Solution 48 to problem e3c2new
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
1
a22=---*a33
3
b22=0
b31=0
b32=0
1
---*a33*m1
3
b33=------------
n1
c12=0
c13=0
c22=0
c23=0
1 2
- ----*a33*m1
12
c33=-----------------
2
n1
n2=0
n3=0
m2=0
m3=0
r6=0
r5=0
3 2 27 2
---*a33 *m1*q1 + ----*k10*m1*n1
4 16
r4=----------------------------------
3
a33
r3=0
r2=0
3 2 27 3
- ---*a33 *n1*q1 - ----*k10*n1
2 8
r1=----------------------------------
3
a33
q20=0
q19=0
q17=0
q16=0
1 2 2 9 2 2
---*a33 *m1 *q1 + ----*k10*m1 *n1
4 16
q15=------------------------------------
2 2
a33 *n1
q14=0
q13=0
2 9 2
- a33 *m1*q1 - ---*k10*m1*n1
4
q12=--------------------------------
2
a33 *n1
q11=0
1 2 2 9 2 2
---*a33 *m1 *q1 + ---*k10*m1 *n1
4 8
q10=-----------------------------------
2 2
a33 *n1
q9=0
q8=0
2 9 2
- a33 *m1*q1 - ---*k10*m1*n1
4
q7=--------------------------------
2
a33 *n1
q5=0
q4=0
2 9 2
a33 *q1 + ---*k10*n1
4
q3=-----------------------
2
a33
q2=0
p56=0
p55=0
3 3 3 2
- ---*k10*m1 + ---*k26*m1*n1
8 4
p54=---------------------------------
2
a33*n1
p53=0
p52=0
3
- ---*k26*n1
2
p51=---------------
a33
p50=0
p49=0
p48=0
p47=0
p46=0
p45=0
3 2 3
---*k10*m1 + ---*k23*m1*n1
4 4
p44=-----------------------------
a33*n1
p43=0
p42=0
p41=0
p40=0
3 3
---*k10*m1 + ---*k23*n1
2 2
p39=-------------------------
a33
p38=0
p37=0
p36=0
p35=0
3 3 3 2
- ----*k10*m1 + ---*k26*m1*n1
16 4
p34=----------------------------------
2
a33*n1
p33=0
p32=0
3 2 3 2
---*k10*m1 - ---*k26*n1
8 2
p31=---------------------------
a33*n1
p30=0
p29=0
3
---*k23*m1
4
p28=------------
a33
p27=0
p26=0
p25=0
p24=0
p23=0
3
---*k23*n1
2
p22=------------
a33
p21=0
3 3 3 2
- ----*k10*m1 + ---*k26*m1*n1
16 4
p20=----------------------------------
2
a33*n1
p19=0
p18=0
3 2 3 3 2
---*k10*m1 + ---*k23*m1*n1 - ---*k26*n1
8 4 2
p17=-------------------------------------------
a33*n1
3
---*k10*m1
4
p16=------------
a33
p15=0
p14=0
3
---*k10*m1
4
p13=------------
a33
p12=0
3 3
---*k10*m1 + ---*k23*n1
4 2
p11=-------------------------
a33
p10=0
p9=0
3
- ---*k10*n1
2
p8=---------------
a33
p7=0
p6=0
p5=0
p4=0
3
---*k10*n1
2
p3=------------
a33
p2=0
3
---*k10*n1
2
p1=------------
a33
k125=0
k124=0
k122=0
k121=0
1 4 1 2 2
- ----*k10*m1 + ---*k26*m1 *n1
16 4
k120=-----------------------------------
4
n1
k119=0
k118=0
1 3 2
---*k10*m1 - k26*m1*n1
4
k117=--------------------------
3
n1
k116=0
1 4 1 2 2
- ----*k10*m1 + ---*k26*m1 *n1
16 4
k115=-----------------------------------
