Solution 4 to problem e3null
Expressions |
Parameters |
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Expressions
The solution is given through the following expressions:
1 2
- ---*b33
4
a22=-------------
c33
1 2
- ----*b33
12
a33=--------------
c33
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
n3=0
4*c33*n1
m1=----------
b33
4*c33*n2
m2=----------
b33
m3=0
r6=0
- 3*c33*n1*n2*p45
r5=--------------------
3
b33
2
- 3*c33*n1 *p45
r4=------------------
3
b33
r3=0
3
- ---*n1*n2*p45
2
r2=------------------
2
b33
3 2
- ---*n1 *p45
2
r1=----------------
2
b33
q19=0
q18=0
q16=0
q15=0
2 2
c33*n1 *p45 - 7*c33*n2 *p45
q14=-----------------------------
2
b33 *n1
- 14*c33*n2*p45
q13=------------------
2
b33
q12=0
2 2
n1 *p45 - n2 *p45
q11=-------------------
b33*n1
- n2*p45
q10=-----------
b33
- 6*c33*n1*p45
q9=-----------------
2
b33
q8=0
- n2*p45
q7=-----------
b33
q5=0
q4=0
1 2 5 2
---*n1 *p45 + ---*n2 *p45
4 4
q3=---------------------------
c33*n1
5
---*n2*p45
2
q2=------------
c33
3
---*n1*p45
2
q1=------------
c33
p50=0
- 2*c33*n2*p45
p49=-----------------
b33*n1
- 2*c33*p45
p48=--------------
b33
p47=0
n2*p45
p46=--------
n1
p44=0
p43=0
8
- ---*n2*p45
3
p42=---------------
n1
p41=0
p40=0
p39=0
p38= - 3*p45
p37=0
p36=0
2
---*b33*n2*p45
3
p35=----------------
c33*n1
p34=0
p33=0
p32=0
p31=0
2
---*c33*n2*p45
3
p30=----------------
b33*n1
2
---*c33*p45
3
p29=-------------
b33
p28=0
1
---*n2*p45
3
p27=------------
n1
1
p26=---*p45
3
2
---*c33*n2*p45
3
p25=----------------
b33*n1
p24=0
1
p23= - ---*p45
3
1
---*b33*n2*p45
2
p22=----------------
c33*n1
p21=0
p20=0
2
---*b33*n2*p45
3
p19=----------------
c33*n1
2
- ---*b33*p45
3
p18=----------------
c33
2
---*b33*n2*p45
3
p17=----------------
c33*n1
2
---*c33*p45
3
p16=-------------
b33
p15=0
1
---*n2*p45
3
p14=------------
n1
1
---*b33*p45
2
p13=-------------
c33
p12=0
2
---*b33*p45
3
p11=-------------
c33
p10=0
1 2
---*b33 *n2*p45
4
p9=-----------------
2
c33 *n1
1 2
---*b33 *p45
4
p8=--------------
2
c33
p7=0
p6=0
p5=0
1 2
- ---*b33 *n2*p45
6
p4=--------------------
2
c33 *n1
1 2
- ---*b33 *p45
6
p3=-----------------
2
c33
1 2
- ---*b33 *n2*p45
6
p2=--------------------
2
c33 *n1
1 2
- ---*b33 *p45
6
p1=-----------------
2
c33
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
2
---*b33*p45
9
k83=-------------
n1
k82=0
k81=0
k80=0
k79=0
k78=0
k77=0
1 2
- ---*b33 *p45
9
k76=-----------------
c33*n1
k75=0
k74=0
k73=0
k72=0
k71=0
2
---*b33*p45
9
k70=-------------
n1
k69=0
k68=0
k67=0
k66=0
1 3
---*b33 *p45
6
k65=--------------
2
