Solution 50 to problem e3c2new
Expressions |
Parameters |
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Back to problem e3c2new
Expressions
The solution is given through the following expressions:
b22=0
b31=0
b32=0
2
- 2*a22 *m1 + 2*a22*a33*m1
b33=-----------------------------
a22*n1 + a33*n1
c12=0
c13=0
c22=0
c23=0
3 2 2 2 2 2
- a22 *m1 + 2*a22 *a33*m1 - a22*a33 *m1
c33=---------------------------------------------
2 2 2 2 2
a22 *n1 + 2*a22*a33*n1 + a33 *n1
n2=0
n3=0
m2=0
m3=0
r6=0
r5=0
r4=0
r3=0
r2=0
r1=0
q20=0
q19=0
q17=0
q16=0
q15=0
q14=0
q13=0
q12=0
q11=0
q10=0
q9=0
q8=0
q7=0
q5=0
q4=0
q3=0
q2=0
q1=0
p56=0
p55=0
k26*m1
p54=-----------
a22 + a33
p53=0
p52=0
k26*n1
p51=-----------
a22 - a33
p50=0
p49=0
p48=0
p47=0
p46=0
p45=0
k23*m1
p44=-----------
a22 + a33
p43=0
p42=0
p41=0
p40=0
- k23*n1
p39=-----------
a22 - a33
p38=0
p37=0
p36=0
p35=0
k26*m1
p34=-----------
a22 + a33
p33=0
p32=0
k26*n1
p31=-----------
a22 - a33
p30=0
p29=0
k23*m1
p28=-----------
a22 + a33
p27=0
p26=0
p25=0
p24=0
p23=0
- k23*n1
p22=-----------
a22 - a33
p21=0
k26*m1
p20=-----------
a22 + a33
p19=0
p18=0
a22*k23*m1 + a22*k26*n1 - a33*k23*m1 + a33*k26*n1
p17=---------------------------------------------------
2 2
a22 - a33
p16=0
p15=0
p14=0
p13=0
p12=0
- k23*n1
p11=-----------
a22 - a33
p10=0
p9=0
p8=0
p7=0
p6=0
p5=0
p4=0
p3=0
p2=0
p1=0
k125=0
k124=0
k122=0
k121=0
2 2 2 2 2
a22 *k26*m1 - 2*a22*a33*k26*m1 + a33 *k26*m1
k120=-------------------------------------------------
2 2 2 2 2
a22 *n1 + 2*a22*a33*n1 + a33 *n1
k119=0
k118=0
2*a22*k26*m1 - 2*a33*k26*m1
k117=-----------------------------
a22*n1 + a33*n1
k116=0
2 2 2 2 2
a22 *k26*m1 - 2*a22*a33*k26*m1 + a33 *k26*m1
k115=-------------------------------------------------
2 2 2 2 2
a22 *n1 + 2*a22*a33*n1 + a33 *n1
k114=0
k113=0
2*a22*k26*m1 - 2*a33*k26*m1
k112=-----------------------------
a22*n1 + a33*n1
k110=0
k109=0
k108=k26
k107=0
k106=k26
k105=0
k104=0
k103=0
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
k95=0
k94=0
k93=0
k92=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
k85=0
k84=0
k83=0
k82=0
k81=0
k80=k23
k79=0
k78=0
k77=0
k76=0
k75=0
k74=0
k73=0
k72=0
k71=0
2 2 2 2 2
a22 *k26*m1 - 2*a22*a33*k26*m1 + a33 *k26*m1
k70=-------------------------------------------------
2 2 2 2 2
a22 *n1 + 2*a22*a33*n1 + a33 *n1