4
n1
k114=0
k113=0
1 3 2
---*k10*m1 - k26*m1*n1
4
k112=--------------------------
3
n1
k110=0
k109=0
1 2 2
- ---*k10*m1 + k26*n1
4
k108=--------------------------
2
n1
k107=0
1 2 2
- ---*k10*m1 + k26*n1
4
k106=--------------------------
2
n1
k105=0
k104=0
k103=0
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
1 2
---*k10*m1
2
k95=-------------
2
n1
k94=0
k93=0
k92=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
k85=0
1 2
---*k10*m1
2
k84=-------------
2
n1
k83=0
k82=0
k81=0
k10*m1 + k23*n1
k80=-----------------
n1
k79=0
k78=0
k77=0
k76=0
k75=0
k74=0
k73=0
k72=0
k71=0
1 4 1 2 2
- ----*k10*m1 + ---*k26*m1 *n1
16 4
k70=-----------------------------------
4
n1
k69=0
k68=0
1 3 2
---*k10*m1 - k26*m1*n1
4
k67=--------------------------
3
n1
k66=0
1 4 1 2 2
- ---*k10*m1 + ---*k26*m1 *n1
8 2
k65=----------------------------------
4
n1
k64=0
k63=0
1 3 2
---*k10*m1 - k26*m1*n1
4
k62=--------------------------
3
n1
k61=0
k60=0
k59=0
k58=k26
k57=0
1 2 2
- ---*k10*m1 + k26*n1
4
k56=--------------------------
2
n1
k55=0
k54=0
1 3 2
---*k10*m1 - k26*m1*n1
4
k53=--------------------------
3
n1
k52=0
k51=0
k50=0
k49=0
k48=0
1 2
---*k10*m1
2
k47=-------------
2
n1
k46=0
k45=0
k44=k23
k43=0
k42=0
k41=0
k40=0
k38=0
k37=0
k36=0
1 4 1 2 2
- ----*k10*m1 + ---*k26*m1 *n1
16 4
k35=-----------------------------------
4
n1
k34=0
k33=0
1 3 2
---*k10*m1 - k26*m1*n1
4
k32=--------------------------
3
n1
k30=0
k29=0
1 2 2
- ---*k10*m1 + k26*n1
4
k28=--------------------------
2
n1
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:
m1,q1,k26,k23,k10,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=( - a33*m1 *v3 + 4*a33*m1*n1*u3*v3 + 4*a33*n1 *u1 + 4*a33*n1 *u2
2 2 2 3 2
+ 12*a33*n1 *u3 + 12*m1*n1 *v1 + 12*n1 *u1)/(12*n1 )
3 4 4 3 4 2 2 3 4 2 2
INT=( - a33 *k10*m1 *v1 - 2*a33 *k10*m1 *v1 *v2 - a33 *k10*m1 *v1 *v3
3 4 4 3 4 2 2 3 3 3
- a33 *k10*m1 *v2 - a33 *k10*m1 *v2 *v3 + 4*a33 *k10*m1 *n1*u1*v1
3 3 2 3 3 2
+ 4*a33 *k10*m1 *n1*u1*v1*v2 + 4*a33 *k10*m1 *n1*u1*v1*v3
3 3 2 3 3 3
+ 4*a33 *k10*m1 *n1*u2*v1 *v2 + 4*a33 *k10*m1 *n1*u2*v2
3 3 2 3 2 2 2 2
+ 4*a33 *k10*m1 *n1*u2*v2*v3 - 4*a33 *k10*m1 *n1 *u1 *v2
3 2 2 2 2 3 2 2
- 4*a33 *k10*m1 *n1 *u1 *v3 + 8*a33 *k10*m1 *n1 *u1*u2*v1*v2
3 2 2 3 2 2 2 2
+ 8*a33 *k10*m1 *n1 *u1*u3*v1*v3 - 4*a33 *k10*m1 *n1 *u2 *v1
3 2 2 2 2 3 2 2
- 4*a33 *k10*m1 *n1 *u2 *v3 + 8*a33 *k10*m1 *n1 *u2*u3*v2*v3
3 3 3 3 4 2 2
+ 16*a33 *k10*m1*n1 *u3 *v3 + 16*a33 *k10*n1 *u1 *u3
