Solution 53 to problem e3quant
Expressions |
Parameters |
Relevance |
Back to problem e3quant
Expressions
The solution is given through the following expressions:
b22=0
b31=0
b32=0
b33=0
c12=0
c13=0
c22=0
c23=0
c33=0
n1=0
n3=0
m1=0
m2=0
m3=0
r6=0
r5=0
r4=0
r3=0
- k1*n2 + n2*q1
r2=------------------
a22 - a33
r1=0
q20=0
q19=0
q18=0
q17=0
q16=0
q14=0
k38*n2
q13=-----------
a22 - a33
q11=0
q10= - k38
q9=0
q8=0
2 2 2 2
q7=( - 4*a22 *k1 + a22 *q1 + 8*a22*a33*k1 - 2*a22*a33*q1 - 4*a33 *k1 + a33 *q1
2 2 2 2
+ k1*n2 - k12*n2 )/(a22 - 2*a22*a33 + a33 )
q6=0
q5=0
q4= - k38
- 2*k1*n2
q3=------------
a22 - a33
q2=0
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
k16*n2
p36=-----------
a22 - a33
p35=0
k16*n2
p34=-----------
a22 - a33
p33=0
p32=0
k16*n2
p31=-----------
a22 - a33
- k38*n2
p30=-----------
a22 - a33
p29=0
p28=2*k38
k12*n2
p27=-----------
a22 - a33
p26=0
- k38*n2
p25=-----------
a22 - a33
p24=0
p23=0
2*k1*n2 - k12*n2
p22=------------------
a22 - a33
p21=0
p20=0
p19=0
p18=0
p17=0
p16=0
p15=0
p14= - 2*k38
p13=0
p12=0
p11=0
p10=0
- k38*n2
p9=-----------
a22 - a33
p8= - 4*k1
p7=0
p6=0
p5=0
p4=0
p3=0
2*k1*n2 - k12*n2
p2=------------------
a22 - a33
p1=0
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=k38
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=k38
k73=0
k72=0
k71=k16
k70=0
k69=k16
k68=0
k67=0
k66=k16
k65=0
k64=0
k63=0
k62=k12
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
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=k16
k20=0
k19=k16
k18=0
k17=0
k15=0
k14=0
k13=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:
q1,k38,k16,k12,k1,n2,a33,a22
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2
HAM=a22*u1 + a22*u2 + a33*u3 + n2*u2
2 4 2 2 2 2 2 2
INT=(a22 *k1*u1 + 2*a22 *k1*u1 *u2*u3 + a22 *k1*u1*u2*v2 - 4*a22 *k1*u1*u3
2 2 2 2 2 2
- 4*a22 *k1*u1*v1 + a22 *k12*u1*v1*v2 + a22 *k12*u2 *u3
2 3 2 3 2
+ a22 *k16*u1 *v1 + a22 *k16*u1 *v3 + a22 *k16*u1*u2*u3*v1
2 3 2 3 2 3 2 2
+ a22 *k16*u1*v2 + a22 *k16*u2 *v1 + a22 *k16*v1*v2 + a22 *k38*u1*u2 *v2
2 2 2 3
- 2*a22 *k38*u1*u3*v1 - a22 *k38*u1*u3 + a22 *k38*u2 *v3
