Solution 13 to problem e3c2new
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
1
a22= - ---*a33
5
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
5 2
---*b33
4
c33=----------
a33
5
- ---*b33*n1
3
m1=---------------
a33
5
- ---*b33*n2
3
m2=---------------
a33
5
---*b33*n3
6
m3=------------
a33
25
- ----*b33*n3*q1
6
r6=-------------------
2
a33
25
- ----*b33*n2*q1
12
r5=-------------------
2
a33
25
- ----*b33*n1*q1
12
r4=-------------------
2
a33
5
- ---*n3*q1
6
r3=--------------
a33
5
- ---*n2*q1
6
r2=--------------
a33
5
- ---*n1*q1
6
r1=--------------
a33
q20=0
q19=0
q17=0
q16=0
25 2
----*b33 *q1
4
q15=--------------
2
a33
q14=0
q13=0
5*b33*q1
q12=----------
a33
q11=0
25 2
----*b33 *q1
4
q10=--------------
2
a33
q9=0
q8=0
5*b33*q1
q7=----------
a33
q5=0
q4=0
q3=q1
q2=0
25
- ----*b33*k26*n3
6
p56=--------------------
2
a33
25
- ----*b33*k26*n2
12
p55=--------------------
2
a33
25
- ----*b33*k26*n1
12
p54=--------------------
2
a33
5
- ---*k26*n3
6
p53=---------------
a33
5
- ---*k26*n2
6
p52=---------------
a33
5
- ---*k26*n1
6
p51=---------------
a33
25
- ----*b33*k26*n3
6
p50=--------------------
2
a33
p49=0
p48=0
p47=0
p46=0
25
- ----*b33*k26*n3
6
p45=--------------------
2
a33
p44=0
p43=0
p42=0
p41=0
p40=0
p39=0
p38=0
p37=0
p36=0
25
- ----*b33*k26*n2
12
p35=--------------------
2
a33
25
- ----*b33*k26*n1
12
p34=--------------------
2
a33
5
- ---*k26*n3
6
p33=---------------
a33
5
- ---*k26*n2
6
p32=---------------
a33
5
- ---*k26*n1
6
p31=---------------
a33
25
- ----*b33*k26*n2
12
p30=--------------------
2
a33
p29=0
p28=0
p27=0
p26=0
p25=0
p24=0
p23=0
p22=0
p21=0
25
- ----*b33*k26*n1
12
p20=--------------------
2
a33
5
- ---*k26*n3
6
p19=---------------
a33
5
- ---*k26*n2
6
p18=---------------
a33
5
- ---*k26*n1
6
p17=---------------
a33
p16=0
p15=0
p14=0
p13=0
p12=0
p11=0
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
25 2
----*b33 *k26
4
k120=---------------
2
a33
k119=0
k118=0
5*b33*k26
k117=-----------
a33
k116=0
25 2
----*b33 *k26
4
k115=---------------
2
a33
k114=0
k113=0
5*b33*k26
k112=-----------
a33
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=0
k79=0
k78=0
k77=0
k76=0
k75=0
k74=0
k73=0
k72=0
k71=0
25 2
----*b33 *k26
4
k70=---------------
2
a33
k69=0
k68=0
5*b33*k26
k67=-----------
a33
k66=0
25 2
----*b33 *k26
2
k65=---------------
2
a33
k64=0
k63=0
5*b33*k26
k62=-----------
a33
k61=0
k60=0
k59=0
k58=k26
k57=0
k56=k26
k55=0
k54=0
5*b33*k26
k53=-----------
a33
k52=0
k51=0
k50=0
k49=0
k48=0
k47=0
k46=0
k45=0
k44=0
k43=0
k42=0
k41=0
k40=0
k38=0
k37=0
k36=0
25 2
----*b33 *k26
4
k35=---------------
2
a33
k34=0
k33=0
5*b33*k26
k32=-----------
a33
k30=0
k29=0
k28=k26
k27=0
k25=0
k24=0
k23=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:
n3,q1,k26,n1,b33,n2,a33