k69=0
k68=0
2*a22*k26*m1 - 2*a33*k26*m1
k67=-----------------------------
a22*n1 + a33*n1
k66=0
2 2 2 2 2
2*a22 *k26*m1 - 4*a22*a33*k26*m1 + 2*a33 *k26*m1
k65=-----------------------------------------------------
2 2 2 2 2
a22 *n1 + 2*a22*a33*n1 + a33 *n1
k64=0
k63=0
2*a22*k26*m1 - 2*a33*k26*m1
k62=-----------------------------
a22*n1 + a33*n1
k61=0
k60=0
k59=0
k58=k26
k57=0
k56=k26
k55=0
k54=0
2*a22*k26*m1 - 2*a33*k26*m1
k53=-----------------------------
a22*n1 + a33*n1
k52=0
k51=0
k50=0
k49=0
k48=0
k47=0
k46=0
k45=0
k44=k23
k43=0
k42=0
k41=0
k40=0
k38=0
k37=0
k36=0
2 2 2 2 2
a22 *k26*m1 - 2*a22*a33*k26*m1 + a33 *k26*m1
k35=-------------------------------------------------
2 2 2 2 2
a22 *n1 + 2*a22*a33*n1 + a33 *n1
k34=0
k33=0
2*a22*k26*m1 - 2*a33*k26*m1
k32=-----------------------------
a22*n1 + a33*n1
k30=0
k29=0
k28=k26
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=0
k11=0
k10=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,k26,k23,n1,a33,a22
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
3 2 2 3 3 2 2 3 2 2
HAM=( - a22 *m1 *v3 - 2*a22 *m1*n1*u3*v3 + a22 *n1 *u1 + a22 *n1 *u2
2 2 2 2 2 2 2 2 2
+ 2*a22 *a33*m1 *v3 + 2*a22 *a33*n1 *u1 + 2*a22 *a33*n1 *u2
2 2 2 2 2 2 3 2 2 2
+ a22 *a33*n1 *u3 + a22 *m1*n1 *v1 + a22 *n1 *u1 - a22*a33 *m1 *v3
2 2 2 2 2 2 2
+ 2*a22*a33 *m1*n1*u3*v3 + a22*a33 *n1 *u1 + a22*a33 *n1 *u2
2 2 2 2 3
+ 2*a22*a33 *n1 *u3 + 2*a22*a33*m1*n1 *v1 + 2*a22*a33*n1 *u1
3 2 2 2 2 2 3 2
+ a33 *n1 *u3 + a33 *m1*n1 *v1 + a33 *n1 *u1)/(n1
2 2
*(a22 + 2*a22*a33 + a33 ))
3 2 2 3 2 2 3 2 3
INT=(a22 *k23*n1 *u1*u3 *v1 + a22 *k23*n1 *u2*u3 *v2 + a22 *k23*n1 *u3 *v3
3 2 4 3 2 2 2 3 2 2 2
+ a22 *k26*m1 *v1 + 2*a22 *k26*m1 *v1 *v2 + a22 *k26*m1 *v1 *v3
3 2 4 3 2 2 2 3 3
+ a22 *k26*m1 *v2 + a22 *k26*m1 *v2 *v3 + 2*a22 *k26*m1*n1*u1*v1
3 2 3 2
+ 2*a22 *k26*m1*n1*u1*v1*v2 + 2*a22 *k26*m1*n1*u1*v1*v3
3 2 3 3
+ 2*a22 *k26*m1*n1*u2*v1 *v2 + 2*a22 *k26*m1*n1*u2*v2
3 2 3 2 2 2 3 2 2 2
+ 2*a22 *k26*m1*n1*u2*v2*v3 + a22 *k26*n1 *u1 *v1 + a22 *k26*n1 *u1 *v2
3 2 2 2 3 2 2 2 3 2 2 2
+ a22 *k26*n1 *u1 *v3 + a22 *k26*n1 *u2 *v1 + a22 *k26*n1 *u2 *v2
3 2 2 2 2 2 2
+ a22 *k26*n1 *u2 *v3 + a22 *a33*k23*n1 *u1*u3 *v1
2 2 2 2 2 3
+ a22 *a33*k23*n1 *u2*u3 *v2 + a22 *a33*k23*n1 *u3 *v3
2 2 4 2 2 2 2
- 3*a22 *a33*k26*m1 *v1 - 6*a22 *a33*k26*m1 *v1 *v2
