Solution 38 to problem e3c2new
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Expressions
The solution is given through the following expressions:
1
a22=---*a33
3
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
3 2
- ---*b33
4
c33=-------------
a33
n2=0
3*b33*n1
m1=----------
a33
m2=0
3
---*b33*n3
2
m3=------------
a33
9 2 81 2
---*a33 *b33*n3*q1 + ----*b33*k10*n1 *n3
2 8
r6=------------------------------------------
4
a33
r5=0
9 2 81 3 81 2
---*a33 *b33*n1*q1 + ----*b33*k10*n1 + ----*b33*k10*n1*n3
4 16 16
r4=-------------------------------------------------------------
4
a33
3 2 27 2 27 3
- ---*a33 *n3*q1 - ----*k10*n1 *n3 + ----*k10*n3
2 8 8
r3=----------------------------------------------------
3
a33
r2=0
3 2 27 3 27 2
- ---*a33 *n1*q1 - ----*k10*n1 + ----*k10*n1*n3
2 8 8
r1=----------------------------------------------------
3
a33
q20=0
81 2
----*b33 *k10*n1*n3
8
q19=---------------------
4
a33
q17=0
27
----*b33*k10*n1*n3
4
q16=--------------------
3
a33
9 2 2 81 2 2
---*a33 *b33 *q1 + ----*b33 *k10*n1
4 16
q15=--------------------------------------
4
a33
q14=0
q13=0
2 27 2 27 2
- 3*a33 *b33*q1 - ----*b33*k10*n1 - ----*b33*k10*n3
4 4
q12=--------------------------------------------------------
3
a33
q11=0
9 2 2 81 2 2
---*a33 *b33 *q1 + ----*b33 *k10*n1
4 8
q10=--------------------------------------
4
a33
q9=0
q8=0
2 27 2 27 2
- 3*a33 *b33*q1 - ----*b33*k10*n1 - ----*b33*k10*n3
4 4
q7=--------------------------------------------------------
3
a33
q5=0
9
- ---*k10*n1*n3
2
q4=------------------
2
a33
2 9 2
a33 *q1 + ---*k10*n1
4
q3=-----------------------
2
a33
q2=0
9 2 81 3
---*a33 *b33*k26*n3 - ----*b33 *k10*n3
2 8
p56=----------------------------------------
4
a33
p55=0
9 2 81 3
---*a33 *b33*k26*n1 - ----*b33 *k10*n1
4 8
p54=----------------------------------------
4
a33
3
- ---*k26*n3
2
p53=---------------
a33
p52=0
3
- ---*k26*n1
2
p51=---------------
a33
9 2 81 3
---*a33 *b33*k26*n3 - ----*b33 *k10*n3
2 8
p50=----------------------------------------
4
a33
p49=0
p48=0
p47=0
p46=0
9 2 81 3
---*a33 *b33*k26*n3 - ----*b33 *k10*n3
2 8
p45=----------------------------------------
4
a33
9 27 2
---*a33*b33*k23*n1 + ----*b33 *k10*n1
4 4
p44=---------------------------------------
3
a33
p43=0
p42=0
3
---*a33*k23*n3 + 9*b33*k10*n3
2
p41=-------------------------------
2
a33
p40=0
3 9
---*a33*k23*n1 + ---*b33*k10*n1
2 2
p39=---------------------------------
2
a33
p38=0
p37=0
p36=0
p35=0
9 2 81 3
---*a33 *b33*k26*n1 - ----*b33 *k10*n1
4 16
p34=----------------------------------------
