Solution 58 to problem e3c2new
Expressions |
Parameters |
Relevance |
Back to problem e3c2new
Expressions
The solution is given through the following expressions:
1
---*b32*n3
2
a22=------------
m2
b32*n3
a33=--------
m2
b22=0
b31=0
b33=0
c12=0
c13=0
c22=0
c23=0
1
- ---*b32*m2
2
c33=---------------
n3
n1=0
n2=0
m1=0
m3=0
r6=0
r5=0
r4=0
r3=0
r2=0
r1=0
q20=0
q19=0
2
- k21*m2
q17=------------
2
b32
q16=0
1 3
- ---*k21*m2
2
q15=----------------
2
b32 *n3
q14=0
q13=0
q12=0
q11=0
1 3
- ---*k21*m2
2
q10=----------------
2
b32 *n3
q9=0
q8=0
q7=0
q5=0
q4=0
1
---*k21*m2*n3
2
q3=---------------
2
b32
q2=0
1
---*k21*m2*n3
2
q1=---------------
2
b32
p56=0
1 3
---*k21*m2
2
p55=-------------
2
b32*n3
p54=0
1 2
---*k21*m2
2
p53=-------------
b32*n3
p52=0
p51=0
p50=0
p49=0
p48=0
2
- k21*m2
p47=------------
b32*n3
p46=0
p45=0
p44=0
p43=0
2
- k21*m2
p42=------------
b32*n3
p41=0
- k21*m2
p40=-----------
b32
p39=0
p38=0
p37=0
p36=0
p35=0
p34=0
p33=0
p32=0
p31=0
p30=0
p29=0
p28=0
p27=0
p26=0
p25=0
p24=0
1
---*k21*m2
2
p23=------------
b32
p22=0
1
- ---*k21*m2
2
p21=---------------
b32
p20=0
p19=0
p18=0
p17=0
p16=0
p15=0
p14=0
p13=0
k21*m2
p12=--------
b32
p11=0
p10=0
p9=0
p8=0
1
---*k21*n3
2
p7=------------
b32
p6=0
1
---*k21*n3
2
p5=------------
b32
p4=0
p3=0
p2=0
p1=0
k125=0
k124=0
k122=0
k121=0
1 3
---*k21*m2
4
k120=-------------
3
n3
k119=0
1 2
---*k21*m2
2
k118=-------------
2
n3
k117=0
k116=0
1 3
---*k21*m2
4
k115=-------------
3
n3
k114=0
k113=0
k112=0
k110=0
k109=0
1
---*k21*m2
4
k108=------------
n3
k107=0
1
---*k21*m2
4
k106=------------
n3
k105=0
k104=0
k103=0
k102=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
- k21*m2
k95=-----------
n3
k94=0
k93=0
k92=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
k85=0
- k21*m2
k84=-----------
n3
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
1 3
---*k21*m2
8
k70=-------------
3
n3
k69=0
k68=0
k67=0
k66=0
1 3
---*k21*m2
4
k65=-------------
3
n3
k64=0
k63=0
k62=0
k61=0
k60=0
k59=0
k58=0
k57=0
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
k43=0
1
k42=---*k21
2
k41=0
1
k40= - ---*k21
2
k38=0
k37=0
k36=0
1 3
---*k21*m2
8
k35=-------------
3
n3
k34=0
k33=0
k32=0
k30=0
k29=0
k28=0
k27=0
k26=0
k25=0
k24=0
k23=0
k22=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
1
- ---*k21*n3
8
k5=---------------
m2
