b33 *k1
128
k104=---------------
3
a33
k103=0
k102=0
k100=0
k99=0
k98=0
k97=0
7
a33*k38 + ----*b33*k1
88
k95=-----------------------
a33
k94=0
k93=0
k91=0
1 2 37 3
- ---*a33 *b33*k16 - ------*b33 *k1
4 5632
k90=--------------------------------------
3
a33
k89=0
1 2 37 3
- ---*a33 *b33*k16 - ------*b33 *k1
4 5632
k88=--------------------------------------
3
a33
k87=0
k86=0
k85=0
k84=0
1 2 37 3
- ---*a33 *b33*k16 - ------*b33 *k1
4 5632
k83=--------------------------------------
3
a33
k82=0
k81=0
81 2
- -----*b33 *k1
704
k80=------------------
2
a33
k79=0
k78=0
k77=0
k76=0
k75=0
k74=k38
k73=0
k72=0
2 37 2
a33 *k16 + ------*b33 *k1
1408
k71=---------------------------
2
a33
k70=0
k69=k16
k68=0
k67=0
2 81 2
a33 *k16 + ------*b33 *k1
1408
k66=---------------------------
2
a33
1
---*b33*k1
2
k65=------------
a33
k64=0
k63=0
7
k62=----*k1
22
k61=0
k59=0
k58=0
k57=k1
k56=0
k55=0
k54=0
k53=0
1 2 37 3
- ---*a33 *b33*k16 - ------*b33 *k1
4 5632
k52=--------------------------------------
3
a33
k51=0
1 2 37 3
- ---*a33 *b33*k16 - ------*b33 *k1
4 5632
k50=--------------------------------------
3
a33
k49=0
k48=0
1 2 37 3
- ---*a33 *b33*k16 - ------*b33 *k1
4 5632
k47=--------------------------------------
3
a33
k46=0
k45=0
k44=0
81 2
- -----*b33 *k1
704
k43=------------------
2
a33
k42=0
k41=0
k40=0
k39=0
k37=0
k36=0
k35=0
k34=0
k33=0
81 2
- -----*b33 *k1
704
k32=------------------
2
a33
k31=0
k30=0
k29=0
k28=0
k27=0
k26=0
k25=0
k24=0
k23=0
k22=0
2 37 2
a33 *k16 + ------*b33 *k1
1408
k21=---------------------------
2
a33
k20=0
2 81 2
a33 *k16 + ------*b33 *k1
1408
k19=---------------------------
2
a33
k18=0
k17=0
1
---*b33*k1
2
k15=------------
a33
k14=0
k13=0
7
k12=----*k1
22
k11=0
k10=0
k9=0
k8=0
k7=2*k1
k6=0
k5=0
k4=0
k3=0
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:
b33,q1,k38,k16,k1,n1,a33
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2 2 2 2
HAM=(192*a33 *u1 + 192*a33 *u2 + 48*a33 *u3 + 48*a33*b33*u3*v3 + 48*a33*n1*u1
2 2
- 3*b33 *v3 - 10*b33*n1*v1)/(48*a33)
4 4 4 2 4 2
INT=(811008*a33 *k1*u1 + 1622016*a33 *k1*u1 *u2*u3 + 811008*a33 *k1*u1*u2*v2
4 2 4 2
- 3244032*a33 *k1*u1*u3 + 258048*a33 *k1*u1*v1*v2
4 4 2 2 4 3
- 3244032*a33 *k1*u1*v1 + 258048*a33 *k1*u2 *u3 + 811008*a33 *k16*u1 *v1
4 3 4
+ 811008*a33 *k16*u1 *v3 + 811008*a33 *k16*u1*u2*u3*v1