4
a33
3 2 27 2
- ---*a33 *k26*n3 + ----*b33 *k10*n3
2 8
p33=---------------------------------------
3
a33
p32=0
3 2 27 2
- ---*a33 *k26*n1 + ----*b33 *k10*n1
2 8
p31=---------------------------------------
3
a33
p30=0
p29=0
9
---*b33*k23*n1
4
p28=----------------
2
a33
p27=0
p26=0
3
---*k23*n3
2
p25=------------
a33
p24=0
p23=0
3
---*k23*n1
2
p22=------------
a33
p21=0
9 2 81 3
---*a33 *b33*k26*n1 - ----*b33 *k10*n1
4 16
p20=----------------------------------------
4
a33
3 2 27 2
- ---*a33 *k26*n3 + ----*b33 *k10*n3
2 8
p19=---------------------------------------
3
a33
p18=0
3 2 9 27 2
- ---*a33 *k26*n1 + ---*a33*b33*k23*n1 + ----*b33 *k10*n1
2 4 8
p17=------------------------------------------------------------
3
a33
9
---*b33*k10*n1
4
p16=----------------
2
a33
p15=0
3
---*k23*n3
2
p14=------------
a33
9
---*b33*k10*n1
4
p13=----------------
2
a33
p12=0
3 9
---*a33*k23*n1 + ---*b33*k10*n1
2 4
p11=---------------------------------
2
a33
3
- ---*k10*n3
2
p10=---------------
a33
p9=0
3
- ---*k10*n1
2
p8=---------------
a33
3
---*k10*n3
2
p7=------------
a33
p6=0
3
---*k10*n3
2
p5=------------
a33
p4=0
3
---*k10*n1
2
p3=------------
a33
p2=0
3
---*k10*n1
2
p1=------------
a33
k125=0
k124=0
k122=0
k121=0
9 2 2 81 4
---*a33 *b33 *k26 - ----*b33 *k10
4 16
k120=-----------------------------------
4
a33
k119=0
k118=0
2 27 3
- 3*a33 *b33*k26 + ----*b33 *k10
4
k117=-----------------------------------
3
a33
k116=0
9 2 2 81 4
---*a33 *b33 *k26 - ----*b33 *k10
4 16
k115=-----------------------------------
4
a33
k114=0
k113=0
2 27 3
- 3*a33 *b33*k26 + ----*b33 *k10
4
k112=-----------------------------------
3
a33
k110=0
k109=0
2 9 2
a33 *k26 - ---*b33 *k10
4
k108=-------------------------
2
a33
k107=0
2 9 2
a33 *k26 - ---*b33 *k10
4
k106=-------------------------
2
a33
k105=0
k104=0
k103=0
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
9 2
---*b33 *k10
2
k95=--------------
2
a33
k94=0
k93=0
k92=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
k85=0
9 2
---*b33 *k10
2
k84=--------------
2
a33
k83=0
k82=0
k81=0
a33*k23 + 3*b33*k10
k80=---------------------
a33
k79=0
k78=0
k77=0
k76=0
k75=0
k74=0
k73=0
k72=0
k71=0
9 2 2 81 4
---*a33 *b33 *k26 - ----*b33 *k10
4 16
k70=-----------------------------------
4
a33
k69=0
k68=0
2 27 3
- 3*a33 *b33*k26 + ----*b33 *k10
4
k67=-----------------------------------
3
a33
k66=0
9 2 2 81 4
---*a33 *b33 *k26 - ----*b33 *k10
2 8
k65=-----------------------------------
4
a33
k64=0
k63=0
2 27 3
- 3*a33 *b33*k26 + ----*b33 *k10
4
k62=-----------------------------------