k4=0
1
- ---*k21*n3
4
k3=---------------
m2
k2=0
1
- ---*k21*n3
8
k1=---------------
m2
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:
k21,m2,n3,b32
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2 2 2 2
HAM=( - b32*m2 *v3 + 2*b32*m2*n3*u3*v2 + b32*n3 *u1 + b32*n3 *u2
2 2 2 2
+ 2*b32*n3 *u3 + 2*m2 *n3*v2 + 2*m2*n3 *u3)/(2*m2*n3)
2 4 4 2 4 2 2 2 4 2 2 2 4 4
INT=(k21*(b32 *m2 *v1 + 2*b32 *m2 *v1 *v2 + 2*b32 *m2 *v1 *v3 + b32 *m2 *v2
2 4 2 2 2 3 2
+ 2*b32 *m2 *v2 *v3 + 4*b32 *m2 *n3*u3*v2*v3
2 2 2 2 2 2 2 2
+ 2*b32 *m2 *n3 *u1 *v3 - 8*b32 *m2 *n3 *u1*u3*v1*v3
2 2 2 2 2 2 2 2
+ 2*b32 *m2 *n3 *u2 *v3 - 8*b32 *m2 *n3 *u2*u3*v2*v3
2 3 2 2 3
- 4*b32 *m2*n3 *u1 *u3*v2 + 8*b32 *m2*n3 *u1*u2*u3*v1
2 3 2 2 4 4 2 4 2 2
+ 4*b32 *m2*n3 *u2 *u3*v2 - b32 *n3 *u1 - 2*b32 *n3 *u1 *u2
2 4 4 4 2 3 2
- b32 *n3 *u2 + 4*b32*m2 *n3*v2*v3 - 8*b32*m2 *n3 *u1*v1*v3
3 2 3 2 2
- 8*b32*m2 *n3 *u2*v2*v3 + 4*b32*m2 *n3 *u3*v3
2 3 2 2 3
- 4*b32*m2 *n3 *u1 *v2 + 8*b32*m2 *n3 *u1*u2*v1
2 3 2 2 3 4 2
+ 4*b32*m2 *n3 *u2 *v2 - 8*b32*m2 *n3 *u2*u3*v3 + 4*b32*m2*n3 *u1 *u3
4 2 4 2 2 4 2 2
+ 4*b32*m2*n3 *u2 *u3 - 4*m2 *n3 *v1 - 4*m2 *n3 *v2
3 3 2 4 2 2 4 2 2 3
- 8*m2 *n3 *u2*v3 + 4*m2 *n3 *u1 + 4*m2 *n3 *u2 ))/(8*b32 *m2*n3 )
And again in machine readable form:
HAM=( - b32*m2**2*v3**2 + 2*b32*m2*n3*u3*v2 + b32*n3**2*u1**2 + b32*n3**2*u2**2
+ 2*b32*n3**2*u3**2 + 2*m2**2*n3*v2 + 2*m2*n3**2*u3)/(2*m2*n3)$
INT=(k21*(b32**2*m2**4*v1**4 + 2*b32**2*m2**4*v1**2*v2**2 + 2*b32**2*m2**4*v1**2
*v3**2 + b32**2*m2**4*v2**4 + 2*b32**2*m2**4*v2**2*v3**2 + 4*b32**2*m2**3*n3*u3*
v2*v3**2 + 2*b32**2*m2**2*n3**2*u1**2*v3**2 - 8*b32**2*m2**2*n3**2*u1*u3*v1*v3 +
2*b32**2*m2**2*n3**2*u2**2*v3**2 - 8*b32**2*m2**2*n3**2*u2*u3*v2*v3 - 4*b32**2*
m2*n3**3*u1**2*u3*v2 + 8*b32**2*m2*n3**3*u1*u2*u3*v1 + 4*b32**2*m2*n3**3*u2**2*
u3*v2 - b32**2*n3**4*u1**4 - 2*b32**2*n3**4*u1**2*u2**2 - b32**2*n3**4*u2**4 + 4
*b32*m2**4*n3*v2*v3**2 - 8*b32*m2**3*n3**2*u1*v1*v3 - 8*b32*m2**3*n3**2*u2*v2*v3
+ 4*b32*m2**3*n3**2*u3*v3**2 - 4*b32*m2**2*n3**3*u1**2*v2 + 8*b32*m2**2*n3**3*
u1*u2*v1 + 4*b32*m2**2*n3**3*u2**2*v2 - 8*b32*m2**2*n3**3*u2*u3*v3 + 4*b32*m2*n3
**4*u1**2*u3 + 4*b32*m2*n3**4*u2**2*u3 - 4*m2**4*n3**2*v1**2 - 4*m2**4*n3**2*v2
**2 - 8*m2**3*n3**3*u2*v3 + 4*m2**2*n3**4*u1**2 + 4*m2**2*n3**4*u2**2))/(8*b32**
2*m2*n3**3)$