2 2 2 2 2 2 4
- 3*a22 *a33*k26*m1 *v1 *v3 - 3*a22 *a33*k26*m1 *v2
2 2 2 2 2 3
- 3*a22 *a33*k26*m1 *v2 *v3 - 2*a22 *a33*k26*m1*n1*u1*v1
2 2 2 2
- 2*a22 *a33*k26*m1*n1*u1*v1*v2 - 2*a22 *a33*k26*m1*n1*u1*v1*v3
2 2 2 3
- 2*a22 *a33*k26*m1*n1*u2*v1 *v2 - 2*a22 *a33*k26*m1*n1*u2*v2
2 2 2 2 2 2
- 2*a22 *a33*k26*m1*n1*u2*v2*v3 + a22 *a33*k26*n1 *u1 *v1
2 2 2 2 2 2 2 2
+ a22 *a33*k26*n1 *u1 *v2 + a22 *a33*k26*n1 *u1 *v3
2 2 2 2 2 2 2 2
+ a22 *a33*k26*n1 *u2 *v1 + a22 *a33*k26*n1 *u2 *v2
2 2 2 2 2 2 2
+ a22 *a33*k26*n1 *u2 *v3 + a22 *k23*m1*n1 *u1*v1
2 2 2 2
+ a22 *k23*m1*n1 *u2*v1*v2 + a22 *k23*m1*n1 *u3*v1*v3
2 3 2 2 3 2 3
- a22 *k23*n1 *u1 *v1 - a22 *k23*n1 *u1*u2*v2 - a22 *k23*n1 *u1*u3*v3
2 2 3 2 2 2 2 2 2
+ a22 *k26*m1*n1 *v1 + a22 *k26*m1*n1 *v1*v2 + a22 *k26*m1*n1 *v1*v3
2 3 2 2 3 2 2 3 2
+ a22 *k26*n1 *u1*v1 + a22 *k26*n1 *u1*v2 + a22 *k26*n1 *u1*v3
2 2 2 2 2 2
- a22*a33 *k23*n1 *u1*u3 *v1 - a22*a33 *k23*n1 *u2*u3 *v2
2 2 3 2 2 4
- a22*a33 *k23*n1 *u3 *v3 + 3*a22*a33 *k26*m1 *v1
2 2 2 2 2 2 2 2
+ 6*a22*a33 *k26*m1 *v1 *v2 + 3*a22*a33 *k26*m1 *v1 *v3
2 2 4 2 2 2 2
+ 3*a22*a33 *k26*m1 *v2 + 3*a22*a33 *k26*m1 *v2 *v3
2 3 2 2
- 2*a22*a33 *k26*m1*n1*u1*v1 - 2*a22*a33 *k26*m1*n1*u1*v1*v2
2 2 2 2
- 2*a22*a33 *k26*m1*n1*u1*v1*v3 - 2*a22*a33 *k26*m1*n1*u2*v1 *v2
2 3 2 2
- 2*a22*a33 *k26*m1*n1*u2*v2 - 2*a22*a33 *k26*m1*n1*u2*v2*v3
2 2 2 2 2 2 2 2
- a22*a33 *k26*n1 *u1 *v1 - a22*a33 *k26*n1 *u1 *v2
2 2 2 2 2 2 2 2
- a22*a33 *k26*n1 *u1 *v3 - a22*a33 *k26*n1 *u2 *v1
2 2 2 2 2 2 2 2
- a22*a33 *k26*n1 *u2 *v2 - a22*a33 *k26*n1 *u2 *v3
3 2 3
- 2*a22*a33*k23*n1 *u1 *v1 - 2*a22*a33*k23*n1 *u1*u2*v2
3 3 2
- 2*a22*a33*k23*n1 *u1*u3*v3 + 2*a22*a33*k26*n1 *u1*v1
3 2 3 2
+ 2*a22*a33*k26*n1 *u1*v2 + 2*a22*a33*k26*n1 *u1*v3
3 2 2 3 2 2 3 2 3
- a33 *k23*n1 *u1*u3 *v1 - a33 *k23*n1 *u2*u3 *v2 - a33 *k23*n1 *u3 *v3
3 2 4 3 2 2 2 3 2 2 2
- a33 *k26*m1 *v1 - 2*a33 *k26*m1 *v1 *v2 - a33 *k26*m1 *v1 *v3
3 2 4 3 2 2 2 3 3
- a33 *k26*m1 *v2 - a33 *k26*m1 *v2 *v3 + 2*a33 *k26*m1*n1*u1*v1
3 2 3 2
+ 2*a33 *k26*m1*n1*u1*v1*v2 + 2*a33 *k26*m1*n1*u1*v1*v3
3 2 3 3
+ 2*a33 *k26*m1*n1*u2*v1 *v2 + 2*a33 *k26*m1*n1*u2*v2
3 2 3 2 2 2 3 2 2 2
+ 2*a33 *k26*m1*n1*u2*v2*v3 - a33 *k26*n1 *u1 *v1 - a33 *k26*n1 *u1 *v2
3 2 2 2 3 2 2 2 3 2 2 2
- a33 *k26*n1 *u1 *v3 - a33 *k26*n1 *u2 *v1 - a33 *k26*n1 *u2 *v2