3
a33
k61=0
k60=0
k59=0
k58=k26
k57=0
2 9 2
a33 *k26 - ---*b33 *k10
4
k56=-------------------------
2
a33
k55=0
k54=0
2 27 3
- 3*a33 *b33*k26 + ----*b33 *k10
4
k53=-----------------------------------
3
a33
k52=0
k51=0
k50=0
k49=0
k48=0
9 2
---*b33 *k10
2
k47=--------------
2
a33
k46=0
k45=0
k44=k23
k43=0
k42=0
k41=0
k40=0
k38=0
k37=0
k36=0
9 2 2 81 4
---*a33 *b33 *k26 - ----*b33 *k10
4 16
k35=-----------------------------------
4
a33
k34=0
k33=0
2 27 3
- 3*a33 *b33*k26 + ----*b33 *k10
4
k32=-----------------------------------
3
a33
k30=0
k29=0
2 9 2
a33 *k26 - ---*b33 *k10
4
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
k12=k10
k11=0
k9=0
k8=0
k7=0
k6=0
k5=0
k4=0
k3=0
k2=0
k1=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,k10,b33,n3,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=(4*a33 *u1 + 4*a33 *u2 + 12*a33 *u3 + 12*a33*b33*u3*v3 + 12*a33*n1*u1
2 2
+ 12*a33*n3*u3 - 9*b33 *v3 + 36*b33*n1*v1 + 18*b33*n3*v3)/(12*a33)
4 2 2 4 2 2 4 2
INT=(16*a33 *k10*u1 *u3 + 16*a33 *k10*u2 *u3 + 16*a33 *k23*u1*u3 *v1
4 2 4 3 4 2 2
+ 16*a33 *k23*u2*u3 *v2 + 16*a33 *k23*u3 *v3 + 16*a33 *k26*u1 *v1
4 2 2 4 2 2 4 2 2
+ 16*a33 *k26*u1 *v2 + 16*a33 *k26*u1 *v3 + 16*a33 *k26*u2 *v1
4 2 2 4 2 2 4 2
+ 16*a33 *k26*u2 *v2 + 16*a33 *k26*u2 *v3 + 16*a33 *q1*u1
4 2 3 3 3 3
+ 16*a33 *q1*u2 + 48*a33 *b33*k10*u3 *v3 - 48*a33 *b33*k26*u1*v1
3 2 3 2
- 48*a33 *b33*k26*u1*v1*v2 - 48*a33 *b33*k26*u1*v1*v3
3 2 3 3
- 48*a33 *b33*k26*u2*v1 *v2 - 48*a33 *b33*k26*u2*v2
3 2 3 3
- 48*a33 *b33*k26*u2*v2*v3 - 48*a33 *b33*q1*u1*v1 - 48*a33 *b33*q1*u2*v2
3 3 3 2 3 2
+ 24*a33 *k10*n1*u1 + 24*a33 *k10*n1*u1*u2 - 24*a33 *k10*n1*u1*u3
3 2 3 2 3 3
+ 24*a33 *k10*n3*u1 *u3 + 24*a33 *k10*n3*u2 *u3 - 24*a33 *k10*n3*u3
3 2 3
+ 24*a33 *k23*n1*u1 *v1 + 24*a33 *k23*n1*u1*u2*v2
3 3
+ 24*a33 *k23*n1*u1*u3*v3 + 24*a33 *k23*n3*u1*u3*v1
3 3 2 3 2
+ 24*a33 *k23*n3*u2*u3*v2 + 24*a33 *k23*n3*u3 *v3 - 24*a33 *k26*n1*u1*v1
3 2 3 2 3 2
- 24*a33 *k26*n1*u1*v2 - 24*a33 *k26*n1*u1*v3 - 24*a33 *k26*n3*u3*v1
3 2 3 2 3
- 24*a33 *k26*n3*u3*v2 - 24*a33 *k26*n3*u3*v3 - 24*a33 *n1*q1*u1
3 2 2 2 2 2 2 2 2
- 24*a33 *n3*q1*u3 - 36*a33 *b33 *k10*u1 *v2 - 36*a33 *b33 *k10*u1 *v3
2 2 2 2
+ 72*a33 *b33 *k10*u1*u2*v1*v2 + 72*a33 *b33 *k10*u1*u3*v1*v3
2 2 2 2 2 2 2 2
- 36*a33 *b33 *k10*u2 *v1 - 36*a33 *b33 *k10*u2 *v3
2 2 2 2 4
+ 72*a33 *b33 *k10*u2*u3*v2*v3 + 36*a33 *b33 *k26*v1
2 2 2 2 2 2 2 2
+ 72*a33 *b33 *k26*v1 *v2 + 36*a33 *b33 *k26*v1 *v3
2 2 4 2 2 2 2 2 2 2
+ 36*a33 *b33 *k26*v2 + 36*a33 *b33 *k26*v2 *v3 + 36*a33 *b33 *q1*v1
2 2 2 2 2
