Solution 18 to problem e3null
Remaining equations |
Expressions |
Parameters |
Relevance |
Back to problem e3null
Equations
The following unsolved equations remain:
2 2
0=b31 + b32
Expressions
The solution is given through the following expressions:
1
a22=---*a33
2
b22=0
b33=0
c12=0
c13=0
c22=0
c23=0
c33=0
n1=0
n2=0
1 2 1 2
- a33*b32*m2 - ---*b31 *n3 - ---*b32 *n3
3 3
m1=-------------------------------------------
a33*b31
m3=0
r6=0
r5=0
r4=0
r3=0
r2=0
r1=0
q19=0
q18=0
8*k1*m2*n3
q16=------------
2
a33
8 2 2 8 2 2
- 8*a33*b32*k1*m2*n3 - ---*b31 *k1*n3 - ---*b32 *k1*n3
3 3
q15=-----------------------------------------------------------
3
a33 *b31
2 2 2 2 2 2 8 2
q14=(4*a33 *b31 *k1*m2 + 4*a33 *b32 *k1*m2 + ---*a33*b31 *b32*k1*m2*n3
3
8 3 8 4 2 4 2 2 2
+ ---*a33*b32 *k1*m2*n3 - ---*b31 *k1*n3 - ---*b31 *b32 *k1*n3
3 9 9
4 4 2 4 2
+ ---*b32 *k1*n3 )/(a33 *b31 )
9
q13=0
q12=0
q11=0
q10=0
2 2 2 2 2 2 8 2
q9=(4*a33 *b31 *k1*m2 + 4*a33 *b32 *k1*m2 + ---*a33*b31 *b32*k1*m2*n3
3
8 3 8 4 2 4 2 2 2
+ ---*a33*b32 *k1*m2*n3 - ---*b31 *k1*n3 - ---*b31 *b32 *k1*n3
3 9 9
4 4 2 4 2
+ ---*b32 *k1*n3 )/(a33 *b31 )
9
q8=0
q7=0
q5=0
q4=0
2
- 4*k1*n3
q3=-------------
2
a33
q2=0
2
- 4*k1*n3
q1=-------------
2
a33
p50=0
p49=0
p48=0
p47=0
p46=0
p45=0
p44=0
p43=0
p42=0
p41=0
p40=0
p39=0
p38=0
p37=0
p36=0
8*a33*k1*m2 + 8*b32*k1*n3
p35=---------------------------
2
a33
20 2 4 2
- 4*a33*b32*k1*m2 + ----*b31 *k1*n3 - ---*b32 *k1*n3
3 3
p34=-------------------------------------------------------
2
a33 *b31
p33=0
p32=0
p31=0
p30=0
p29=0
p28=0
p27=0
p26=0
p25=0
p24=0
p23=0
p22=0
p21=0
p20=0
4*k1*m2
p19=---------
a33
4 2 4 2
4*a33*b32*k1*m2 + ---*b31 *k1*n3 + ---*b32 *k1*n3
3 3
p18=---------------------------------------------------
2
a33 *b31
4*k1*m2
p17=---------
a33
p16=0
p15=0
p14=0
p13=0
p12=0
4 2 4 2
- 4*a33*b32*k1*m2 - ---*b31 *k1*n3 - ---*b32 *k1*n3
3 3
p11=------------------------------------------------------
2
a33 *b31
p10=0
p9=0
p8=0
- 4*k1*n3
p7=------------
a33
p6=0
- 4*k1*n3
p5=------------
a33
p4=0
p3=0
p2=0
p1=0
k104=0
k103=0
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
k95=0
k94=0
k93=0
k92=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
k70=0
k69=0
k68=0
k67=0
k66=0
k65=0
8*b32*k1
k64=----------
a33
4*b31*k1
k63=----------
a33
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
k42=0
k41=0
k40=0
k39=0
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
4*b32*k1
k32=----------
a33
- 4*b31*k1
k31=-------------
a33
4*b32*k1
k30=----------
a33
k28=0
k27=0
k26=0
k25=0
k24=0
k23=0
k21=0
k20=0
k19=0
k18=0
4*b31*k1
k17=----------
a33
k16=0
k14=0
k13=0
k12=0
k11=0
k10=0
k9=0
k8=0
k7=0
k6=0
k5=k1
k4=0
k3=2*k1
k2=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,m2,b32,k1,b31,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=(3*a33 *b31*u1 + 3*a33 *b31*u2 + 6*a33 *b31*u3 + 6*a33*b31 *u3*v1
+ 6*a33*b31*b32*u3*v2 + 6*a33*b31*m2*v2 + 6*a33*b31*n3*u3
2 2
- 6*a33*b32*m2*v1 - 2*b31 *n3*v1 - 2*b32 *n3*v1)/(6*a33*b31)
4 2 4 4 2 2 2 4 2 4
