Solution 42 to problem e3quant
Remaining equations |
Expressions |
Parameters |
Relevance |
Back to problem e3quant
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
m3=0
392 2 88 2
- 16*a33*k1*m1*m2*n2 + -----*k1*m1 *n2*n3 + ----*k1*m2 *n2*n3
19 19
r6=----------------------------------------------------------------
3
a33 *m2
2 2 2 2 2 2
- 3*a33 *k1*m2 + 4*a33*k1*m1*m2*n3 + 8*k1*m1 *n2 - 8*k1*m2 *n2
r5=--------------------------------------------------------------------
3
a33 *m2
r4=
2 2 3 196 3 2 500 2 2
- 3*a33 *k1*m1*m2 - 4*a33*k1*m2 *n3 + -----*k1*m1 *n2 + -----*k1*m1*m2 *n2
19 19
--------------------------------------------------------------------------------
3 2
a33 *m2
2 2 2 588 2 2
r3=( - a33 *k1*m2 *n3 + 4*a33*k1*m1*m2*n2 + -----*k1*m1 *n2 *n3
19
588 2 2 3 2
+ -----*k1*m2 *n2 *n3)/(a33 *m2 )
19
5668 2 3920 2
a33*k1*m1*m2*n2 - ------*k1*m1 *n2*n3 - ------*k1*m2 *n2*n3
437 437
r2=-------------------------------------------------------------
2
a33 *m1*m2
3920 2 2172 2
3*a33*k1*m1*m2*n2 + ------*k1*m1 *n2*n3 + ------*k1*m2 *n2*n3
437 437
r1=---------------------------------------------------------------
2 2
a33 *m2
q20=0
2 2
4*k1*m1 + 4*k1*m2
q19=---------------------
2
a33
q18=0
q17=0
2 2
4*k1*m1 + 4*k1*m2
q16=---------------------
2
a33
4*k1*m1
q14=---------
a33
4*k1*m2
q13=---------
a33
8*a33*k1*m1 + 8*k1*m2*n3
q11=--------------------------
2
a33
392 2 696 2
-----*k1*m1 *n2 + -----*k1*m2 *n2
19 19
q10=-----------------------------------
2
a33 *m2
- 16*k1*m1*n2
q9=----------------
2
a33
- 4*a33*k1*m1*n2 + 8*k1*m2*n2*n3
q8=-----------------------------------
2
a33 *m2
7 2 2 294 2 2 370 2 2 2 2
- ---*a33 *k1*m2 + -----*k1*m1 *n2 + -----*k1*m2 *n2 - 4*k1*m2 *n3
2 19 19
q7=-------------------------------------------------------------------------
2 2
a33 *m2
- 8*a33*k1*m2 + 8*k1*m1*n3
q6=-----------------------------
2
a33
16*k1*m1*n2
q5=-------------
2
a33
16*k1*m2*n2
q4=-------------
2
a33
4*a33*k1*m2*n2 - 8*k1*m1*n2*n3
q3=--------------------------------
2
a33 *m2
2
- 8*k1*m1*n2
q2=----------------
2
a33 *m2
1 2 2 370 2 2 294 2 2 2 2
---*a33 *k1*m2 + -----*k1*m1 *n2 + -----*k1*m2 *n2 - 4*k1*m2 *n3
2 19 19
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
- 4*k1*m2
p25=------------
a33
4*k1*m1
p24=---------
a33
- 4*k1*n3
p23=------------
a33
- 4*k1*n2
p22=------------
a33
p21=0
p20=0
p19=0
p18=0
p17=0
p16=0
p15=0
p14=0
p13=0
p12=0
p11=0
- 8*k1*m1
