Solution 11 to problem e3quant
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Expressions
The solution is given through the following expressions:
1
a22=---*a33
2
b22=0
b31=0
b33=0
c12=0
c13=0
c22=0
c23=0
1 2
- ---*b32
2
c33=-------------
a33
n1=0
n2=0
a33*m2
n3=--------
b32
m3=0
r6=0
3 1
---*b32*k28*m2 - ---*k28*m1*m2
8 2
r5=--------------------------------
2
b32
3 1 2
---*b32*k28*m1 + ---*k28*m2
8 2
r4=------------------------------
2
b32
1
---*a33*k28*m2
8
r3=----------------
2
b32
r2=0
r1=0
q20=0
15 2 1 1 2 1 2
----*b32 *k28 + ---*b32*k28*m1 - ---*k28*m1 - ---*k28*m2
16 2 2 2
q19=------------------------------------------------------------
a33*b32
q18=0
q17=0
15 2 1 1 2 1 2
----*b32 *k28 + ---*b32*k28*m1 - ---*k28*m1 - ---*k28*m2
16 2 2 2
q16=------------------------------------------------------------
a33*b32
11 1
----*b32*k28 - ---*k28*m1
8 2
q14=---------------------------
b32
q13=0
2 2
b32 *k28 - b32*k28*m1 - k28*m2
q11=---------------------------------
2
b32
q10=0
q9=0
q8=0
7 2 1 2
----*a33*b32 *k28 + ---*a33*k28*m2
16 2
q7=-------------------------------------
3
b32
2*b32*k28*m2 - k28*m1*m2
q6=--------------------------
2
b32
q5=0
q4=0
q3=0
q2=0
1 2 1 2
- ----*a33*b32 *k28 + ---*a33*k28*m2
16 2
q1=----------------------------------------
3
b32
p56=0
1
---*b32*k28*m2
2
p55=----------------
2
a33
p54=0
p53=0
1
---*b32*k28*m1
2
p52=----------------
2
a33
p51=0
p50=0
p49=0
p48=0
p47=0
1
---*k28*m2
2
p46=------------
a33
p45=0
p44=0
p43=0
p42=0
p41=0
p40=0
p39=0
1
p38= - ---*k28
2
p37=0
p36=0
- k28*m2
p35=-----------
a33
p34=0
3
- ---*b32*k28
2
p33=----------------
a33
p32=0
p31=0
- k28*m2
p30=-----------
b32
p29=0
p28=0
p27=0
p26=0
1
---*k28*m2
2
p25=------------
b32
1
b32*k28 - ---*k28*m1
2
p24=----------------------
b32
1
---*a33*k28*m2
2
p23=----------------
2
b32
p22=0
p21=0
3
---*b32*k28
2
p20=-------------
a33
p19=0
- k28*m2
p18=-----------
a33
p17=0
p16=0
p15=k28
p14=0
p13=0
p12=0
p11=0
- 2*b32*k28 + k28*m1
p10=-----------------------
b32
k28*m2
p9=--------
b32
1
---*a33*k28
2
p8=-------------
b32
p7=0
p6=0
1
- ---*k28*m2
2
p5=---------------
b32
1
- b32*k28 + ---*k28*m1
2
p4=-------------------------
b32
1
---*a33*k28*m2
2
p3=----------------
2
b32
p2=0
p1=0
k125=0
1 3
---*b32 *k28
4
k124=--------------
3
a33
k123=0
1 3
---*b32 *k28
8
k122=--------------
3
a33
k121=0
k120=0
k119=0
k118=0
1 3
---*b32 *k28
4
k117=--------------
3
a33
k116=0
1 3
