Solution 31 to problem e3quant
Expressions |
Parameters |
Relevance |
Back to problem e3quant
Expressions
The solution is given through the following expressions:
1
a22=---*a33
2
b22=0
b32=0
b33=0
c12=0
c13=0
c22=0
c23=0
1 2
- ---*b31
2
c33=-------------
a33
n1=0
n2=0
a33*m1
n3=--------
b31
m3=0
r6=0
2
- 3*b31*k1*m2 + 4*k1*m1
r5=---------------------------
a33*b31
- 3*b31*k1*m1 - 4*k1*m1*m2
r4=-----------------------------
a33*b31
- k1*m1
r3=----------
b31
r2=0
r1=0
q20=0
15 2 2 2
- ----*b31 *k1 + 4*b31*k1*m2 + 4*k1*m1 + 4*k1*m2
2
q19=-----------------------------------------------------
2
a33
q18=0
q17=0
15 2 2 2
- ----*b31 *k1 + 4*b31*k1*m2 + 4*k1*m1 + 4*k1*m2
2
q16=-----------------------------------------------------
2
a33
8*k1*m1
q14=---------
a33
5*b31*k1 + 4*k1*m2
q13=--------------------
a33
16*b31*k1*m1 + 8*k1*m1*m2
q11=---------------------------
a33*b31
q10=0
q9=0
q8=0
7 2 2
- ---*b31 *k1 - 4*k1*m1
2
q7=---------------------------
2
b31
2 2
- 8*b31 *k1 - 8*b31*k1*m2 + 8*k1*m1
q6=---------------------------------------
a33*b31
q5=0
q4=0
q3=0
q2=0
1 2 2
---*b31 *k1 - 4*k1*m1
2
q1=------------------------
2
b31
p56=0
2
- 4*b31 *k1*m2
p55=-----------------
3
a33
p54=0
p53=0
2
- 4*b31 *k1*m1
p52=-----------------
3
a33
p51=0
p50=0
p49=0
p48=0
p47=0
- 4*b31*k1*m1
p46=----------------
2
a33
p45=0
p44=0
p43=0
p42=0
p41=0
p40=0
4*b31*k1
p39=----------
a33
p38=0
p37=0
p36=0
8*b31*k1*m1
p35=-------------
2
a33
p34=0
2
12*b31 *k1
p33=------------
2
a33
p32=0
p31=0
8*b31*k1
p30=----------
a33
p29=0
p28=0
p27=0
p26=0
- 8*b31*k1 - 4*k1*m2
p25=-----------------------
a33
4*k1*m1
p24=---------
a33
- 4*k1*m1
p23=------------
b31
p22=0
p21=0
2
- 12*b31 *k1
p20=---------------
2
a33
p19=0
8*b31*k1*m1
p18=-------------
2
a33
p17=0
p16=0
8*k1*m1
p15=---------
a33
p14=0
p13=0
p12=0
p11=0
- 8*k1*m1
p10=------------
a33
- 16*b31*k1 - 8*k1*m2
p9=------------------------
a33
p8= - 4*k1
p7=0
p6=0
8*b31*k1 + 4*k1*m2
p5=--------------------
a33
- 4*k1*m1
p4=------------
a33
- 4*k1*m1
p3=------------
b31
p2=0
p1=0
k125=0
4
- 2*b31 *k1
k124=--------------
4
a33
k123=0
4
- b31 *k1
k122=------------
4
a33
k121=0
k120=0
k119=0
k118=0
4
- 2*b31 *k1
k117=--------------
4
a33
k116=0
4
- 2*b31 *k1
k115=--------------
4
a33
k114=0
k113=0
4
- b31 *k1
k112=------------
4
a33
k110=0
k109=0
k108=0
3
- 4*b31 *k1
k107=--------------
3
a33