p10=------------
a33
- 8*k1*m2
p9=------------
a33
p8= - 4*k1
4*k1*m1*n2
p7=------------
a33*m2
p6=0
4*k1*m2
p5=---------
a33
- 4*k1*m1
p4=------------
a33
- 4*k1*n3
p3=------------
a33
- 4*k1*n2
p2=------------
a33
4*k1*m1*n2
p1=------------
a33*m2
k125=0
k124=0
k123=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
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
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
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
k43=0
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
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
k15=0
k14=0
k13=0
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:
k1,m2,m1,n3,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*n3*u3)/(2*m2)
3 2 4 3 2 2 3 2 2
INT=(k1*(874*a33 *m1*m2 *u1 + 1748*a33 *m1*m2 *u1 *u2*u3 + 437*a33 *m1*m2 *u1
3 2 2 3 2 2
+ 874*a33 *m1*m2 *u1*u2*v2 - 3496*a33 *m1*m2 *u1*u3
3 2 2 2 2
- 3059*a33 *m1*m2 *u1*v1 + 3496*a33 *m1 *m2 *u1*u3*v2
2 2 2 2 2 2 3
+ 6992*a33 *m1 *m2 *u1*v2 - 3496*a33 *m1 *m2 *u2
2 2 2 3 2 2 2
- 6992*a33 *m1 *m2 *u3 + 3496*a33 *m1 *m2 *v1*v2
2 2 2 2 2 3
- 2622*a33 *m1 *m2 *v1 + 3496*a33 *m1 *m2*n2*u1
2 2 2 2 2
+ 2622*a33 *m1 *m2*n2*u1 + 3496*a33 *m1 *m2*n2*u2 *u3
2 2 2 3 2
- 3496*a33 *m1 *m2*n2*u2*v1 + 3496*a33 *m1*m2 *u1 *u3
2 3 2 2 3
- 6992*a33 *m1*m2 *u2*u3 - 3496*a33 *m1*m2 *u2*u3*v2
2 3 2 2 3 2 3
- 6992*a33 *m1*m2 *u3 + 3496*a33 *m1*m2 *u3*v2 - 2622*a33 *m1*m2 *v2
2 2 2 2 2
- 3496*a33 *m1*m2 *n2*u1 *u2 - 3496*a33 *m1*m2 *n2*u1*u2*v2
2 2 2 2 2
+ 3496*a33 *m1*m2 *n2*u2 + 874*a33 *m1*m2 *n2*u2
2 2 2 2 2 2
- 3496*a33 *m1*m2 *n3*u1*u2 - 3496*a33 *m1*m2 *n3*u2 *v2
2 2 3 2
- 874*a33 *m1*m2 *n3*u3 + 3496*a33*m1 *m2 *u1*v3
3 2 3 2
+ 3496*a33*m1 *m2 *v1*v3 + 18032*a33*m1 *m2*n2*v1
3 2 2 3 2
+ 17020*a33*m1 *n2 *u1 + 13524*a33*m1 *n2 *u1*v1
3 2 2
+ 7840*a33*m1 *n2*n3*u1 + 13984*a33*m1 *m2 *n2*u2*u3
2 2 2 2
- 13984*a33*m1 *m2 *n2*u3*v1 - 13984*a33*m1 *m2 *n2*v3
2 2 2 2 2
+ 6992*a33*m1 *m2 *n3*u3 + 3496*a33*m1 *m2 *n3*v2
2 2 2 2
- 6992*a33*m1 *m2*n2 *u1*u2 + 3496*a33*m1 *m2*n2 *u3
2 2 2
- 6992*a33*m1 *m2*n2*n3*u2 - 11336*a33*m1 *m2*n2*n3*u2
4 4
+ 3496*a33*m1*m2 *u1*v3 + 3496*a33*m1*m2 *v1*v3
3 3 2
+ 13984*a33*m1*m2 *n2*u1*u3 + 32016*a33*m1*m2 *n2*v1
3 3
+ 6992*a33*m1*m2 *n3*u1*v2 - 3496*a33*m1*m2 *n3*v1
2 2 2 2 2
+ 13524*a33*m1*m2 *n2 *u1 + 17020*a33*m1*m2 *n2 *u1*v1
2 2
+ 4344*a33*m1*m2 *n2*n3*u1 + 6992*a33*m1*m2 *n2*n3*u2*v1
