Solution 16 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
1
- ---*a33*m3
2
n1=---------------
b31
1
---*a33*b32*m3
2
n2=----------------
2
b31
n3=0
m1=0
m2=0
2 2 1 2
---*b31 *m3 - ---*b32 *m3
3 3
m3=---------------------------
2
b31
r6=0
r5=0
r4=0
r3=0
2 3 3 3
b31 *b32*k1*m3 + b32 *k1*m3
r2=-------------------------------
6
b31
2 3 2 3
- b31 *k1*m3 - b32 *k1*m3
r1=------------------------------
5
b31
q19=0
q18=0
q16=0
q15=0
q14=0
q13=0
2 2 3 2
3*b31 *b32*k1*m3 - b32 *k1*m3
q12=---------------------------------
4
a33*b31
q11=0
q10=0
q9=0
2 2 2 2
- b31 *k1*m3 + 3*b32 *k1*m3
q8=--------------------------------
3
a33*b31
q7=0
q5=0
q4=0
2
- k1*m3
q3=-----------
2
b31
2
- 2*b32*k1*m3
q2=-----------------
3
b31
2 2
- b32 *k1*m3
q1=----------------
4
b31
p50=0
p49=0
p48=0
p47=0
p46=0
p45=0
p44=0
p43=0
8 2 8 3
- ---*b31 *b32*k1*m3 - ---*b32 *k1*m3
3 3
p42=----------------------------------------
2 2
a33 *b31
p41=0
p40=0
p39=0
8 2 8 2
---*b31 *k1*m3 + ---*b32 *k1*m3
3 3
p38=---------------------------------
2
a33 *b31
p37=0
2 2
4*b31 *k1*m3 - 4*b32 *k1*m3
p36=-----------------------------
2
a33*b31
p35=0
p34=0
4 2 4 2
- ---*b31 *k1*m3 - ---*b32 *k1*m3
3 3
p33=------------------------------------
2
a33*b31
p32=0
4 2 4 2
- ---*b31 *k1*m3 - ---*b32 *k1*m3
3 3
p31=------------------------------------
2
a33*b31
p30=0
p29=0
p28=0
p27=0
p26=0
p25=0
p24=0
p23=0
p22=0
p21=0
8*b32*k1*m3
p20=-------------
a33*b31
p19=0
p18=0
p17=0
p16=0
p15=0
p14=0
p13=0
- 8*b32*k1*m3
p12=----------------
a33*b31
p11=0
p10=0
p9=0
p8=0
p7=0
p6=0
p5=0
- 2*b32*k1*m3
p4=----------------
2
b31
2*k1*m3
p3=---------
b31
- 2*b32*k1*m3
p2=----------------
2
b31
2*k1*m3
p1=---------
b31
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:
m3,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
HAM=(3*a33*b31 *u1 + 3*a33*b31 *u2 + 6*a33*b31 *u3 - 3*a33*b31*m3*u1
3 2 2
+ 3*a33*b32*m3*u2 + 6*b31 *u3*v1 + 6*b31 *b32*u3*v2 + 4*b31 *m3*v3
2 2
- 2*b32 *m3*v3)/(6*b31 )
2 6 4 2 6 2 2 2 6 4
INT=(k1*(3*a33 *b31 *u1 + 6*a33 *b31 *u1 *u2 + 3*a33 *b31 *u2
2 5 3 2 5 2
+ 6*a33 *b31 *m3*u1 + 6*a33 *b31 *m3*u1*u2
2 4 2 2 4 3
- 6*a33 *b31 *b32*m3*u1 *u2 - 6*a33 *b31 *b32*m3*u2
2 4 2 2 2 3 2 2 3 3