---*b32 *k28
4
k115=--------------
3
a33
k114=0
k113=0
1 3
---*b32 *k28
8
k112=--------------
3
a33
1 2
---*b32 *k28
2
k110=--------------
2
a33
k109=0
k108=0
k107=0
k106=0
k105=0
k104=0
k103=0
k102=0
k100=0
k99=0
k98=0
k97=0
k95=0
k94=0
k93=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
k85=0
k84=0
k83=0
k82=0
k81=0
- b32*k28
k80=------------
a33
k79=0
k78=0
k77=0
k76=0
k75=0
k74=0
k73=0
k72=0
1
---*b32*k28
4
k71=-------------
a33
k70=0
k69=0
k68=0
k67=0
k66=0
k65=0
1
k64=---*k28
2
k63=0
k62=0
k61=0
k59=0
k58=0
1
- ---*a33*k28
8
k57=----------------
b32
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
- b32*k28
k43=------------
a33
k42=0
k41=0
k40=0
k39=0
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=0
k31=0
k30=0
k29=0
k27=0
k26=0
k25=0
k24=0
k23=0
k22=0
1
---*b32*k28
4
k21=-------------
a33
k20=0
k19=0
k18=0
k17=0
k16=0
k15=0
1
k14= - ---*k28
2
k13=0
k12=0
k11=0
k10=0
k9=0
k8=0
1
- ---*a33*k28
4
k7=----------------
b32
k6=0
k5=0
k4=0
k3=0
k2=0
1
- ---*a33*k28
8
k1=----------------
b32
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:
m1,m2,k28,b32,a33
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2 2 2 2 2
HAM=(a33 *b32*u1 + a33 *b32*u2 + 2*a33 *b32*u3 + 2*a33 *m2*u3
2 3 2
+ 2*a33*b32 *u3*v2 + 2*a33*b32*m1*v1 + 2*a33*b32*m2*v2 - b32 *v3 )/(2*a33
*b32)
4 2 4 4 2 2 4 2 2
INT=(k28*( - 2*a33 *b32 *u1 - 4*a33 *b32 *u1 *u2*u3 - a33 *b32 *u1
4 2 2 4 2 2 4 2
- 2*a33 *b32 *u1*u2*v2 + 8*a33 *b32 *u1*u3 + 7*a33 *b32 *u1*v1
4 2 4 2 4
+ 8*a33 *b32*m2*u1*u2 + 8*a33 *b32*m2*u2 *v2 + 2*a33 *b32*m2*u3
4 2 2 4 2 3 3
+ 8*a33 *m2 *u1 + 8*a33 *m2 *u1*v1 + 16*a33 *b32 *u1*u3*v2
3 3 3 3 3 3 3 2 2
+ 16*a33 *b32 *u1*v2 - 16*a33 *b32 *u2 + 16*a33 *b32 *u2 *v1
3 3 2 3 3 3 3 3
- 8*a33 *b32 *u2 *v3 - 8*a33 *b32 *u2*u3 + 16*a33 *b32 *u2*u3*v1
3 3 3 3 3 2 3 3
- 32*a33 *b32 *u3 + 8*a33 *b32 *u3*v1*v2 + 22*a33 *b32 *v1*v2
3 2 3 2
- 8*a33 *b32 *m1*u1*u3*v2 - 16*a33 *b32 *m1*u1*v2
3 2 3 3 2 3 3 2
+ 8*a33 *b32 *m1*u2 + 16*a33 *b32 *m1*u3 - 8*a33 *b32 *m1*v1*v2
3 2 3 2 2 3 2 2
+ 6*a33 *b32 *m1*v1 - 8*a33 *b32 *m2*u1 *u3 + 16*a33 *b32 *m2*u2*u3
3 2 3 2 2
+ 8*a33 *b32 *m2*u2*u3*v2 + 32*a33 *b32 *m2*u3
3 2 2 3 2 3 2
- 16*a33 *b32 *m2*v1 *v2 + 6*a33 *b32 *m2*v2 - 16*a33 *b32*m1*m2*u3
3 3 2 3 2
- 8*a33 *b32*m1*m2*v2 - 16*a33 *b32*m2 *u1*v2 + 8*a33 *b32*m2 *v1
2 4 3 2 4
+ 4*a33 *b32 *u1 *v3 + 4*a33 *b32 *u1*u2*u3*v1
2 4 2 2 4 2 4 3
- 16*a33 *b32 *u1*u3 *v2 + 15*a33 *b32 *u1*v3 - 16*a33 *b32 *u3 *v3