2 2 2 2 2 2
+ a22 *k38*u2*u3*v2*v3 + 2*a22 *k38*u2*v1*v2 - a22 *k38*v1 + a22 *q1*u1
2 4 2
+ a22 *q1*u1*v1 - 2*a22*a33*k1*u1 - 4*a22*a33*k1*u1 *u2*u3
2 2
- 2*a22*a33*k1*u1*u2*v2 + 8*a22*a33*k1*u1*u3 + 8*a22*a33*k1*u1*v1
2 2 2 3
- 2*a22*a33*k12*u1*v1*v2 - 2*a22*a33*k12*u2 *u3 - 2*a22*a33*k16*u1 *v1
3 3
- 2*a22*a33*k16*u1 *v3 - 2*a22*a33*k16*u1*u2*u3*v1 - 2*a22*a33*k16*u1*v2
3 3 2
- 2*a22*a33*k16*u2 *v1 - 2*a22*a33*k16*v1*v2 - 2*a22*a33*k38*u1*u2 *v2
3
+ 4*a22*a33*k38*u1*u3*v1 + 2*a22*a33*k38*u1*u3 - 2*a22*a33*k38*u2 *v3
2
- 2*a22*a33*k38*u2*u3*v2*v3 - 4*a22*a33*k38*u2*v1*v2 + 2*a22*a33*k38*v1
2 2
- 2*a22*a33*q1*u1 - 2*a22*a33*q1*u1*v1 + 2*a22*k1*n2*u1 *u2
2
+ 2*a22*k1*n2*u1*u2*v2 - 2*a22*k1*n2*u2 - a22*k1*n2*u2
2
- a22*k12*n2*u1 *u2 - a22*k12*n2*u1*u2*v2 + a22*k12*n2*u1*v1*v2
2 2 2
+ a22*k16*n2*u1 *v3 + a22*k16*n2*u1*v2 + a22*k16*n2*v1*v2
2
- a22*k38*n2*u2*u3 - a22*k38*n2*u2*u3*v2 + a22*k38*n2*u3*v2
2 2 4 2 2
- a22*k38*n2*v1 *v2 + a22*n2*q1*u2 + a33 *k1*u1 + 2*a33 *k1*u1 *u2*u3
2 2 2 2 2
+ a33 *k1*u1*u2*v2 - 4*a33 *k1*u1*u3 - 4*a33 *k1*u1*v1
2 2 2 2 2 2 3
+ a33 *k12*u1*v1*v2 + a33 *k12*u2 *u3 + a33 *k16*u1 *v1
2 3 2 2 3
+ a33 *k16*u1 *v3 + a33 *k16*u1*u2*u3*v1 + a33 *k16*u1*v2
2 3 2 3 2 2
+ a33 *k16*u2 *v1 + a33 *k16*v1*v2 + a33 *k38*u1*u2 *v2
2 2 2 3
- 2*a33 *k38*u1*u3*v1 - a33 *k38*u1*u3 + a33 *k38*u2 *v3
2 2 2 2 2 2
+ a33 *k38*u2*u3*v2*v3 + 2*a33 *k38*u2*v1*v2 - a33 *k38*v1 + a33 *q1*u1
2 2
+ a33 *q1*u1*v1 - 2*a33*k1*n2*u1 *u2 - 2*a33*k1*n2*u1*u2*v2
2 2
+ 2*a33*k1*n2*u2 + a33*k1*n2*u2 + a33*k12*n2*u1 *u2 + a33*k12*n2*u1*u2*v2
2 2
- a33*k12*n2*u1*v1*v2 - a33*k16*n2*u1 *v3 - a33*k16*n2*u1*v2
2 2
- a33*k16*n2*v1*v2 + a33*k38*n2*u2*u3 + a33*k38*n2*u2*u3*v2
2 2
- a33*k38*n2*u3*v2 + a33*k38*n2*v1 *v2 - a33*n2*q1*u2 + k1*n2 *u1*v1
2 2 2
- k12*n2 *u1*v1)/(a22 - 2*a22*a33 + a33 )
And again in machine readable form:
HAM=a22*u1**2 + a22*u2**2 + a33*u3**2 + n2*u2$
INT=(a22**2*k1*u1**4 + 2*a22**2*k1*u1**2*u2*u3 + a22**2*k1*u1*u2*v2**2 - 4*a22**
2*k1*u1*u3**2 - 4*a22**2*k1*u1*v1 + a22**2*k12*u1*v1*v2**2 + a22**2*k12*u2**2*u3
