Solution 70 to problem e3null
Expressions |
Parameters |
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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=0
m2=0
m3=0
r6=0
r5=0
r4=0
2 2 8 2 8 3 8 2
r3=( - ---*a33 *n3*q1 + ----*k1*n1 *n3 - ----*k1*n3 - ----*k10*n1 *n3
3 27 27 27
8 3 3
+ ----*k10*n3 )/a33
27
2 2 8 2 8 2 8 2
r2=( - ---*a33 *n2*q1 + ----*k1*n1 *n2 - ----*k1*n2*n3 - ----*k10*n1 *n2
3 27 27 27
8 2 3
+ ----*k10*n2*n3 )/a33
27
2 2 8 3 8 2 8 3
r1=( - ---*a33 *n1*q1 + ----*k1*n1 - ----*k1*n1*n3 - ----*k10*n1
3 27 27 27
8 2 3
+ ----*k10*n1*n3 )/a33
27
q19=0
q18=0
q16=0
q15=0
q14=0
q13=0
q12=0
q11=0
q10=0
q9=0
q8=0
q7=0
8 8
---*k1*n2*n3 - ---*k10*n2*n3
9 9
q5=------------------------------
2
a33
8 8
---*k1*n1*n3 - ---*k10*n1*n3
9 9
q4=------------------------------
2
a33
2 4 2 4 2 4 2 4 2
a33 *q1 - ---*k1*n1 + ---*k1*n2 + ---*k10*n1 - ---*k10*n2
9 9 9 9
q3=---------------------------------------------------------------
2
a33
8 8
---*k1*n1*n2 - ---*k10*n1*n2
9 9
q2=------------------------------
2
a33
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
p23=0
p22=0
p21=0
p20=0
p19=0
p18=0
p17=0
p16=0
p15=0
p14=0
p13=0
p12=0
p11=0
2
- ---*k10*n3
3
p10=---------------
a33
2
- ---*k10*n2
3
p9=---------------
a33
2
- ---*k10*n1
3
p8=---------------
a33
4 2
- ---*k1*n3 + ---*k10*n3
3 3
p7=---------------------------
a33
p6=0
4 2
- ---*k1*n3 + ---*k10*n3
3 3
p5=---------------------------
a33
4 2
- ---*k1*n2 + ---*k10*n2
3 3
p4=---------------------------
a33
4 2
- ---*k1*n1 + ---*k10*n1
3 3
p3=---------------------------
a33
4 2
- ---*k1*n2 + ---*k10*n2
3 3
p2=---------------------------
a33
4 2
- ---*k1*n1 + ---*k10*n1
3 3
p1=---------------------------
a33
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
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
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
k25=0
k24=0
k23=0
k21=0
k20=0
k19=0
k18=0
k17=0
k16=0
k14=0
k13=0
k12=k10
k11=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:
n3,q1,k1,n1,n2,k10,a33
Relevance for the application:
The following expression INT is a first
integral for the Hamiltonian HAM:
2 2 2
- a33*u1 - a33*u2 + 2*a33*u3 + 2*n1*u1 + 2*n2*u2 + 2*n3*u3
HAM=----------------------------------------------------------------
2
3 4 3 2 2 3 4 3 2 2