INT=(k1*(9*a33 *b31 *u1 + 18*a33 *b31 *u1 *u2 + 9*a33 *b31 *u2
3 3 3 3 2
- 36*a33 *b31 *u1*u2*u3*v2 + 36*a33 *b31 *u1*u3 *v3
3 3 2 3 2 2
+ 36*a33 *b31 *u2 *u3*v1 + 36*a33 *b31 *b32*u1 *u3*v2
3 2 2 3 2 2
+ 36*a33 *b31 *b32*u2 *u3*v2 + 72*a33 *b31 *b32*u2*u3 *v3
3 2 2 3 2 2
+ 36*a33 *b31 *m2*u1 *v2 + 36*a33 *b31 *m2*u2 *v2
3 2 3 2 2
+ 72*a33 *b31 *m2*u2*u3*v3 - 36*a33 *b31 *n3*u1 *u3
3 2 2 3
- 36*a33 *b31 *n3*u2 *u3 + 36*a33 *b31*b32*m2*u1*u2*v2
3 3 2
- 36*a33 *b31*b32*m2*u1*u3*v3 - 36*a33 *b31*b32*m2*u2 *v1
2 3 2 3
+ 12*a33 *b31 *n3*u1*u2*v2 + 60*a33 *b31 *n3*u1*u3*v3
2 3 2 2 2
- 12*a33 *b31 *n3*u2 *v1 + 72*a33 *b31 *b32*n3*u2*u3*v3
2 2 2 2 2 2 2 2
+ 36*a33 *b31 *m2 *v1 + 36*a33 *b31 *m2 *v2
2 2 2 2 2 2
+ 72*a33 *b31 *m2*n3*u2*v3 - 36*a33 *b31 *n3 *u1
2 2 2 2 2 2
- 36*a33 *b31 *n3 *u2 + 12*a33 *b31*b32 *n3*u1*u2*v2
2 2 2 2 2
- 12*a33 *b31*b32 *n3*u1*u3*v3 - 12*a33 *b31*b32 *n3*u2 *v1
2 2 2 2 2
- 72*a33 *b31*b32*m2*n3*u1*v3 + 36*a33 *b32 *m2 *v1
2 2 2 2 3 2
+ 36*a33 *b32 *m2 *v2 - 24*a33*b31 *n3 *u1*v3
2 2 2 2
+ 24*a33*b31 *b32*m2*n3*v1 + 24*a33*b31 *b32*m2*n3*v2
2 2 3 2
- 24*a33*b31*b32 *n3 *u1*v3 + 24*a33*b32 *m2*n3*v1
3 2 4 2 2 4 2 2
+ 24*a33*b32 *m2*n3*v2 - 8*b31 *n3 *v1 - 8*b31 *n3 *v2
2 2 2 2 2 2 2 2 4 2 2
- 4*b31 *b32 *n3 *v1 - 4*b31 *b32 *n3 *v2 + 4*b32 *n3 *v1
4 2 2 4 2
+ 4*b32 *n3 *v2 ))/(9*a33 *b31 )
And again in machine readable form:
HAM=(3*a33**2*b31*u1**2 + 3*a33**2*b31*u2**2 + 6*a33**2*b31*u3**2 + 6*a33*b31**2
*u3*v1 + 6*a33*b31*b32*u3*v2 + 6*a33*b31*m2*v2 + 6*a33*b31*n3*u3 - 6*a33*b32*m2*
v1 - 2*b31**2*n3*v1 - 2*b32**2*n3*v1)/(6*a33*b31)$
INT=(k1*(9*a33**4*b31**2*u1**4 + 18*a33**4*b31**2*u1**2*u2**2 + 9*a33**4*b31**2*
u2**4 - 36*a33**3*b31**3*u1*u2*u3*v2 + 36*a33**3*b31**3*u1*u3**2*v3 + 36*a33**3*
b31**3*u2**2*u3*v1 + 36*a33**3*b31**2*b32*u1**2*u3*v2 + 36*a33**3*b31**2*b32*u2
**2*u3*v2 + 72*a33**3*b31**2*b32*u2*u3**2*v3 + 36*a33**3*b31**2*m2*u1**2*v2 + 36
*a33**3*b31**2*m2*u2**2*v2 + 72*a33**3*b31**2*m2*u2*u3*v3 - 36*a33**3*b31**2*n3*
u1**2*u3 - 36*a33**3*b31**2*n3*u2**2*u3 + 36*a33**3*b31*b32*m2*u1*u2*v2 - 36*a33
**3*b31*b32*m2*u1*u3*v3 - 36*a33**3*b31*b32*m2*u2**2*v1 + 12*a33**2*b31**3*n3*u1
*u2*v2 + 60*a33**2*b31**3*n3*u1*u3*v3 - 12*a33**2*b31**3*n3*u2**2*v1 + 72*a33**2
*b31**2*b32*n3*u2*u3*v3 + 36*a33**2*b31**2*m2**2*v1**2 + 36*a33**2*b31**2*m2**2*
v2**2 + 72*a33**2*b31**2*m2*n3*u2*v3 - 36*a33**2*b31**2*n3**2*u1**2 - 36*a33**2*
b31**2*n3**2*u2**2 + 12*a33**2*b31*b32**2*n3*u1*u2*v2 - 12*a33**2*b31*b32**2*n3*
u1*u3*v3 - 12*a33**2*b31*b32**2*n3*u2**2*v1 - 72*a33**2*b31*b32*m2*n3*u1*v3 + 36
*a33**2*b32**2*m2**2*v1**2 + 36*a33**2*b32**2*m2**2*v2**2 - 24*a33*b31**3*n3**2*
u1*v3 + 24*a33*b31**2*b32*m2*n3*v1**2 + 24*a33*b31**2*b32*m2*n3*v2**2 - 24*a33*
b31*b32**2*n3**2*u1*v3 + 24*a33*b32**3*m2*n3*v1**2 + 24*a33*b32**3*m2*n3*v2**2 -
8*b31**4*n3**2*v1**2 - 8*b31**4*n3**2*v2**2 - 4*b31**2*b32**2*n3**2*v1**2 - 4*
b31**2*b32**2*n3**2*v2**2 + 4*b32**4*n3**2*v1**2 + 4*b32**4*n3**2*v2**2))/(9*a33
**4*b31**2)$