4 3 4 3 4 3
+ 811008*a33 *k16*u1*v2 + 811008*a33 *k16*u2 *v1 + 811008*a33 *k16*v1*v2
4 2 4
+ 811008*a33 *k38*u1*u2 *v2 - 1622016*a33 *k38*u1*u3*v1
4 4 3
- 811008*a33 *k38*u1*u3 + 811008*a33 *k38*u2 *v3
4 4
+ 811008*a33 *k38*u2*u3*v2*v3 + 1622016*a33 *k38*u2*v1*v2
4 2 4 2 4
- 811008*a33 *k38*v1 + 811008*a33 *q1*u1 + 811008*a33 *q1*u1*v1
3 3
+ 405504*a33 *b33*k1*u1*u3*v1 + 1216512*a33 *b33*k1*u1*u3
3 3
+ 64512*a33 *b33*k1*u2*u3*v2*v3 - 405504*a33 *b33*k1*u2*v1*v2
3 4 3 2 2
+ 405504*a33 *b33*k1*u3 + 405504*a33 *b33*k1*v1 *v2
3 2 3
+ 1216512*a33 *b33*k1*v1 - 202752*a33 *b33*k16*u1*u2*v1*v2
3 2 3 2
- 202752*a33 *b33*k16*u1*v1 *v2 - 202752*a33 *b33*k16*u2 *v1*v3
3 3 2
- 202752*a33 *b33*k16*u2*u3*v1*v2 - 202752*a33 *b33*k16*u2*v1 *v3
3 3 3
- 202752*a33 *b33*k16*v1 *v3 - 202752*a33 *b33*q1*u1*u3
3 2 3 3
- 202752*a33 *b33*q1*v1 + 454656*a33 *k1*n1*u1
3 3
+ 86016*a33 *k1*n1*u1*u2*v1 + 184320*a33 *k1*n1*u1
3 2 3
+ 454656*a33 *k1*n1*u2 *u3 - 368640*a33 *k1*n1*u2*v1
3 2 3 2
+ 270336*a33 *k16*n1*u1 *v2 + 270336*a33 *k16*n1*u3 *v1
3 2 3 3
+ 270336*a33 *k16*n1*u3*v1 - 270336*a33 *k38*n1*u2
3 3 3
- 270336*a33 *k38*n1*u2*u3*v1 - 270336*a33 *k38*n1*u3
3 2 2 3
+ 270336*a33 *n1*q1*u1 + 21312*a33 *b33 *k1*u1 *v3
2 2 2 2 2
+ 21312*a33 *b33 *k1*u1*u2*u3*v1 - 93312*a33 *b33 *k1*u1*u3 *v2
2 2 3 2 2
- 93312*a33 *b33 *k1*u1*v1 + 46656*a33 *b33 *k1*u1*v1*v3
2 2 3 2 2
+ 46656*a33 *b33 *k1*u1*v2 - 101376*a33 *b33 *k1*u1*v3
2 2 3 2 2 3
+ 46656*a33 *b33 *k1*u2 *v1 - 93312*a33 *b33 *k1*u3 *v3
2 2 2 2 2 3
+ 4032*a33 *b33 *k1*u3*v2 - 4032*a33 *b33 *k1*v1
2 2 2 2 2
- 101376*a33 *b33 *k1*v1*v3 + 12672*a33 *b33 *k16*u1*v1*v3
2 2 2 2 2 3
+ 12672*a33 *b33 *k16*u2*v2*v3 + 12672*a33 *b33 *k16*u2*v3
2 2 2 2 2 2 3
+ 25344*a33 *b33 *k16*v1 *v3 + 12672*a33 *b33 *k16*v1*v3
2 2 2 2
+ 12672*a33 *b33 *q1*u1*v3 + 12672*a33 *b33 *q1*v1*v3
2 2
- 78336*a33 *b33*k1*n1*u1*u3*v2 - 21504*a33 *b33*k1*n1*u1*v2
2 3 2 2
- 78336*a33 *b33*k1*n1*u2 - 10752*a33 *b33*k1*n1*u2 *v3
2 2
+ 113664*a33 *b33*k1*n1*u2*u3*v1 + 67584*a33 *b33*k1*n1*v1*v2
2 2
+ 179712*a33 *b33*k1*n1*v1 - 33792*a33 *b33*k16*n1*u2*v2*v3
2 2 2 2
- 33792*a33 *b33*k16*n1*u2*v3 - 33792*a33 *b33*k16*n1*v2 *v3
2 2 2
- 33792*a33 *b33*k38*n1*u2*v1*v3 - 33792*a33 *b33*k38*n1*u2*v2
2 2 2