3 4 2 2 3 4 2
+ 16*a33 *k10*n1 *u2 *u3 + 16*a33 *k23*n1 *u1*u3 *v1
3 4 2 3 4 3
+ 16*a33 *k23*n1 *u2*u3 *v2 + 16*a33 *k23*n1 *u3 *v3
3 2 2 4 3 2 2 2 2
+ 4*a33 *k26*m1 *n1 *v1 + 8*a33 *k26*m1 *n1 *v1 *v2
3 2 2 2 2 3 2 2 4
+ 4*a33 *k26*m1 *n1 *v1 *v3 + 4*a33 *k26*m1 *n1 *v2
3 2 2 2 2 3 3 3
+ 4*a33 *k26*m1 *n1 *v2 *v3 - 16*a33 *k26*m1*n1 *u1*v1
3 3 2 3 3 2
- 16*a33 *k26*m1*n1 *u1*v1*v2 - 16*a33 *k26*m1*n1 *u1*v1*v3
3 3 2 3 3 3
- 16*a33 *k26*m1*n1 *u2*v1 *v2 - 16*a33 *k26*m1*n1 *u2*v2
3 3 2 3 4 2 2
- 16*a33 *k26*m1*n1 *u2*v2*v3 + 16*a33 *k26*n1 *u1 *v1
3 4 2 2 3 4 2 2
+ 16*a33 *k26*n1 *u1 *v2 + 16*a33 *k26*n1 *u1 *v3
3 4 2 2 3 4 2 2
+ 16*a33 *k26*n1 *u2 *v1 + 16*a33 *k26*n1 *u2 *v2
3 4 2 2 3 2 2 2 3 2 2 2
+ 16*a33 *k26*n1 *u2 *v3 + 4*a33 *m1 *n1 *q1*v1 + 4*a33 *m1 *n1 *q1*v2
3 3 3 3 3 4 2
- 16*a33 *m1*n1 *q1*u1*v1 - 16*a33 *m1*n1 *q1*u2*v2 + 16*a33 *n1 *q1*u1
3 4 2 2 3 2 3 2 3 2 2
+ 16*a33 *n1 *q1*u2 - 3*a33 *k10*m1 *n1 *v1 - 3*a33 *k10*m1 *n1 *v1*v2
2 3 2 2 2 2 3 2
- 6*a33 *k10*m1 *n1 *v1*v3 + 6*a33 *k10*m1 *n1 *u1*v1
2 2 3 2 2 2 3
+ 6*a33 *k10*m1 *n1 *u1*v2 + 12*a33 *k10*m1 *n1 *u3*v1*v3
2 4 2 2 4
+ 12*a33 *k10*m1*n1 *u1 *v1 + 24*a33 *k10*m1*n1 *u1*u3*v3
2 4 2 2 4 2
+ 12*a33 *k10*m1*n1 *u2 *v1 + 12*a33 *k10*m1*n1 *u3 *v1
2 5 3 2 5 2 2 5 2
+ 24*a33 *k10*n1 *u1 + 24*a33 *k10*n1 *u1*u2 - 24*a33 *k10*n1 *u1*u3
2 4 2 2 4
+ 12*a33 *k23*m1*n1 *u1*v1 + 12*a33 *k23*m1*n1 *u2*v1*v2
2 4 2 5 2
+ 12*a33 *k23*m1*n1 *u3*v1*v3 + 24*a33 *k23*n1 *u1 *v1
2 5 2 5
+ 24*a33 *k23*n1 *u1*u2*v2 + 24*a33 *k23*n1 *u1*u3*v3
2 4 3 2 4 2
+ 12*a33 *k26*m1*n1 *v1 + 12*a33 *k26*m1*n1 *v1*v2
2 4 2 2 5 2
+ 12*a33 *k26*m1*n1 *v1*v3 - 24*a33 *k26*n1 *u1*v1
2 5 2 2 5 2 2 4
- 24*a33 *k26*n1 *u1*v2 - 24*a33 *k26*n1 *u1*v3 + 12*a33 *m1*n1 *q1*v1
2 5 2 4 2 2 4 2
- 24*a33 *n1 *q1*u1 + 18*a33*k10*m1 *n1 *v1 + 9*a33*k10*m1 *n1 *v2
5 5 6 2
- 36*a33*k10*m1*n1 *u1*v1 - 36*a33*k10*m1*n1 *u2*v2 + 36*a33*k10*n1 *u2
6 7 3 4
+ 27*k10*m1*n1 *v1 - 54*k10*n1 *u1)/(16*a33 *n1 )
And again in machine readable form:
HAM=( - a33*m1**2*v3**2 + 4*a33*m1*n1*u3*v3 + 4*a33*n1**2*u1**2 + 4*a33*n1**2*u2
**2 + 12*a33*n1**2*u3**2 + 12*m1*n1**2*v1 + 12*n1**3*u1)/(12*n1**2)$
INT=( - a33**3*k10*m1**4*v1**4 - 2*a33**3*k10*m1**4*v1**2*v2**2 - a33**3*k10*m1
**4*v1**2*v3**2 - a33**3*k10*m1**4*v2**4 - a33**3*k10*m1**4*v2**2*v3**2 + 4*a33
**3*k10*m1**3*n1*u1*v1**3 + 4*a33**3*k10*m1**3*n1*u1*v1*v2**2 + 4*a33**3*k10*m1
**3*n1*u1*v1*v3**2 + 4*a33**3*k10*m1**3*n1*u2*v1**2*v2 + 4*a33**3*k10*m1**3*n1*
u2*v2**3 + 4*a33**3*k10*m1**3*n1*u2*v2*v3**2 - 4*a33**3*k10*m1**2*n1**2*u1**2*v2