3 2 2 2 2 2 2 2 2
- a33 *k26*n1 *u2 *v3 - a33 *k23*m1*n1 *u1*v1 - a33 *k23*m1*n1 *u2*v1*v2
2 2 2 3 2 2 3
- a33 *k23*m1*n1 *u3*v1*v3 - a33 *k23*n1 *u1 *v1 - a33 *k23*n1 *u1*u2*v2
2 3 2 2 3 2 2 2
- a33 *k23*n1 *u1*u3*v3 - a33 *k26*m1*n1 *v1 - a33 *k26*m1*n1 *v1*v2
2 2 2 2 3 2 2 3 2
- a33 *k26*m1*n1 *v1*v3 + a33 *k26*n1 *u1*v1 + a33 *k26*n1 *u1*v2
2 3 2 2 3 2 2 3
+ a33 *k26*n1 *u1*v3 )/(n1 *(a22 + a22 *a33 - a22*a33 - a33 ))
And again in machine readable form:
HAM=( - a22**3*m1**2*v3**2 - 2*a22**3*m1*n1*u3*v3 + a22**3*n1**2*u1**2 + a22**3*
n1**2*u2**2 + 2*a22**2*a33*m1**2*v3**2 + 2*a22**2*a33*n1**2*u1**2 + 2*a22**2*a33
*n1**2*u2**2 + a22**2*a33*n1**2*u3**2 + a22**2*m1*n1**2*v1 + a22**2*n1**3*u1 -
a22*a33**2*m1**2*v3**2 + 2*a22*a33**2*m1*n1*u3*v3 + a22*a33**2*n1**2*u1**2 + a22
*a33**2*n1**2*u2**2 + 2*a22*a33**2*n1**2*u3**2 + 2*a22*a33*m1*n1**2*v1 + 2*a22*
a33*n1**3*u1 + a33**3*n1**2*u3**2 + a33**2*m1*n1**2*v1 + a33**2*n1**3*u1)/(n1**2
*(a22**2 + 2*a22*a33 + a33**2))$
INT=(a22**3*k23*n1**2*u1*u3**2*v1 + a22**3*k23*n1**2*u2*u3**2*v2 + a22**3*k23*n1
**2*u3**3*v3 + a22**3*k26*m1**2*v1**4 + 2*a22**3*k26*m1**2*v1**2*v2**2 + a22**3*
k26*m1**2*v1**2*v3**2 + a22**3*k26*m1**2*v2**4 + a22**3*k26*m1**2*v2**2*v3**2 +
2*a22**3*k26*m1*n1*u1*v1**3 + 2*a22**3*k26*m1*n1*u1*v1*v2**2 + 2*a22**3*k26*m1*
n1*u1*v1*v3**2 + 2*a22**3*k26*m1*n1*u2*v1**2*v2 + 2*a22**3*k26*m1*n1*u2*v2**3 +
2*a22**3*k26*m1*n1*u2*v2*v3**2 + a22**3*k26*n1**2*u1**2*v1**2 + a22**3*k26*n1**2
*u1**2*v2**2 + a22**3*k26*n1**2*u1**2*v3**2 + a22**3*k26*n1**2*u2**2*v1**2 + a22
**3*k26*n1**2*u2**2*v2**2 + a22**3*k26*n1**2*u2**2*v3**2 + a22**2*a33*k23*n1**2*
u1*u3**2*v1 + a22**2*a33*k23*n1**2*u2*u3**2*v2 + a22**2*a33*k23*n1**2*u3**3*v3 -
3*a22**2*a33*k26*m1**2*v1**4 - 6*a22**2*a33*k26*m1**2*v1**2*v2**2 - 3*a22**2*
a33*k26*m1**2*v1**2*v3**2 - 3*a22**2*a33*k26*m1**2*v2**4 - 3*a22**2*a33*k26*m1**
2*v2**2*v3**2 - 2*a22**2*a33*k26*m1*n1*u1*v1**3 - 2*a22**2*a33*k26*m1*n1*u1*v1*
v2**2 - 2*a22**2*a33*k26*m1*n1*u1*v1*v3**2 - 2*a22**2*a33*k26*m1*n1*u2*v1**2*v2
- 2*a22**2*a33*k26*m1*n1*u2*v2**3 - 2*a22**2*a33*k26*m1*n1*u2*v2*v3**2 + a22**2*
a33*k26*n1**2*u1**2*v1**2 + a22**2*a33*k26*n1**2*u1**2*v2**2 + a22**2*a33*k26*n1
**2*u1**2*v3**2 + a22**2*a33*k26*n1**2*u2**2*v1**2 + a22**2*a33*k26*n1**2*u2**2*
v2**2 + a22**2*a33*k26*n1**2*u2**2*v3**2 + a22**2*k23*m1*n1**2*u1*v1**2 + a22**2