- 33792*a33 *b33*k38*n1*u3 *v1 - 33792*a33 *b33*n1*q1*v1
2 2 3
- 61440*a33 *k1*n1 *u1*v1 - 5328*a33*b33 *k1*u1*u2*v1*v2
3 2 3 2
+ 6336*a33*b33 *k1*u1*u3*v3 - 5328*a33*b33 *k1*u1*v1 *v2
3 2 3
- 5328*a33*b33 *k1*u2 *v1*v3 - 5328*a33*b33 *k1*u2*u3*v1*v2
3 2 3 3
- 5328*a33*b33 *k1*u2*v1 *v3 - 5328*a33*b33 *k1*v1 *v3
3 2 2 2
+ 6336*a33*b33 *k1*v1*v2 *v3 + 8448*a33*b33 *k1*n1*u1 *v2
2 2 2
- 19584*a33*b33 *k1*n1*u2*v1*v3 + 15552*a33*b33 *k1*n1*u3 *v1
2 2 2
+ 15552*a33*b33 *k1*n1*u3*v1 + 2688*a33*b33 *k1*n1*v2*v3
2 2 2
- 7168*a33*b33*k1*n1 *u1*u3 + 15360*a33*b33*k1*n1 *v1
3 4 2 4 2
- 20480*a33*k1*n1 *u1 + 531*b33 *k1*u1*v1*v3 + 333*b33 *k1*u2*v2*v3
4 3 4 2 2 4 3
+ 531*b33 *k1*u2*v3 + 1062*b33 *k1*v1 *v3 + 333*b33 *k1*v1*v3
3 3 2
- 1944*b33 *k1*n1*u2*v2*v3 - 720*b33 *k1*n1*u2*v3
3 2 2 2 2 2
- 1944*b33 *k1*n1*v2 *v3 + 896*b33 *k1*n1 *u1*v3 - 960*b33 *k1*n1 *v1*v3
3 4
+ 2560*b33*k1*n1 *v1)/(811008*a33 )
And again in machine readable form:
HAM=(192*a33**2*u1**2 + 192*a33**2*u2**2 + 48*a33**2*u3**2 + 48*a33*b33*u3*v3 +
48*a33*n1*u1 - 3*b33**2*v3**2 - 10*b33*n1*v1)/(48*a33)$
INT=(811008*a33**4*k1*u1**4 + 1622016*a33**4*k1*u1**2*u2*u3 + 811008*a33**4*k1*
u1*u2*v2**2 - 3244032*a33**4*k1*u1*u3**2 + 258048*a33**4*k1*u1*v1*v2**2 -
3244032*a33**4*k1*u1*v1 + 258048*a33**4*k1*u2**2*u3**2 + 811008*a33**4*k16*u1**3
*v1 + 811008*a33**4*k16*u1**3*v3 + 811008*a33**4*k16*u1*u2*u3*v1 + 811008*a33**4
*k16*u1*v2**3 + 811008*a33**4*k16*u2**3*v1 + 811008*a33**4*k16*v1*v2**3 + 811008
*a33**4*k38*u1*u2**2*v2 - 1622016*a33**4*k38*u1*u3*v1 - 811008*a33**4*k38*u1*u3
+ 811008*a33**4*k38*u2**3*v3 + 811008*a33**4*k38*u2*u3*v2*v3 + 1622016*a33**4*
k38*u2*v1*v2 - 811008*a33**4*k38*v1**2 + 811008*a33**4*q1*u1**2 + 811008*a33**4*
q1*u1*v1 + 405504*a33**3*b33*k1*u1*u3*v1 + 1216512*a33**3*b33*k1*u1*u3 + 64512*
a33**3*b33*k1*u2*u3*v2*v3 - 405504*a33**3*b33*k1*u2*v1*v2 + 405504*a33**3*b33*k1
*u3**4 + 405504*a33**3*b33*k1*v1**2*v2**2 + 1216512*a33**3*b33*k1*v1**2 - 202752
*a33**3*b33*k16*u1*u2*v1*v2 - 202752*a33**3*b33*k16*u1*v1**2*v2 - 202752*a33**3*
b33*k16*u2**2*v1*v3 - 202752*a33**3*b33*k16*u2*u3*v1*v2 - 202752*a33**3*b33*k16*
u2*v1**2*v3 - 202752*a33**3*b33*k16*v1**3*v3 - 202752*a33**3*b33*q1*u1*u3 -
202752*a33**3*b33*q1*v1**2 + 454656*a33**3*k1*n1*u1**3 + 86016*a33**3*k1*n1*u1*