k106=0
k105=0
k104=0
k103=0
k102=0
k100=0
k99=0
k98=0
k97=0
k95=0
k94=0
k93=0
k91=0
k90=0
k89=0
k88=0
k87=0
k86=0
k85=0
k84=0
k83=0
k82=0
k81=0
2
8*b31 *k1
k80=-----------
2
a33
k79=0
k78=0
k77=0
k76=0
k75=0
k74=0
k73=0
k72=0
2
- 2*b31 *k1
k71=--------------
2
a33
k70=0
k69=0
k68=0
k67=0
k66=0
k65=0
k64=0
4*b31*k1
k63=----------
a33
k62=0
k61=0
k59=0
k58=0
k57=k1
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
2
8*b31 *k1
k43=-----------
2
a33
k42=0
k41=0
k40=0
k39=0
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=0
k31=0
k30=0
- 8*b31*k1
k29=-------------
a33
k28=0
k27=0
k26=0
k25=0
k24=0
k23=0
k22=0
2
- 2*b31 *k1
k21=--------------
2
a33
k20=0
k19=0
k18=0
k17=0
k16=0
k15=0
k14=0
- 4*b31*k1
k13=-------------
a33
k12=0
k11=0
k10=0
k9=0
k8=0
k7=2*k1
k6=0
k5=0
k4=0
k3=0
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:
m1,m2,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=(a33 *b31*u1 + a33 *b31*u2 + 2*a33 *b31*u3 + 2*a33 *m1*u3
2 3 2
+ 2*a33*b31 *u3*v1 + 2*a33*b31*m1*v1 + 2*a33*b31*m2*v2 - b31 *v3 )/(2*a33
*b31)
4 2 4 4 2 2 4 2 2
INT=(k1*(2*a33 *b31 *u1 + 4*a33 *b31 *u1 *u2*u3 + a33 *b31 *u1
4 2 2 4 2 2 4 2
+ 2*a33 *b31 *u1*u2*v2 - 8*a33 *b31 *u1*u3 - 7*a33 *b31 *u1*v1
4 2 4 2 4
- 8*a33 *b31*m1*u1*u2 - 8*a33 *b31*m1*u2 *v2 - 2*a33 *b31*m1*u3
4 2 2 4 2 3 3 2
- 8*a33 *m1 *u1 - 8*a33 *m1 *u1*v1 + 16*a33 *b31 *u1 *u3
3 3 3 3 3 2 3 3
- 8*a33 *b31 *u1*u3 - 16*a33 *b31 *u1*u3*v1 + 8*a33 *b31 *u1*u3*v3
3 3 2 3 3 3 3 2
- 32*a33 *b31 *u2*u3 - 16*a33 *b31 *u2*u3*v2 + 8*a33 *b31 *u2*v1*v2
3 3 2 3 3 3 3 2
- 16*a33 *b31 *u3 + 10*a33 *b31 *u3*v2 + 16*a33 *b31 *v1 *v2
3 2 3 2 3 2 3
+ 8*a33 *b31 *m1*u1*u3*v2 + 32*a33 *b31 *m1*u1*v2 - 8*a33 *b31 *m1*u2
3 2 3 2 3
+ 16*a33 *b31 *m1*u2*u3*v1 - 16*a33 *b31 *m1*u3
3 2 3 2 3 2 2
+ 16*a33 *b31 *m1*v1*v2 - 6*a33 *b31 *m1*v1 + 8*a33 *b31 *m2*u1 *u3
3 2 2 3 2
- 16*a33 *b31 *m2*u2*u3 - 8*a33 *b31 *m2*u2*u3*v2
3 2 2 3 2 3 2
- 16*a33 *b31 *m2*u3 + 8*a33 *b31 *m2*u3*v2 - 6*a33 *b31 *m2*v2
3 2 2 3 2 3
+ 16*a33 *b31*m1 *u3 + 8*a33 *b31*m1 *v2 + 16*a33 *b31*m1*m2*u1*v2
3 2 4 3 2 4
- 8*a33 *b31*m1*m2*v1 - 4*a33 *b31 *u1 *v3 - 4*a33 *b31 *u1*u2*u3*v1
2 4 2 2 4 2 4 3
+ 16*a33 *b31 *u1*u3 *v2 - 15*a33 *b31 *u1*v3 + 16*a33 *b31 *u3 *v3
2 4 2 2 4 3 2 4
+ 24*a33 *b31 *u3*v2 - 24*a33 *b31 *v1 - 15*a33 *b31 *v1*v3
2 3 2 3 2
- 8*a33 *b31 *m1*u1*v2*v3 + 16*a33 *b31 *m1*u2*v1