*k23*m1*n1**2*u2*v1*v2 + a22**2*k23*m1*n1**2*u3*v1*v3 - a22**2*k23*n1**3*u1**2*
v1 - a22**2*k23*n1**3*u1*u2*v2 - a22**2*k23*n1**3*u1*u3*v3 + a22**2*k26*m1*n1**2
*v1**3 + a22**2*k26*m1*n1**2*v1*v2**2 + a22**2*k26*m1*n1**2*v1*v3**2 + a22**2*
k26*n1**3*u1*v1**2 + a22**2*k26*n1**3*u1*v2**2 + a22**2*k26*n1**3*u1*v3**2 - a22
*a33**2*k23*n1**2*u1*u3**2*v1 - a22*a33**2*k23*n1**2*u2*u3**2*v2 - a22*a33**2*
k23*n1**2*u3**3*v3 + 3*a22*a33**2*k26*m1**2*v1**4 + 6*a22*a33**2*k26*m1**2*v1**2
*v2**2 + 3*a22*a33**2*k26*m1**2*v1**2*v3**2 + 3*a22*a33**2*k26*m1**2*v2**4 + 3*
a22*a33**2*k26*m1**2*v2**2*v3**2 - 2*a22*a33**2*k26*m1*n1*u1*v1**3 - 2*a22*a33**
2*k26*m1*n1*u1*v1*v2**2 - 2*a22*a33**2*k26*m1*n1*u1*v1*v3**2 - 2*a22*a33**2*k26*
m1*n1*u2*v1**2*v2 - 2*a22*a33**2*k26*m1*n1*u2*v2**3 - 2*a22*a33**2*k26*m1*n1*u2*
v2*v3**2 - a22*a33**2*k26*n1**2*u1**2*v1**2 - a22*a33**2*k26*n1**2*u1**2*v2**2 -
a22*a33**2*k26*n1**2*u1**2*v3**2 - a22*a33**2*k26*n1**2*u2**2*v1**2 - a22*a33**
2*k26*n1**2*u2**2*v2**2 - a22*a33**2*k26*n1**2*u2**2*v3**2 - 2*a22*a33*k23*n1**3
*u1**2*v1 - 2*a22*a33*k23*n1**3*u1*u2*v2 - 2*a22*a33*k23*n1**3*u1*u3*v3 + 2*a22*
a33*k26*n1**3*u1*v1**2 + 2*a22*a33*k26*n1**3*u1*v2**2 + 2*a22*a33*k26*n1**3*u1*
v3**2 - a33**3*k23*n1**2*u1*u3**2*v1 - a33**3*k23*n1**2*u2*u3**2*v2 - a33**3*k23
*n1**2*u3**3*v3 - a33**3*k26*m1**2*v1**4 - 2*a33**3*k26*m1**2*v1**2*v2**2 - a33
**3*k26*m1**2*v1**2*v3**2 - a33**3*k26*m1**2*v2**4 - a33**3*k26*m1**2*v2**2*v3**
2 + 2*a33**3*k26*m1*n1*u1*v1**3 + 2*a33**3*k26*m1*n1*u1*v1*v2**2 + 2*a33**3*k26*
m1*n1*u1*v1*v3**2 + 2*a33**3*k26*m1*n1*u2*v1**2*v2 + 2*a33**3*k26*m1*n1*u2*v2**3
+ 2*a33**3*k26*m1*n1*u2*v2*v3**2 - a33**3*k26*n1**2*u1**2*v1**2 - a33**3*k26*n1
**2*u1**2*v2**2 - a33**3*k26*n1**2*u1**2*v3**2 - a33**3*k26*n1**2*u2**2*v1**2 -
a33**3*k26*n1**2*u2**2*v2**2 - a33**3*k26*n1**2*u2**2*v3**2 - a33**2*k23*m1*n1**
2*u1*v1**2 - a33**2*k23*m1*n1**2*u2*v1*v2 - a33**2*k23*m1*n1**2*u3*v1*v3 - a33**
2*k23*n1**3*u1**2*v1 - a33**2*k23*n1**3*u1*u2*v2 - a33**2*k23*n1**3*u1*u3*v3 -
a33**2*k26*m1*n1**2*v1**3 - a33**2*k26*m1*n1**2*v1*v2**2 - a33**2*k26*m1*n1**2*
v1*v3**2 + a33**2*k26*n1**3*u1*v1**2 + a33**2*k26*n1**3*u1*v2**2 + a33**2*k26*n1
**3*u1*v3**2)/(n1**2*(a22**3 + a22**2*a33 - a22*a33**2 - a33**3))$