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2 2 2 2
HAM=( - 12*a33 *u1 - 12*a33 *u2 + 60*a33 *u3 + 60*a33*b33*u3*v3
2 2
+ 60*a33*n1*u1 + 60*a33*n2*u2 + 60*a33*n3*u3 + 75*b33 *v3 - 100*b33*n1*v1
- 100*b33*n2*v2 + 50*b33*n3*v3)/(60*a33)
2 2 2 2 2 2 2 2 2
INT=(12*a33 *k26*u1 *v1 + 12*a33 *k26*u1 *v2 + 12*a33 *k26*u1 *v3
2 2 2 2 2 2 2 2 2
+ 12*a33 *k26*u2 *v1 + 12*a33 *k26*u2 *v2 + 12*a33 *k26*u2 *v3
2 2 2 2 3
+ 12*a33 *q1*u1 + 12*a33 *q1*u2 + 60*a33*b33*k26*u1*v1
2 2
+ 60*a33*b33*k26*u1*v1*v2 + 60*a33*b33*k26*u1*v1*v3
2 3
+ 60*a33*b33*k26*u2*v1 *v2 + 60*a33*b33*k26*u2*v2
2
+ 60*a33*b33*k26*u2*v2*v3 + 60*a33*b33*q1*u1*v1 + 60*a33*b33*q1*u2*v2
2 2 2
- 10*a33*k26*n1*u1*v1 - 10*a33*k26*n1*u1*v2 - 10*a33*k26*n1*u1*v3
2 2 2
- 10*a33*k26*n2*u2*v1 - 10*a33*k26*n2*u2*v2 - 10*a33*k26*n2*u2*v3
2 2 2
- 10*a33*k26*n3*u3*v1 - 10*a33*k26*n3*u3*v2 - 10*a33*k26*n3*u3*v3
2 4
- 10*a33*n1*q1*u1 - 10*a33*n2*q1*u2 - 10*a33*n3*q1*u3 + 75*b33 *k26*v1
2 2 2 2 2 2 2 4
+ 150*b33 *k26*v1 *v2 + 75*b33 *k26*v1 *v3 + 75*b33 *k26*v2
2 2 2 2 2 2 2
+ 75*b33 *k26*v2 *v3 + 75*b33 *q1*v1 + 75*b33 *q1*v2
3 2 2
- 25*b33*k26*n1*v1 - 25*b33*k26*n1*v1*v2 - 25*b33*k26*n1*v1*v3
2 3 2
- 25*b33*k26*n2*v1 *v2 - 25*b33*k26*n2*v2 - 25*b33*k26*n2*v2*v3
2 2 3
- 50*b33*k26*n3*v1 *v3 - 50*b33*k26*n3*v2 *v3 - 50*b33*k26*n3*v3
2
- 25*b33*n1*q1*v1 - 25*b33*n2*q1*v2 - 50*b33*n3*q1*v3)/(12*a33 )
And again in machine readable form:
HAM=( - 12*a33**2*u1**2 - 12*a33**2*u2**2 + 60*a33**2*u3**2 + 60*a33*b33*u3*v3 +
60*a33*n1*u1 + 60*a33*n2*u2 + 60*a33*n3*u3 + 75*b33**2*v3**2 - 100*b33*n1*v1 -
100*b33*n2*v2 + 50*b33*n3*v3)/(60*a33)$
INT=(12*a33**2*k26*u1**2*v1**2 + 12*a33**2*k26*u1**2*v2**2 + 12*a33**2*k26*u1**2
*v3**2 + 12*a33**2*k26*u2**2*v1**2 + 12*a33**2*k26*u2**2*v2**2 + 12*a33**2*k26*
u2**2*v3**2 + 12*a33**2*q1*u1**2 + 12*a33**2*q1*u2**2 + 60*a33*b33*k26*u1*v1**3
+ 60*a33*b33*k26*u1*v1*v2**2 + 60*a33*b33*k26*u1*v1*v3**2 + 60*a33*b33*k26*u2*v1
**2*v2 + 60*a33*b33*k26*u2*v2**3 + 60*a33*b33*k26*u2*v2*v3**2 + 60*a33*b33*q1*u1
*v1 + 60*a33*b33*q1*u2*v2 - 10*a33*k26*n1*u1*v1**2 - 10*a33*k26*n1*u1*v2**2 - 10
*a33*k26*n1*u1*v3**2 - 10*a33*k26*n2*u2*v1**2 - 10*a33*k26*n2*u2*v2**2 - 10*a33*
k26*n2*u2*v3**2 - 10*a33*k26*n3*u3*v1**2 - 10*a33*k26*n3*u3*v2**2 - 10*a33*k26*
n3*u3*v3**2 - 10*a33*n1*q1*u1 - 10*a33*n2*q1*u2 - 10*a33*n3*q1*u3 + 75*b33**2*
k26*v1**4 + 150*b33**2*k26*v1**2*v2**2 + 75*b33**2*k26*v1**2*v3**2 + 75*b33**2*
k26*v2**4 + 75*b33**2*k26*v2**2*v3**2 + 75*b33**2*q1*v1**2 + 75*b33**2*q1*v2**2
- 25*b33*k26*n1*v1**3 - 25*b33*k26*n1*v1*v2**2 - 25*b33*k26*n1*v1*v3**2 - 25*b33
*k26*n2*v1**2*v2 - 25*b33*k26*n2*v2**3 - 25*b33*k26*n2*v2*v3**2 - 50*b33*k26*n3*
v1**2*v3 - 50*b33*k26*n3*v2**2*v3 - 50*b33*k26*n3*v3**3 - 25*b33*n1*q1*v1 - 25*
b33*n2*q1*v2 - 50*b33*n3*q1*v3)/(12*a33**2)$