Solution 28 to problem e3c2new
Remaining equations |
Expressions |
Parameters |
Relevance |
Back to problem e3c2new
Equations
The following unsolved equations remain:
2 2
0=m1 + m2
Expressions
The solution is given through the following expressions:
1
a22=---*a33
2
b22=0
b31=0
b32=0
b33=0
c12=0
c13=0
c22=0
c23=0
c33=0
- m1*n2
n1=----------
m2
n3=0
m3=0
r6=0
2 2 2 2
8*k1*m1 *n2 - 8*k1*m2 *n2
r5=-----------------------------
3
a33 *m2
3 2 2 2
- 8*k1*m1 *n2 + 8*k1*m1*m2 *n2
r4=-----------------------------------
3 2
a33 *m2
r3=0
2 3 2 3
- 24*k1*m1 *n2 - 24*k1*m2 *n2
r2=----------------------------------
3 2
a33 *m2
3 3 2 3
- 24*k1*m1 *n2 - 24*k1*m1*m2 *n2
r1=-------------------------------------
3 3
a33 *m2
q20=0
q19=0
q17=0
q16=0
2 2
- 8*k1*m1 - 8*k1*m2
q15=------------------------
2
a33
q14=0
q13=0
2
- 16*k1*m1 *n2
q12=-----------------
2
a33 *m2
16*k1*m1*n2
q11=-------------
2
a33
2 2
- 8*k1*m1 - 8*k1*m2
q10=------------------------
2
a33
q9=0
- 16*k1*m1*n2
q8=----------------
2
a33
16*k1*m2*n2
q7=-------------
2
a33
q5=0
q4=0
2 2 2 2
- 12*k1*m1 *n2 - 8*k1*m2 *n2
q3=---------------------------------
2 2
a33 *m2
2
- 8*k1*m1*n2
q2=----------------
2
a33 *m2
2 2 2 2
16*k1*m1 *n2 + 12*k1*m2 *n2
q1=-------------------------------
2 2
a33 *m2
p56=0
p55=0
p54=0
p53=0
p52=0
p51=0
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
p35=0
p34=0
p33=0
p32=0
p31=0
p30=0
p29=0
p28=0
p27=0
p26=0
p25=0
p24=0
- 4*k1*m2
p23=------------
a33
- 8*k1*m1
p22=------------
a33
4*k1*m2
p21=---------
a33
p20=0
p19=0
p18=0
p17=0
p16=0
p15=0
p14=0
4*k1*m1
p13=---------
a33
- 8*k1*m2
p12=------------
a33
- 4*k1*m1
p11=------------
a33
p10=0
p9=0
p8=0
p7=0
p6=0
p5=0
- 4*k1*n2
p4=------------
a33
4*k1*m1*n2
p3=------------
a33*m2
- 4*k1*n2
p2=------------
a33
4*k1*m1*n2
p1=------------
a33*m2
k125=0
k124=0
k122=0
k121=0
k120=0
k119=0
k118=0
k117=0
k116=0
k115=0
k114=0
k113=0
k112=0
k110=0
k109=0
k108=0
k107=0
k106=0
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
k70=0
k69=0
k68=0
k67=0
k66=0
k65=0
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
k42=0
k41=0
k40=0
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=0
k30=0
k29=0
k28=0
k27=0