2 3 3 2 3 2 3
+ 16*a33 *b31 *m1*v2 + 8*a33 *b31 *m2*u1*v3 + 8*a33 *b31 *m2*v1*v3
2 2 2 2 2 2
+ 8*a33 *b31 *m1 *u1*v3 + 8*a33 *b31 *m1 *v1*v3
2 2 2 2 2 2 5 2
+ 8*a33 *b31 *m2 *u1*v3 + 8*a33 *b31 *m2 *v1*v3 - 8*a33*b31 *u1*u2*v3
4 2 4 2 6 2
- 8*a33*b31 *m1*u2*v3 - 8*a33*b31 *m2*v2*v3 - 2*b31 *u1*v1*v3
6 2 6 3 6 2 2 6 3
- 4*b31 *u2*v2*v3 - 2*b31 *u2*v3 - 4*b31 *v1 *v3 - 4*b31 *v1*v3 ))/
4 2
(2*a33 *b31 )
And again in machine readable form:
HAM=(a33**2*b31*u1**2 + a33**2*b31*u2**2 + 2*a33**2*b31*u3**2 + 2*a33**2*m1*u3 +
2*a33*b31**2*u3*v1 + 2*a33*b31*m1*v1 + 2*a33*b31*m2*v2 - b31**3*v3**2)/(2*a33*
b31)$
INT=(k1*(2*a33**4*b31**2*u1**4 + 4*a33**4*b31**2*u1**2*u2*u3 + a33**4*b31**2*u1
**2 + 2*a33**4*b31**2*u1*u2*v2**2 - 8*a33**4*b31**2*u1*u3**2 - 7*a33**4*b31**2*
u1*v1 - 8*a33**4*b31*m1*u1*u2**2 - 8*a33**4*b31*m1*u2**2*v2 - 2*a33**4*b31*m1*u3
- 8*a33**4*m1**2*u1**2 - 8*a33**4*m1**2*u1*v1 + 16*a33**3*b31**3*u1**2*u3 - 8*
a33**3*b31**3*u1*u3**3 - 16*a33**3*b31**3*u1*u3*v1**2 + 8*a33**3*b31**3*u1*u3*v3
- 32*a33**3*b31**3*u2*u3**2 - 16*a33**3*b31**3*u2*u3*v2 + 8*a33**3*b31**3*u2*v1
*v2**2 - 16*a33**3*b31**3*u3**2 + 10*a33**3*b31**3*u3*v2 + 16*a33**3*b31**3*v1**
2*v2 + 8*a33**3*b31**2*m1*u1*u3*v2 + 32*a33**3*b31**2*m1*u1*v2 - 8*a33**3*b31**2
*m1*u2**3 + 16*a33**3*b31**2*m1*u2*u3*v1 - 16*a33**3*b31**2*m1*u3**3 + 16*a33**3
*b31**2*m1*v1*v2 - 6*a33**3*b31**2*m1*v1 + 8*a33**3*b31**2*m2*u1**2*u3 - 16*a33
**3*b31**2*m2*u2*u3**2 - 8*a33**3*b31**2*m2*u2*u3*v2 - 16*a33**3*b31**2*m2*u3**2
+ 8*a33**3*b31**2*m2*u3*v2 - 6*a33**3*b31**2*m2*v2 + 16*a33**3*b31*m1**2*u3**2
+ 8*a33**3*b31*m1**2*v2 + 16*a33**3*b31*m1*m2*u1*v2 - 8*a33**3*b31*m1*m2*v1 - 4*
a33**2*b31**4*u1**3*v3 - 4*a33**2*b31**4*u1*u2*u3*v1 + 16*a33**2*b31**4*u1*u3**2
*v2 - 15*a33**2*b31**4*u1*v3 + 16*a33**2*b31**4*u3**3*v3 + 24*a33**2*b31**4*u3*
v2**2 - 24*a33**2*b31**4*v1**3 - 15*a33**2*b31**4*v1*v3 - 8*a33**2*b31**3*m1*u1*
v2*v3 + 16*a33**2*b31**3*m1*u2*v1**2 + 16*a33**2*b31**3*m1*v2**3 + 8*a33**2*b31
**3*m2*u1*v3 + 8*a33**2*b31**3*m2*v1*v3 + 8*a33**2*b31**2*m1**2*u1*v3 + 8*a33**2
*b31**2*m1**2*v1*v3 + 8*a33**2*b31**2*m2**2*u1*v3 + 8*a33**2*b31**2*m2**2*v1*v3
- 8*a33*b31**5*u1*u2*v3**2 - 8*a33*b31**4*m1*u2*v3**2 - 8*a33*b31**4*m2*v2*v3**2
- 2*b31**6*u1*v1*v3**2 - 4*b31**6*u2*v2*v3**2 - 2*b31**6*u2*v3**3 - 4*b31**6*v1
**2*v3**2 - 4*b31**6*v1*v3**3))/(2*a33**4*b31**2)$