**2 - 4*a33**3*k10*m1**2*n1**2*u1**2*v3**2 + 8*a33**3*k10*m1**2*n1**2*u1*u2*v1*
v2 + 8*a33**3*k10*m1**2*n1**2*u1*u3*v1*v3 - 4*a33**3*k10*m1**2*n1**2*u2**2*v1**2
- 4*a33**3*k10*m1**2*n1**2*u2**2*v3**2 + 8*a33**3*k10*m1**2*n1**2*u2*u3*v2*v3 +
16*a33**3*k10*m1*n1**3*u3**3*v3 + 16*a33**3*k10*n1**4*u1**2*u3**2 + 16*a33**3*
k10*n1**4*u2**2*u3**2 + 16*a33**3*k23*n1**4*u1*u3**2*v1 + 16*a33**3*k23*n1**4*u2
*u3**2*v2 + 16*a33**3*k23*n1**4*u3**3*v3 + 4*a33**3*k26*m1**2*n1**2*v1**4 + 8*
a33**3*k26*m1**2*n1**2*v1**2*v2**2 + 4*a33**3*k26*m1**2*n1**2*v1**2*v3**2 + 4*
a33**3*k26*m1**2*n1**2*v2**4 + 4*a33**3*k26*m1**2*n1**2*v2**2*v3**2 - 16*a33**3*
k26*m1*n1**3*u1*v1**3 - 16*a33**3*k26*m1*n1**3*u1*v1*v2**2 - 16*a33**3*k26*m1*n1
**3*u1*v1*v3**2 - 16*a33**3*k26*m1*n1**3*u2*v1**2*v2 - 16*a33**3*k26*m1*n1**3*u2
*v2**3 - 16*a33**3*k26*m1*n1**3*u2*v2*v3**2 + 16*a33**3*k26*n1**4*u1**2*v1**2 +
16*a33**3*k26*n1**4*u1**2*v2**2 + 16*a33**3*k26*n1**4*u1**2*v3**2 + 16*a33**3*
k26*n1**4*u2**2*v1**2 + 16*a33**3*k26*n1**4*u2**2*v2**2 + 16*a33**3*k26*n1**4*u2
**2*v3**2 + 4*a33**3*m1**2*n1**2*q1*v1**2 + 4*a33**3*m1**2*n1**2*q1*v2**2 - 16*
a33**3*m1*n1**3*q1*u1*v1 - 16*a33**3*m1*n1**3*q1*u2*v2 + 16*a33**3*n1**4*q1*u1**
2 + 16*a33**3*n1**4*q1*u2**2 - 3*a33**2*k10*m1**3*n1**2*v1**3 - 3*a33**2*k10*m1
**3*n1**2*v1*v2**2 - 6*a33**2*k10*m1**3*n1**2*v1*v3**2 + 6*a33**2*k10*m1**2*n1**
3*u1*v1**2 + 6*a33**2*k10*m1**2*n1**3*u1*v2**2 + 12*a33**2*k10*m1**2*n1**3*u3*v1
*v3 + 12*a33**2*k10*m1*n1**4*u1**2*v1 + 24*a33**2*k10*m1*n1**4*u1*u3*v3 + 12*a33
**2*k10*m1*n1**4*u2**2*v1 + 12*a33**2*k10*m1*n1**4*u3**2*v1 + 24*a33**2*k10*n1**
5*u1**3 + 24*a33**2*k10*n1**5*u1*u2**2 - 24*a33**2*k10*n1**5*u1*u3**2 + 12*a33**
2*k23*m1*n1**4*u1*v1**2 + 12*a33**2*k23*m1*n1**4*u2*v1*v2 + 12*a33**2*k23*m1*n1
**4*u3*v1*v3 + 24*a33**2*k23*n1**5*u1**2*v1 + 24*a33**2*k23*n1**5*u1*u2*v2 + 24*
a33**2*k23*n1**5*u1*u3*v3 + 12*a33**2*k26*m1*n1**4*v1**3 + 12*a33**2*k26*m1*n1**
4*v1*v2**2 + 12*a33**2*k26*m1*n1**4*v1*v3**2 - 24*a33**2*k26*n1**5*u1*v1**2 - 24
*a33**2*k26*n1**5*u1*v2**2 - 24*a33**2*k26*n1**5*u1*v3**2 + 12*a33**2*m1*n1**4*
q1*v1 - 24*a33**2*n1**5*q1*u1 + 18*a33*k10*m1**2*n1**4*v1**2 + 9*a33*k10*m1**2*
n1**4*v2**2 - 36*a33*k10*m1*n1**5*u1*v1 - 36*a33*k10*m1*n1**5*u2*v2 + 36*a33*k10
*n1**6*u2**2 + 27*k10*m1*n1**6*v1 - 54*k10*n1**7*u1)/(16*a33**3*n1**4)$