k26=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=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:
k1,m1,m2,n2,a33
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2 2
HAM=(a33*m2*u1 + a33*m2*u2 + 2*a33*m2*u3 + 2*m1*m2*v1 - 2*m1*n2*u1 + 2*m2 *v2
+ 2*m2*n2*u2)/(2*m2)
3 3 4 3 3 2 2 3 3 4 2 3 2
INT=(k1*(a33 *m2 *u1 + 2*a33 *m2 *u1 *u2 + a33 *m2 *u2 - 4*a33 *m1*m2 *u1 *v1
2 3 2 3 2 2 2 3
- 8*a33 *m1*m2 *u1*u2*v2 + 4*a33 *m1*m2 *u2 *v1 + 4*a33 *m1*m2 *n2*u1
2 2 2 2 4 2 2 4
+ 4*a33 *m1*m2 *n2*u1*u2 + 4*a33 *m2 *u1 *v2 - 8*a33 *m2 *u1*u2*v1
2 4 2 2 3 2 2 3 3
- 4*a33 *m2 *u2 *v2 - 4*a33 *m2 *n2*u1 *u2 - 4*a33 *m2 *n2*u2
2 3 2 2 3 2 2 2
- 8*a33*m1 *m2 *v1 - 8*a33*m1 *m2 *v2 - 16*a33*m1 *m2 *n2*u2*v2
2 2 2 2 2 2
+ 16*a33*m1 *m2*n2 *u1 - 12*a33*m1 *m2*n2 *u2
3 3
+ 16*a33*m1*m2 *n2*u1*v2 - 16*a33*m1*m2 *n2*u2*v1
2 2 5 2 5 2
- 8*a33*m1*m2 *n2 *u1*u2 - 8*a33*m2 *v1 - 8*a33*m2 *v2
4 3 2 2 3 2 2
+ 16*a33*m2 *n2*u1*v1 + 12*a33*m2 *n2 *u1 - 8*a33*m2 *n2 *u2
3 2 3 3 2 2 2
- 8*m1 *m2*n2 *v1 - 24*m1 *n2 *u1 + 8*m1 *m2 *n2 *v2
2 3 3 2 2 3 4 2
- 24*m1 *m2*n2 *u2 + 8*m1*m2 *n2 *v1 - 24*m1*m2 *n2 *u1 - 8*m2 *n2 *v2
3 3 3 3
- 24*m2 *n2 *u2))/(a33 *m2 )
And again in machine readable form:
HAM=(a33*m2*u1**2 + a33*m2*u2**2 + 2*a33*m2*u3**2 + 2*m1*m2*v1 - 2*m1*n2*u1 + 2*
m2**2*v2 + 2*m2*n2*u2)/(2*m2)$
INT=(k1*(a33**3*m2**3*u1**4 + 2*a33**3*m2**3*u1**2*u2**2 + a33**3*m2**3*u2**4 -
4*a33**2*m1*m2**3*u1**2*v1 - 8*a33**2*m1*m2**3*u1*u2*v2 + 4*a33**2*m1*m2**3*u2**
2*v1 + 4*a33**2*m1*m2**2*n2*u1**3 + 4*a33**2*m1*m2**2*n2*u1*u2**2 + 4*a33**2*m2
**4*u1**2*v2 - 8*a33**2*m2**4*u1*u2*v1 - 4*a33**2*m2**4*u2**2*v2 - 4*a33**2*m2**
3*n2*u1**2*u2 - 4*a33**2*m2**3*n2*u2**3 - 8*a33*m1**2*m2**3*v1**2 - 8*a33*m1**2*
m2**3*v2**2 - 16*a33*m1**2*m2**2*n2*u2*v2 + 16*a33*m1**2*m2*n2**2*u1**2 - 12*a33
*m1**2*m2*n2**2*u2**2 + 16*a33*m1*m2**3*n2*u1*v2 - 16*a33*m1*m2**3*n2*u2*v1 - 8*
a33*m1*m2**2*n2**2*u1*u2 - 8*a33*m2**5*v1**2 - 8*a33*m2**5*v2**2 + 16*a33*m2**4*
n2*u1*v1 + 12*a33*m2**3*n2**2*u1**2 - 8*a33*m2**3*n2**2*u2**2 - 8*m1**3*m2*n2**2
*v1 - 24*m1**3*n2**3*u1 + 8*m1**2*m2**2*n2**2*v2 - 24*m1**2*m2*n2**3*u2 + 8*m1*
m2**3*n2**2*v1 - 24*m1*m2**2*n2**3*u1 - 8*m2**4*n2**2*v2 - 24*m2**3*n2**3*u2))/(
a33**3*m2**3)$