+ 36*a33 *b33 *q1*v2 + 36*a33 *b33*k10*n1*u1 *v1
2 2 2
+ 72*a33 *b33*k10*n1*u1*u3*v3 + 36*a33 *b33*k10*n1*u2 *v1
2 2 2 2
+ 36*a33 *b33*k10*n1*u3 *v1 + 144*a33 *b33*k10*n3*u3 *v3
2 2 2
+ 36*a33 *b33*k23*n1*u1*v1 + 36*a33 *b33*k23*n1*u2*v1*v2
2 2 3
+ 36*a33 *b33*k23*n1*u3*v1*v3 + 36*a33 *b33*k26*n1*v1
2 2 2 2
+ 36*a33 *b33*k26*n1*v1*v2 + 36*a33 *b33*k26*n1*v1*v3
2 2 2 2
+ 72*a33 *b33*k26*n3*v1 *v3 + 72*a33 *b33*k26*n3*v2 *v3
2 3 2 2
+ 72*a33 *b33*k26*n3*v3 + 36*a33 *b33*n1*q1*v1 + 72*a33 *b33*n3*q1*v3
2 2 2 2 3 3
+ 36*a33 *k10*n1 *u2 - 72*a33 *k10*n1*n3*u1*u3 + 108*a33*b33 *k10*u1*v1
3 2 3 2
+ 108*a33*b33 *k10*u1*v1*v2 + 108*a33*b33 *k10*u1*v1*v3
3 2 3 3
+ 108*a33*b33 *k10*u2*v1 *v2 + 108*a33*b33 *k10*u2*v2
3 2 2 2
+ 108*a33*b33 *k10*u2*v2*v3 + 54*a33*b33 *k10*n1*u1*v1
2 2 2
+ 54*a33*b33 *k10*n1*u1*v2 + 108*a33*b33 *k10*n1*u3*v1*v3
2 2 2 2
+ 54*a33*b33 *k10*n3*u3*v1 + 54*a33*b33 *k10*n3*u3*v2
2 2
- 108*a33*b33*k10*n1 *u1*v1 - 108*a33*b33*k10*n1 *u2*v2
2
+ 108*a33*b33*k10*n1*n3*u1*v3 - 108*a33*b33*k10*n3 *u1*v1
2 3 2
- 108*a33*b33*k10*n3 *u2*v2 - 54*a33*k10*n1 *u1 - 54*a33*k10*n1 *n3*u3
2 3 4 4
+ 54*a33*k10*n1*n3 *u1 + 54*a33*k10*n3 *u3 - 81*b33 *k10*v1
4 2 2 4 2 2 4 4
- 162*b33 *k10*v1 *v2 - 81*b33 *k10*v1 *v3 - 81*b33 *k10*v2
4 2 2 3 3 3 2
- 81*b33 *k10*v2 *v3 - 81*b33 *k10*n1*v1 - 81*b33 *k10*n1*v1*v2
3 2 3 2 3 2
- 162*b33 *k10*n1*v1*v3 - 162*b33 *k10*n3*v1 *v3 - 162*b33 *k10*n3*v2 *v3
3 3 2 2 2 2 2 2
- 162*b33 *k10*n3*v3 + 162*b33 *k10*n1 *v1 + 81*b33 *k10*n1 *v2
2 3 2
+ 162*b33 *k10*n1*n3*v1*v3 + 81*b33*k10*n1 *v1 + 162*b33*k10*n1 *n3*v3
2 4
+ 81*b33*k10*n1*n3 *v1)/(16*a33 )
And again in machine readable form:
HAM=(4*a33**2*u1**2 + 4*a33**2*u2**2 + 12*a33**2*u3**2 + 12*a33*b33*u3*v3 + 12*
a33*n1*u1 + 12*a33*n3*u3 - 9*b33**2*v3**2 + 36*b33*n1*v1 + 18*b33*n3*v3)/(12*a33
)$
INT=(16*a33**4*k10*u1**2*u3**2 + 16*a33**4*k10*u2**2*u3**2 + 16*a33**4*k23*u1*u3
**2*v1 + 16*a33**4*k23*u2*u3**2*v2 + 16*a33**4*k23*u3**3*v3 + 16*a33**4*k26*u1**
2*v1**2 + 16*a33**4*k26*u1**2*v2**2 + 16*a33**4*k26*u1**2*v3**2 + 16*a33**4*k26*
u2**2*v1**2 + 16*a33**4*k26*u2**2*v2**2 + 16*a33**4*k26*u2**2*v3**2 + 16*a33**4*
q1*u1**2 + 16*a33**4*q1*u2**2 + 48*a33**3*b33*k10*u3**3*v3 - 48*a33**3*b33*k26*
u1*v1**3 - 48*a33**3*b33*k26*u1*v1*v2**2 - 48*a33**3*b33*k26*u1*v1*v3**2 - 48*
a33**3*b33*k26*u2*v1**2*v2 - 48*a33**3*b33*k26*u2*v2**3 - 48*a33**3*b33*k26*u2*
v2*v3**2 - 48*a33**3*b33*q1*u1*v1 - 48*a33**3*b33*q1*u2*v2 + 24*a33**3*k10*n1*u1
**3 + 24*a33**3*k10*n1*u1*u2**2 - 24*a33**3*k10*n1*u1*u3**2 + 24*a33**3*k10*n3*
u1**2*u3 + 24*a33**3*k10*n3*u2**2*u3 - 24*a33**3*k10*n3*u3**3 + 24*a33**3*k23*n1