c33 *n1
k64=0
k63=0
1 3
----*b33 *p45
18
k62=---------------
2
c33 *n1
k61=0
1 3
----*b33 *p45
18
k60=---------------
2
c33 *n1
k59=0
k58=0
k57=0
k56=0
1
- ---*c33*p45
9
k55=----------------
n1
k54=0
k53=0
k52=0
k51=0
2
- ---*c33*p45
9
k50=----------------
n1
k49=0
k48=0
k47=0
k46=0
k45=0
1 2
- ---*b33 *p45
9
k44=-----------------
c33*n1
k43=0
1 2
- ----*b33 *p45
18
k42=------------------
c33*n1
k41=0
k40=0
k39=0
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=0
k31=0
k30=0
k28=0
k27=0
k26=0
1
- ---*c33*p45
9
k25=----------------
n1
k24=0
k23=0
k21=0
1 2
- ----*b33 *p45
18
k20=------------------
c33*n1
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
1 4
- ----*b33 *p45
24
k12=------------------
3
c33 *n1
k11=0
1 4
- ----*b33 *p45
24
k10=------------------
3
c33 *n1
k9=0
k8=0
k7=0
k6=0
1 4
- -----*b33 *p45
144
k5=-------------------
3
c33 *n1
k4=0
1 4
- ----*b33 *p45
72
k3=------------------
3
c33 *n1
k2=0
1 4
- -----*b33 *p45
144
k1=-------------------
3
c33 *n1
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:
c33,p45,n2,b33,n1
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
3 2 3 2 3 2 2 2 2
HAM=( - 3*b33 *u1 - 3*b33 *u2 - b33 *u3 + 12*b33 *c33*u3*v3 + 12*b33*c33 *v3
2 2
+ 12*b33*c33*n1*u1 + 12*b33*c33*n2*u2 + 48*c33 *n1*v1 + 48*c33 *n2*v2)/(12
*b33*c33)
7 4 7 2 2 7 2 2 7 4
INT=(p45*( - b33 *u1 - 2*b33 *u1 *u2 - 6*b33 *u1 *u3 - b33 *u2
7 2 2 6 2 6 2
- 6*b33 *u2 *u3 + 8*b33 *c33*u1 *u3*v3 + 8*b33 *c33*u2 *u3*v3
6 3 5 2 2 2 5 2 2 2
+ 24*b33 *c33*u3 *v3 - 8*b33 *c33 *u1 *v2 - 8*b33 *c33 *u2 *v1
5 2 2 2 5 2
- 16*b33 *c33 *u2 *v2 - 16*b33 *c33 *u2*u3*v2*v3
5 3 5 2 5 2
- 24*b33 *c33*n1*u1 - 24*b33 *c33*n1*u1*u2 + 36*b33 *c33*n1*u1*u3
5 2 5 3 5 2
- 24*b33 *c33*n2*u1 *u2 - 24*b33 *c33*n2*u2 + 36*b33 *c33*n2*u2*u3
4 3 2 4 3 2
+ 32*b33 *c33 *u3*v1 *v3 + 32*b33 *c33 *u3*v2 *v3
4 2 4 2 2
- 96*b33 *c33 *n1*u1*u2*v2 + 96*b33 *c33 *n1*u2 *v1
4 2 2 4 2 2
+ 72*b33 *c33 *n1*u3 *v1 + 96*b33 *c33 *n2*u1 *v2
4 2 2 4 2
+ 96*b33 *c33 *n2*u2 *v2 + 96*b33 *c33 *n2*u2*u3*v3
4 2 2 3 4 4 3 4 2 2
+ 72*b33 *c33 *n2*u3 *v2 - 16*b33 *c33 *v1 - 32*b33 *c33 *v1 *v2
3 4 4 3 3 2 3 3 2
- 16*b33 *c33 *v2 + 48*b33 *c33 *n1*u1*v2 + 144*b33 *c33 *n1*u1*v3
3 3 3 3
- 48*b33 *c33 *n1*u2*v1*v2 - 432*b33 *c33 *n1*u3*v1*v3
3 3 2 3 3 2
+ 48*b33 *c33 *n2*u2*v1 + 48*b33 *c33 *n2*u2*v2
3 3 2 3 3
+ 144*b33 *c33 *n2*u2*v3 - 384*b33 *c33 *n2*u3*v2*v3
3 2 2 2 3 2 2 2
+ 216*b33 *c33 *n1 *u1 + 36*b33 *c33 *n1 *u2
3 2 3 2 2 2
+ 360*b33 *c33 *n1*n2*u1*u2 + 180*b33 *c33 *n2 *u2
2 4 3 2 4 2
+ 96*b33 *c33 *n1*v1 + 96*b33 *c33 *n1*v1*v2