INT=(27*a33 *k1*u1 + 54*a33 *k1*u1 *u2 + 27*a33 *k1*u2 + 27*a33 *k10*u1 *u3
3 2 2 3 2 3 2
+ 27*a33 *k10*u2 *u3 + 27*a33 *q1*u1 + 27*a33 *q1*u2
2 3 2 2 2 2
- 36*a33 *k1*n1*u1 - 36*a33 *k1*n1*u1*u2 - 36*a33 *k1*n2*u1 *u2
2 3 2 2 2 2
- 36*a33 *k1*n2*u2 - 36*a33 *k1*n3*u1 *u3 - 36*a33 *k1*n3*u2 *u3
2 3 2 2 2 2
+ 18*a33 *k10*n1*u1 + 18*a33 *k10*n1*u1*u2 - 18*a33 *k10*n1*u1*u3
2 2 2 3 2 2
+ 18*a33 *k10*n2*u1 *u2 + 18*a33 *k10*n2*u2 - 18*a33 *k10*n2*u2*u3
2 2 2 2 2 3
+ 18*a33 *k10*n3*u1 *u3 + 18*a33 *k10*n3*u2 *u3 - 18*a33 *k10*n3*u3
2 2 2
- 18*a33 *n1*q1*u1 - 18*a33 *n2*q1*u2 - 18*a33 *n3*q1*u3
2 2
- 12*a33*k1*n1 *u2 + 24*a33*k1*n1*n2*u1*u2 + 24*a33*k1*n1*n3*u1*u3
2 2 2 2
+ 12*a33*k1*n2 *u2 + 24*a33*k1*n2*n3*u2*u3 + 12*a33*k10*n1 *u2
2 2
- 24*a33*k10*n1*n2*u1*u2 - 24*a33*k10*n1*n3*u1*u3 - 12*a33*k10*n2 *u2
3 2 2
- 24*a33*k10*n2*n3*u2*u3 + 8*k1*n1 *u1 + 8*k1*n1 *n2*u2 + 8*k1*n1 *n3*u3
2 2 3 3
- 8*k1*n1*n3 *u1 - 8*k1*n2*n3 *u2 - 8*k1*n3 *u3 - 8*k10*n1 *u1
2 2 2 2
- 8*k10*n1 *n2*u2 - 8*k10*n1 *n3*u3 + 8*k10*n1*n3 *u1 + 8*k10*n2*n3 *u2
3 3
+ 8*k10*n3 *u3)/(27*a33 )
And again in machine readable form:
HAM=( - a33*u1**2 - a33*u2**2 + 2*a33*u3**2 + 2*n1*u1 + 2*n2*u2 + 2*n3*u3)/2$
INT=(27*a33**3*k1*u1**4 + 54*a33**3*k1*u1**2*u2**2 + 27*a33**3*k1*u2**4 + 27*a33
**3*k10*u1**2*u3**2 + 27*a33**3*k10*u2**2*u3**2 + 27*a33**3*q1*u1**2 + 27*a33**3
*q1*u2**2 - 36*a33**2*k1*n1*u1**3 - 36*a33**2*k1*n1*u1*u2**2 - 36*a33**2*k1*n2*
u1**2*u2 - 36*a33**2*k1*n2*u2**3 - 36*a33**2*k1*n3*u1**2*u3 - 36*a33**2*k1*n3*u2
**2*u3 + 18*a33**2*k10*n1*u1**3 + 18*a33**2*k10*n1*u1*u2**2 - 18*a33**2*k10*n1*
u1*u3**2 + 18*a33**2*k10*n2*u1**2*u2 + 18*a33**2*k10*n2*u2**3 - 18*a33**2*k10*n2
*u2*u3**2 + 18*a33**2*k10*n3*u1**2*u3 + 18*a33**2*k10*n3*u2**2*u3 - 18*a33**2*
k10*n3*u3**3 - 18*a33**2*n1*q1*u1 - 18*a33**2*n2*q1*u2 - 18*a33**2*n3*q1*u3 - 12
*a33*k1*n1**2*u2**2 + 24*a33*k1*n1*n2*u1*u2 + 24*a33*k1*n1*n3*u1*u3 + 12*a33*k1*
n2**2*u2**2 + 24*a33*k1*n2*n3*u2*u3 + 12*a33*k10*n1**2*u2**2 - 24*a33*k10*n1*n2*
u1*u2 - 24*a33*k10*n1*n3*u1*u3 - 12*a33*k10*n2**2*u2**2 - 24*a33*k10*n2*n3*u2*u3
+ 8*k1*n1**3*u1 + 8*k1*n1**2*n2*u2 + 8*k1*n1**2*n3*u3 - 8*k1*n1*n3**2*u1 - 8*k1
*n2*n3**2*u2 - 8*k1*n3**3*u3 - 8*k10*n1**3*u1 - 8*k10*n1**2*n2*u2 - 8*k10*n1**2*
n3*u3 + 8*k10*n1*n3**2*u1 + 8*k10*n2*n3**2*u2 + 8*k10*n3**3*u3)/(27*a33**3)$