2 2 2 2 2
- 3496*a33*m1*m2 *n3 *u1 - 3496*a33*m1*m2 *n3 *u1*v1
3 4 2 3 2
- 7840*a33*m2 *n2*n3*u2 + 9016*m1 *n2 *v1 + 6992*m1 *m2*n2 *v2
3 3 2 2 2 2
+ 18032*m1 *m2*n2*n3*v3 + 27048*m1 *n2 *n3*u3 + 23000*m1 *m2 *n2 *v1
3 2 3 2 2
- 6992*m1*m2 *n2 *v2 + 4048*m1*m2 *n2*n3*v3 + 27048*m1*m2 *n2 *n3*u3))
3 2
/(874*a33 *m1*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*n3*u3)/(2*m2)$
INT=(k1*(874*a33**3*m1*m2**2*u1**4 + 1748*a33**3*m1*m2**2*u1**2*u2*u3 + 437*a33
**3*m1*m2**2*u1**2 + 874*a33**3*m1*m2**2*u1*u2*v2**2 - 3496*a33**3*m1*m2**2*u1*
u3**2 - 3059*a33**3*m1*m2**2*u1*v1 + 3496*a33**2*m1**2*m2**2*u1*u3*v2 + 6992*a33
**2*m1**2*m2**2*u1*v2 - 3496*a33**2*m1**2*m2**2*u2**3 - 6992*a33**2*m1**2*m2**2*
u3**3 + 3496*a33**2*m1**2*m2**2*v1*v2 - 2622*a33**2*m1**2*m2**2*v1 + 3496*a33**2
*m1**2*m2*n2*u1**3 + 2622*a33**2*m1**2*m2*n2*u1 + 3496*a33**2*m1**2*m2*n2*u2**2*
u3 - 3496*a33**2*m1**2*m2*n2*u2*v1 + 3496*a33**2*m1*m2**3*u1**2*u3 - 6992*a33**2
*m1*m2**3*u2*u3**2 - 3496*a33**2*m1*m2**3*u2*u3*v2 - 6992*a33**2*m1*m2**3*u3**2
+ 3496*a33**2*m1*m2**3*u3*v2 - 2622*a33**2*m1*m2**3*v2 - 3496*a33**2*m1*m2**2*n2
*u1**2*u2 - 3496*a33**2*m1*m2**2*n2*u1*u2*v2 + 3496*a33**2*m1*m2**2*n2*u2**2 +
874*a33**2*m1*m2**2*n2*u2 - 3496*a33**2*m1*m2**2*n3*u1*u2**2 - 3496*a33**2*m1*m2
**2*n3*u2**2*v2 - 874*a33**2*m1*m2**2*n3*u3 + 3496*a33*m1**3*m2**2*u1*v3 + 3496*
a33*m1**3*m2**2*v1*v3 + 18032*a33*m1**3*m2*n2*v1**2 + 17020*a33*m1**3*n2**2*u1**
2 + 13524*a33*m1**3*n2**2*u1*v1 + 7840*a33*m1**3*n2*n3*u1 + 13984*a33*m1**2*m2**
2*n2*u2*u3 - 13984*a33*m1**2*m2**2*n2*u3*v1 - 13984*a33*m1**2*m2**2*n2*v3 + 6992
*a33*m1**2*m2**2*n3*u3**2 + 3496*a33*m1**2*m2**2*n3*v2 - 6992*a33*m1**2*m2*n2**2
*u1*u2 + 3496*a33*m1**2*m2*n2**2*u3 - 6992*a33*m1**2*m2*n2*n3*u2**2 - 11336*a33*
m1**2*m2*n2*n3*u2 + 3496*a33*m1*m2**4*u1*v3 + 3496*a33*m1*m2**4*v1*v3 + 13984*
a33*m1*m2**3*n2*u1*u3 + 32016*a33*m1*m2**3*n2*v1**2 + 6992*a33*m1*m2**3*n3*u1*v2
- 3496*a33*m1*m2**3*n3*v1 + 13524*a33*m1*m2**2*n2**2*u1**2 + 17020*a33*m1*m2**2
*n2**2*u1*v1 + 4344*a33*m1*m2**2*n2*n3*u1 + 6992*a33*m1*m2**2*n2*n3*u2*v1 - 3496
*a33*m1*m2**2*n3**2*u1**2 - 3496*a33*m1*m2**2*n3**2*u1*v1 - 7840*a33*m2**3*n2*n3
*u2 + 9016*m1**4*n2**2*v1 + 6992*m1**3*m2*n2**2*v2 + 18032*m1**3*m2*n2*n3*v3 +
27048*m1**3*n2**2*n3*u3 + 23000*m1**2*m2**2*n2**2*v1 - 6992*m1*m2**3*n2**2*v2 +
4048*m1*m2**3*n2*n3*v3 + 27048*m1*m2**2*n2**2*n3*u3))/(874*a33**3*m1*m2**2)$