2 4 2 2 4 2
- 288*b33 *c33 *n1*v1*v3 + 96*b33 *c33 *n2*v1 *v2
2 4 3 2 4 2
+ 96*b33 *c33 *n2*v2 - 288*b33 *c33 *n2*v2*v3
2 3 2 2 3
+ 144*b33 *c33 *n1 *u2*v2 - 144*b33 *c33 *n1*n2*u1*v2
2 3 2 3 2
- 144*b33 *c33 *n1*n2*u2*v1 - 144*b33 *c33 *n2 *u2*v2
4 2 2 4 2 2
- 864*b33*c33 *n1 *v1 + 144*b33*c33 *n1 *v2
4 4 2 2
- 2016*b33*c33 *n1*n2*v1*v2 - 1008*b33*c33 *n2 *v2
3 3 3 2 4 3
- 216*b33*c33 *n1 *u1 - 216*b33*c33 *n1 *n2*u2 - 432*c33 *n1 *v1
4 2 3 3
- 432*c33 *n1 *n2*v2))/(144*b33 *c33 *n1)
And again in machine readable form:
HAM=( - 3*b33**3*u1**2 - 3*b33**3*u2**2 - b33**3*u3**2 + 12*b33**2*c33*u3*v3 +
12*b33*c33**2*v3**2 + 12*b33*c33*n1*u1 + 12*b33*c33*n2*u2 + 48*c33**2*n1*v1 + 48
*c33**2*n2*v2)/(12*b33*c33)$
INT=(p45*( - b33**7*u1**4 - 2*b33**7*u1**2*u2**2 - 6*b33**7*u1**2*u3**2 - b33**7
*u2**4 - 6*b33**7*u2**2*u3**2 + 8*b33**6*c33*u1**2*u3*v3 + 8*b33**6*c33*u2**2*u3
*v3 + 24*b33**6*c33*u3**3*v3 - 8*b33**5*c33**2*u1**2*v2**2 - 8*b33**5*c33**2*u2
**2*v1**2 - 16*b33**5*c33**2*u2**2*v2**2 - 16*b33**5*c33**2*u2*u3*v2*v3 - 24*b33
**5*c33*n1*u1**3 - 24*b33**5*c33*n1*u1*u2**2 + 36*b33**5*c33*n1*u1*u3**2 - 24*
b33**5*c33*n2*u1**2*u2 - 24*b33**5*c33*n2*u2**3 + 36*b33**5*c33*n2*u2*u3**2 + 32
*b33**4*c33**3*u3*v1**2*v3 + 32*b33**4*c33**3*u3*v2**2*v3 - 96*b33**4*c33**2*n1*
u1*u2*v2 + 96*b33**4*c33**2*n1*u2**2*v1 + 72*b33**4*c33**2*n1*u3**2*v1 + 96*b33
**4*c33**2*n2*u1**2*v2 + 96*b33**4*c33**2*n2*u2**2*v2 + 96*b33**4*c33**2*n2*u2*
u3*v3 + 72*b33**4*c33**2*n2*u3**2*v2 - 16*b33**3*c33**4*v1**4 - 32*b33**3*c33**4
*v1**2*v2**2 - 16*b33**3*c33**4*v2**4 + 48*b33**3*c33**3*n1*u1*v2**2 + 144*b33**
3*c33**3*n1*u1*v3**2 - 48*b33**3*c33**3*n1*u2*v1*v2 - 432*b33**3*c33**3*n1*u3*v1
*v3 + 48*b33**3*c33**3*n2*u2*v1**2 + 48*b33**3*c33**3*n2*u2*v2**2 + 144*b33**3*
c33**3*n2*u2*v3**2 - 384*b33**3*c33**3*n2*u3*v2*v3 + 216*b33**3*c33**2*n1**2*u1
**2 + 36*b33**3*c33**2*n1**2*u2**2 + 360*b33**3*c33**2*n1*n2*u1*u2 + 180*b33**3*
c33**2*n2**2*u2**2 + 96*b33**2*c33**4*n1*v1**3 + 96*b33**2*c33**4*n1*v1*v2**2 -
288*b33**2*c33**4*n1*v1*v3**2 + 96*b33**2*c33**4*n2*v1**2*v2 + 96*b33**2*c33**4*
n2*v2**3 - 288*b33**2*c33**4*n2*v2*v3**2 + 144*b33**2*c33**3*n1**2*u2*v2 - 144*
b33**2*c33**3*n1*n2*u1*v2 - 144*b33**2*c33**3*n1*n2*u2*v1 - 144*b33**2*c33**3*n2
**2*u2*v2 - 864*b33*c33**4*n1**2*v1**2 + 144*b33*c33**4*n1**2*v2**2 - 2016*b33*
c33**4*n1*n2*v1*v2 - 1008*b33*c33**4*n2**2*v2**2 - 216*b33*c33**3*n1**3*u1 - 216
*b33*c33**3*n1**2*n2*u2 - 432*c33**4*n1**3*v1 - 432*c33**4*n1**2*n2*v2))/(144*
b33**3*c33**3*n1)$