u2*v1 + 184320*a33**3*k1*n1*u1 + 454656*a33**3*k1*n1*u2**2*u3 - 368640*a33**3*k1
*n1*u2*v1 + 270336*a33**3*k16*n1*u1**2*v2 + 270336*a33**3*k16*n1*u3**2*v1 +
270336*a33**3*k16*n1*u3*v1**2 - 270336*a33**3*k38*n1*u2**3 - 270336*a33**3*k38*
n1*u2*u3*v1 - 270336*a33**3*k38*n1*u3**3 + 270336*a33**3*n1*q1*u1 + 21312*a33**2
*b33**2*k1*u1**3*v3 + 21312*a33**2*b33**2*k1*u1*u2*u3*v1 - 93312*a33**2*b33**2*
k1*u1*u3**2*v2 - 93312*a33**2*b33**2*k1*u1*v1**3 + 46656*a33**2*b33**2*k1*u1*v1*
v3 + 46656*a33**2*b33**2*k1*u1*v2**3 - 101376*a33**2*b33**2*k1*u1*v3 + 46656*a33
**2*b33**2*k1*u2**3*v1 - 93312*a33**2*b33**2*k1*u3**3*v3 + 4032*a33**2*b33**2*k1
*u3*v2**2 - 4032*a33**2*b33**2*k1*v1**3 - 101376*a33**2*b33**2*k1*v1*v3 + 12672*
a33**2*b33**2*k16*u1*v1*v3**2 + 12672*a33**2*b33**2*k16*u2*v2*v3**2 + 12672*a33
**2*b33**2*k16*u2*v3**3 + 25344*a33**2*b33**2*k16*v1**2*v3**2 + 12672*a33**2*b33
**2*k16*v1*v3**3 + 12672*a33**2*b33**2*q1*u1*v3 + 12672*a33**2*b33**2*q1*v1*v3 -
78336*a33**2*b33*k1*n1*u1*u3*v2 - 21504*a33**2*b33*k1*n1*u1*v2 - 78336*a33**2*
b33*k1*n1*u2**3 - 10752*a33**2*b33*k1*n1*u2**2*v3 + 113664*a33**2*b33*k1*n1*u2*
u3*v1 + 67584*a33**2*b33*k1*n1*v1*v2 + 179712*a33**2*b33*k1*n1*v1 - 33792*a33**2
*b33*k16*n1*u2*v2*v3 - 33792*a33**2*b33*k16*n1*u2*v3**2 - 33792*a33**2*b33*k16*
n1*v2**2*v3 - 33792*a33**2*b33*k38*n1*u2*v1*v3 - 33792*a33**2*b33*k38*n1*u2*v2**
2 - 33792*a33**2*b33*k38*n1*u3**2*v1 - 33792*a33**2*b33*n1*q1*v1 - 61440*a33**2*
k1*n1**2*u1*v1 - 5328*a33*b33**3*k1*u1*u2*v1*v2 + 6336*a33*b33**3*k1*u1*u3*v3**2
- 5328*a33*b33**3*k1*u1*v1**2*v2 - 5328*a33*b33**3*k1*u2**2*v1*v3 - 5328*a33*
b33**3*k1*u2*u3*v1*v2 - 5328*a33*b33**3*k1*u2*v1**2*v3 - 5328*a33*b33**3*k1*v1**
3*v3 + 6336*a33*b33**3*k1*v1*v2**2*v3 + 8448*a33*b33**2*k1*n1*u1**2*v2 - 19584*
a33*b33**2*k1*n1*u2*v1*v3 + 15552*a33*b33**2*k1*n1*u3**2*v1 + 15552*a33*b33**2*
k1*n1*u3*v1**2 + 2688*a33*b33**2*k1*n1*v2*v3 - 7168*a33*b33*k1*n1**2*u1*u3 +
15360*a33*b33*k1*n1**2*v1**2 - 20480*a33*k1*n1**3*u1 + 531*b33**4*k1*u1*v1*v3**2
+ 333*b33**4*k1*u2*v2*v3**2 + 531*b33**4*k1*u2*v3**3 + 1062*b33**4*k1*v1**2*v3
**2 + 333*b33**4*k1*v1*v3**3 - 1944*b33**3*k1*n1*u2*v2*v3 - 720*b33**3*k1*n1*u2*
v3**2 - 1944*b33**3*k1*n1*v2**2*v3 + 896*b33**2*k1*n1**2*u1*v3 - 960*b33**2*k1*
n1**2*v1*v3 + 2560*b33*k1*n1**3*v1)/(811008*a33**4)$