- 3*a33 *b31 *m3 *u2 - 6*a33 *b31 *b32*m3 *u1*u2 - 3*a33 *b31 *m3 *u1
2 2 2 2 2 2 2 3
- 3*a33 *b31 *b32 *m3 *u1 + 3*a33 *b31 *b32*m3 *u2
2 2 3 2 3 3
- 3*a33 *b31*b32 *m3 *u1 + 3*a33 *b32 *m3 *u2
7 7 2
- 12*a33*b31 *u1*u2*u3*v2 + 12*a33*b31 *u1*u3 *v3
7 2 6 2
+ 12*a33*b31 *u2 *u3*v1 + 12*a33*b31 *b32*u1 *u3*v2
6 2 6 2
+ 12*a33*b31 *b32*u2 *u3*v2 + 24*a33*b31 *b32*u2*u3 *v3
6 2 6 2 6 2
- 4*a33*b31 *m3*u1 *v3 - 4*a33*b31 *m3*u2 *v3 + 12*a33*b31 *m3*u3 *v3
5 5
+ 24*a33*b31 *b32*m3*u1*u3*v2 - 24*a33*b31 *b32*m3*u2*u3*v1
5 2 4 2 2
- 3*a33*b31 *m3 *u3*v1 - 4*a33*b31 *b32 *m3*u1 *v3
4 2 2 4 2 2
- 4*a33*b31 *b32 *m3*u2 *v3 - 12*a33*b31 *b32 *m3*u3 *v3
4 2 3 2 2
+ 9*a33*b31 *b32*m3 *u3*v2 + 9*a33*b31 *b32 *m3 *u3*v1
2 3 2 7
- 3*a33*b31 *b32 *m3 *u3*v2 + 8*b31 *m3*u3*v1*v3
6 5 2
- 8*b31 *b32*m3*u3*v2*v3 + 8*b31 *b32 *m3*u3*v1*v3
4 3 2 6
- 8*b31 *b32 *m3*u3*v2*v3))/(3*a33 *b31 )
And again in machine readable form:
HAM=(3*a33*b31**2*u1**2 + 3*a33*b31**2*u2**2 + 6*a33*b31**2*u3**2 - 3*a33*b31*m3
*u1 + 3*a33*b32*m3*u2 + 6*b31**3*u3*v1 + 6*b31**2*b32*u3*v2 + 4*b31**2*m3*v3 - 2
*b32**2*m3*v3)/(6*b31**2)$
INT=(k1*(3*a33**2*b31**6*u1**4 + 6*a33**2*b31**6*u1**2*u2**2 + 3*a33**2*b31**6*
u2**4 + 6*a33**2*b31**5*m3*u1**3 + 6*a33**2*b31**5*m3*u1*u2**2 - 6*a33**2*b31**4
*b32*m3*u1**2*u2 - 6*a33**2*b31**4*b32*m3*u2**3 - 3*a33**2*b31**4*m3**2*u2**2 -
6*a33**2*b31**3*b32*m3**2*u1*u2 - 3*a33**2*b31**3*m3**3*u1 - 3*a33**2*b31**2*b32
**2*m3**2*u1**2 + 3*a33**2*b31**2*b32*m3**3*u2 - 3*a33**2*b31*b32**2*m3**3*u1 +
3*a33**2*b32**3*m3**3*u2 - 12*a33*b31**7*u1*u2*u3*v2 + 12*a33*b31**7*u1*u3**2*v3
+ 12*a33*b31**7*u2**2*u3*v1 + 12*a33*b31**6*b32*u1**2*u3*v2 + 12*a33*b31**6*b32
*u2**2*u3*v2 + 24*a33*b31**6*b32*u2*u3**2*v3 - 4*a33*b31**6*m3*u1**2*v3 - 4*a33*
b31**6*m3*u2**2*v3 + 12*a33*b31**6*m3*u3**2*v3 + 24*a33*b31**5*b32*m3*u1*u3*v2 -
24*a33*b31**5*b32*m3*u2*u3*v1 - 3*a33*b31**5*m3**2*u3*v1 - 4*a33*b31**4*b32**2*
m3*u1**2*v3 - 4*a33*b31**4*b32**2*m3*u2**2*v3 - 12*a33*b31**4*b32**2*m3*u3**2*v3
+ 9*a33*b31**4*b32*m3**2*u3*v2 + 9*a33*b31**3*b32**2*m3**2*u3*v1 - 3*a33*b31**2
*b32**3*m3**2*u3*v2 + 8*b31**7*m3*u3*v1*v3 - 8*b31**6*b32*m3*u3*v2*v3 + 8*b31**5
*b32**2*m3*u3*v1*v3 - 8*b31**4*b32**3*m3*u3*v2*v3))/(3*a33**2*b31**6)$