2 4 2 2 4 3 2 4
- 24*a33 *b32 *u3*v2 + 24*a33 *b32 *v1 + 15*a33 *b32 *v1*v3
2 3 2 3
+ 8*a33 *b32 *m1*u1*v3 + 8*a33 *b32 *m1*v1*v3
2 3 2 3 2
+ 8*a33 *b32 *m2*u1*v2*v3 - 16*a33 *b32 *m2*u2*v1
2 3 3 2 2 2 2 2 2
- 16*a33 *b32 *m2*v2 - 8*a33 *b32 *m1 *u1*v3 - 8*a33 *b32 *m1 *v1*v3
2 2 2 2 2 2
- 8*a33 *b32 *m2 *u1*v3 - 8*a33 *b32 *m2 *v1*v3
5 2 4 2 4 2
+ 8*a33*b32 *u2*u3*v3 + 8*a33*b32 *m1*u2*v3 + 8*a33*b32 *m2*v2*v3
6 2 6 2 6 3
+ 2*b32 *u1*v1*v3 + 4*b32 *u2*v2*v3 + 2*b32 *u2*v3
6 2 2 6 3 3 3
+ 4*b32 *v1 *v3 + 4*b32 *v1*v3 ))/(16*a33 *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**2*m2*u3 +
2*a33*b32**2*u3*v2 + 2*a33*b32*m1*v1 + 2*a33*b32*m2*v2 - b32**3*v3**2)/(2*a33*
b32)$
INT=(k28*( - 2*a33**4*b32**2*u1**4 - 4*a33**4*b32**2*u1**2*u2*u3 - a33**4*b32**2
*u1**2 - 2*a33**4*b32**2*u1*u2*v2**2 + 8*a33**4*b32**2*u1*u3**2 + 7*a33**4*b32**
2*u1*v1 + 8*a33**4*b32*m2*u1*u2**2 + 8*a33**4*b32*m2*u2**2*v2 + 2*a33**4*b32*m2*
u3 + 8*a33**4*m2**2*u1**2 + 8*a33**4*m2**2*u1*v1 + 16*a33**3*b32**3*u1*u3*v2 +
16*a33**3*b32**3*u1*v2 - 16*a33**3*b32**3*u2**3 + 16*a33**3*b32**3*u2**2*v1**2 -
8*a33**3*b32**3*u2**2*v3 - 8*a33**3*b32**3*u2*u3**3 + 16*a33**3*b32**3*u2*u3*v1
- 32*a33**3*b32**3*u3**3 + 8*a33**3*b32**3*u3*v1*v2**2 + 22*a33**3*b32**3*v1*v2
- 8*a33**3*b32**2*m1*u1*u3*v2 - 16*a33**3*b32**2*m1*u1*v2 + 8*a33**3*b32**2*m1*
u2**3 + 16*a33**3*b32**2*m1*u3**3 - 8*a33**3*b32**2*m1*v1*v2 + 6*a33**3*b32**2*
m1*v1 - 8*a33**3*b32**2*m2*u1**2*u3 + 16*a33**3*b32**2*m2*u2*u3**2 + 8*a33**3*
b32**2*m2*u2*u3*v2 + 32*a33**3*b32**2*m2*u3**2 - 16*a33**3*b32**2*m2*v1**2*v2 +
6*a33**3*b32**2*m2*v2 - 16*a33**3*b32*m1*m2*u3**2 - 8*a33**3*b32*m1*m2*v2 - 16*
a33**3*b32*m2**2*u1*v2 + 8*a33**3*b32*m2**2*v1 + 4*a33**2*b32**4*u1**3*v3 + 4*
a33**2*b32**4*u1*u2*u3*v1 - 16*a33**2*b32**4*u1*u3**2*v2 + 15*a33**2*b32**4*u1*
v3 - 16*a33**2*b32**4*u3**3*v3 - 24*a33**2*b32**4*u3*v2**2 + 24*a33**2*b32**4*v1
**3 + 15*a33**2*b32**4*v1*v3 + 8*a33**2*b32**3*m1*u1*v3 + 8*a33**2*b32**3*m1*v1*
v3 + 8*a33**2*b32**3*m2*u1*v2*v3 - 16*a33**2*b32**3*m2*u2*v1**2 - 16*a33**2*b32
**3*m2*v2**3 - 8*a33**2*b32**2*m1**2*u1*v3 - 8*a33**2*b32**2*m1**2*v1*v3 - 8*a33
**2*b32**2*m2**2*u1*v3 - 8*a33**2*b32**2*m2**2*v1*v3 + 8*a33*b32**5*u2*u3*v3**2
+ 8*a33*b32**4*m1*u2*v3**2 + 8*a33*b32**4*m2*v2*v3**2 + 2*b32**6*u1*v1*v3**2 + 4
*b32**6*u2*v2*v3**2 + 2*b32**6*u2*v3**3 + 4*b32**6*v1**2*v3**2 + 4*b32**6*v1*v3
**3))/(16*a33**3*b32**3)$