**2 + a22**2*k16*u1**3*v1 + a22**2*k16*u1**3*v3 + a22**2*k16*u1*u2*u3*v1 + a22**
2*k16*u1*v2**3 + a22**2*k16*u2**3*v1 + a22**2*k16*v1*v2**3 + a22**2*k38*u1*u2**2
*v2 - 2*a22**2*k38*u1*u3*v1 - a22**2*k38*u1*u3 + a22**2*k38*u2**3*v3 + a22**2*
k38*u2*u3*v2*v3 + 2*a22**2*k38*u2*v1*v2 - a22**2*k38*v1**2 + a22**2*q1*u1**2 +
a22**2*q1*u1*v1 - 2*a22*a33*k1*u1**4 - 4*a22*a33*k1*u1**2*u2*u3 - 2*a22*a33*k1*
u1*u2*v2**2 + 8*a22*a33*k1*u1*u3**2 + 8*a22*a33*k1*u1*v1 - 2*a22*a33*k12*u1*v1*
v2**2 - 2*a22*a33*k12*u2**2*u3**2 - 2*a22*a33*k16*u1**3*v1 - 2*a22*a33*k16*u1**3
*v3 - 2*a22*a33*k16*u1*u2*u3*v1 - 2*a22*a33*k16*u1*v2**3 - 2*a22*a33*k16*u2**3*
v1 - 2*a22*a33*k16*v1*v2**3 - 2*a22*a33*k38*u1*u2**2*v2 + 4*a22*a33*k38*u1*u3*v1
+ 2*a22*a33*k38*u1*u3 - 2*a22*a33*k38*u2**3*v3 - 2*a22*a33*k38*u2*u3*v2*v3 - 4*
a22*a33*k38*u2*v1*v2 + 2*a22*a33*k38*v1**2 - 2*a22*a33*q1*u1**2 - 2*a22*a33*q1*
u1*v1 + 2*a22*k1*n2*u1**2*u2 + 2*a22*k1*n2*u1*u2*v2 - 2*a22*k1*n2*u2**2 - a22*k1
*n2*u2 - a22*k12*n2*u1**2*u2 - a22*k12*n2*u1*u2*v2 + a22*k12*n2*u1*v1*v2 + a22*
k16*n2*u1**2*v3 + a22*k16*n2*u1*v2**2 + a22*k16*n2*v1*v2**2 - a22*k38*n2*u2*u3**
2 - a22*k38*n2*u2*u3*v2 + a22*k38*n2*u3*v2 - a22*k38*n2*v1**2*v2 + a22*n2*q1*u2
+ a33**2*k1*u1**4 + 2*a33**2*k1*u1**2*u2*u3 + a33**2*k1*u1*u2*v2**2 - 4*a33**2*
k1*u1*u3**2 - 4*a33**2*k1*u1*v1 + a33**2*k12*u1*v1*v2**2 + a33**2*k12*u2**2*u3**
2 + a33**2*k16*u1**3*v1 + a33**2*k16*u1**3*v3 + a33**2*k16*u1*u2*u3*v1 + a33**2*
k16*u1*v2**3 + a33**2*k16*u2**3*v1 + a33**2*k16*v1*v2**3 + a33**2*k38*u1*u2**2*
v2 - 2*a33**2*k38*u1*u3*v1 - a33**2*k38*u1*u3 + a33**2*k38*u2**3*v3 + a33**2*k38
*u2*u3*v2*v3 + 2*a33**2*k38*u2*v1*v2 - a33**2*k38*v1**2 + a33**2*q1*u1**2 + a33
**2*q1*u1*v1 - 2*a33*k1*n2*u1**2*u2 - 2*a33*k1*n2*u1*u2*v2 + 2*a33*k1*n2*u2**2 +
a33*k1*n2*u2 + a33*k12*n2*u1**2*u2 + a33*k12*n2*u1*u2*v2 - a33*k12*n2*u1*v1*v2
- a33*k16*n2*u1**2*v3 - a33*k16*n2*u1*v2**2 - a33*k16*n2*v1*v2**2 + a33*k38*n2*
u2*u3**2 + a33*k38*n2*u2*u3*v2 - a33*k38*n2*u3*v2 + a33*k38*n2*v1**2*v2 - a33*n2
*q1*u2 + k1*n2**2*u1*v1 - k12*n2**2*u1*v1)/(a22**2 - 2*a22*a33 + a33**2)$