*u1**2*v1 + 24*a33**3*k23*n1*u1*u2*v2 + 24*a33**3*k23*n1*u1*u3*v3 + 24*a33**3*
k23*n3*u1*u3*v1 + 24*a33**3*k23*n3*u2*u3*v2 + 24*a33**3*k23*n3*u3**2*v3 - 24*a33
**3*k26*n1*u1*v1**2 - 24*a33**3*k26*n1*u1*v2**2 - 24*a33**3*k26*n1*u1*v3**2 - 24
*a33**3*k26*n3*u3*v1**2 - 24*a33**3*k26*n3*u3*v2**2 - 24*a33**3*k26*n3*u3*v3**2
- 24*a33**3*n1*q1*u1 - 24*a33**3*n3*q1*u3 - 36*a33**2*b33**2*k10*u1**2*v2**2 -
36*a33**2*b33**2*k10*u1**2*v3**2 + 72*a33**2*b33**2*k10*u1*u2*v1*v2 + 72*a33**2*
b33**2*k10*u1*u3*v1*v3 - 36*a33**2*b33**2*k10*u2**2*v1**2 - 36*a33**2*b33**2*k10
*u2**2*v3**2 + 72*a33**2*b33**2*k10*u2*u3*v2*v3 + 36*a33**2*b33**2*k26*v1**4 +
72*a33**2*b33**2*k26*v1**2*v2**2 + 36*a33**2*b33**2*k26*v1**2*v3**2 + 36*a33**2*
b33**2*k26*v2**4 + 36*a33**2*b33**2*k26*v2**2*v3**2 + 36*a33**2*b33**2*q1*v1**2
+ 36*a33**2*b33**2*q1*v2**2 + 36*a33**2*b33*k10*n1*u1**2*v1 + 72*a33**2*b33*k10*
n1*u1*u3*v3 + 36*a33**2*b33*k10*n1*u2**2*v1 + 36*a33**2*b33*k10*n1*u3**2*v1 +
144*a33**2*b33*k10*n3*u3**2*v3 + 36*a33**2*b33*k23*n1*u1*v1**2 + 36*a33**2*b33*
k23*n1*u2*v1*v2 + 36*a33**2*b33*k23*n1*u3*v1*v3 + 36*a33**2*b33*k26*n1*v1**3 +
36*a33**2*b33*k26*n1*v1*v2**2 + 36*a33**2*b33*k26*n1*v1*v3**2 + 72*a33**2*b33*
k26*n3*v1**2*v3 + 72*a33**2*b33*k26*n3*v2**2*v3 + 72*a33**2*b33*k26*n3*v3**3 +
36*a33**2*b33*n1*q1*v1 + 72*a33**2*b33*n3*q1*v3 + 36*a33**2*k10*n1**2*u2**2 - 72
*a33**2*k10*n1*n3*u1*u3 + 108*a33*b33**3*k10*u1*v1**3 + 108*a33*b33**3*k10*u1*v1
*v2**2 + 108*a33*b33**3*k10*u1*v1*v3**2 + 108*a33*b33**3*k10*u2*v1**2*v2 + 108*
a33*b33**3*k10*u2*v2**3 + 108*a33*b33**3*k10*u2*v2*v3**2 + 54*a33*b33**2*k10*n1*
u1*v1**2 + 54*a33*b33**2*k10*n1*u1*v2**2 + 108*a33*b33**2*k10*n1*u3*v1*v3 + 54*
a33*b33**2*k10*n3*u3*v1**2 + 54*a33*b33**2*k10*n3*u3*v2**2 - 108*a33*b33*k10*n1
**2*u1*v1 - 108*a33*b33*k10*n1**2*u2*v2 + 108*a33*b33*k10*n1*n3*u1*v3 - 108*a33*
b33*k10*n3**2*u1*v1 - 108*a33*b33*k10*n3**2*u2*v2 - 54*a33*k10*n1**3*u1 - 54*a33
*k10*n1**2*n3*u3 + 54*a33*k10*n1*n3**2*u1 + 54*a33*k10*n3**3*u3 - 81*b33**4*k10*
v1**4 - 162*b33**4*k10*v1**2*v2**2 - 81*b33**4*k10*v1**2*v3**2 - 81*b33**4*k10*
v2**4 - 81*b33**4*k10*v2**2*v3**2 - 81*b33**3*k10*n1*v1**3 - 81*b33**3*k10*n1*v1
*v2**2 - 162*b33**3*k10*n1*v1*v3**2 - 162*b33**3*k10*n3*v1**2*v3 - 162*b33**3*
k10*n3*v2**2*v3 - 162*b33**3*k10*n3*v3**3 + 162*b33**2*k10*n1**2*v1**2 + 81*b33
**2*k10*n1**2*v2**2 + 162*b33**2*k10*n1*n3*v1*v3 + 81*b33*k10*n1**3*v1 + 162*b33
*k10*n1**2*n3*v3 + 81*b33*k10*n1*n3**2*v1)/(16*a33**4)$