Solution 6 to problem e3null
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem e3null
Expressions
The solution is given through the following expressions:
b22=0
b31=0
b32=0
c12=0
c13=0
c22=0
c23=0
n1=0
n2=0
m1=0
m2=0
m3*q1
r6=-------
a22
r5=0
r4=0
r2=0
r1=0
q19=0
q18=0
b33*q1 + m3*p5
q17=----------------
a22
q16=0
q15=0
2
- a22*c33*q1 - k1*m3
q14=------------------------
2
a22
q13=0
q12=0
q11=0
q10=0
2
- a22*c33*q1 - k1*m3
q9=------------------------
2
a22
q8=0
q7=0
q5=0
q4=0
q3=q1
q2=0
a22*k20*m3 + 2*c33*k1*m3
p50=--------------------------
2
a22
p49=0
p48=0
2
a22 *p15 + a22*c33*p5 + 2*b33*k1*m3
p47=-------------------------------------
2
a22
p46=0
p45=0
k20*m3
p44=--------
a22
p43=0
p42=0
p41=0
p40=0
k20*m3
p39=--------
a22
p38=0
p37=0
b33*p5 + k10*m3
p36=-----------------
a22
p35=0
p34=0
2*k1*m3
p33=---------
a22
p32=0
2*k1*m3
p31=---------
a22
p30=0
p29=0
p28=p15
p27=0
p26=0
p25=0
p24=0
p23=0
p22=0
p21=0
p20=0
p19=0
p18=0
p17=0
p16=0
p14=0
p13=0
p12=0
p11=0
p9=0
p8=0
p7=p5
p6=0
p4=0
p3=0
p2=0
p1=0
k104=0
k103=0
a22*b33*k20 + 2*b33*c33*k1
k102=----------------------------
2
a22
k101=0
k100=0
2
- a22*c33*k20 - 2*c33 *k1
k99=----------------------------
2
a22
k98=0
k97=0
k96=0
k95=0
2
- a22*c33*k20 - 2*c33 *k1
k94=----------------------------
2
a22
k93=0
k92=0
2 2
a22 *k20 + a22*c33*k10 + b33 *k1
k91=----------------------------------
2
a22
k90=0
k89=0
a22*k20 + 2*c33*k1
k88=--------------------
a22
k87=0
a22*k20 + 2*c33*k1
k86=--------------------
a22
k85=0
k84=0
b33*k20
k83=---------
a22
k82=0
k81=0
k80=0
k79=0
k78=0
k77=0
k76=2*k20
k75=0
k74=0
k73=0
k72=0
k71=0
b33*k20
k70=---------
a22
k69=0
k68=0
k67=0
k66=0
b33*k10
k65=---------
a22
k64=0
k63=0
2*b33*k1
k62=----------
a22
k61=0
2*b33*k1
k60=----------
a22
k59=0
k58=0
k57=0
k56=0
2
- a22*c33*k20 - c33 *k1
k55=--------------------------
2
a22
k54=0
k53=0
k52=0
k51=0
2
- 2*a22*c33*k20 - 2*c33 *k1
k50=------------------------------
2
a22
k49=0
k48=0
k47=0
k46=0
k45=0
k44=2*k20
k43=0
k42=k20
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
2
- a22*c33*k20 - c33 *k1
k25=--------------------------
2
a22
k24=0
k23=0
k21=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:
b33, c33, m3, r3, q1, p15, p10, p5, k20, k10, k1, n3,
a33, a22
Inequalities
In the following not identically vanishing expressions are shown.
Any auxiliary variables g00?? are used to express that at least
one of their coefficients must not vanish, e.g. g0019*p4 + g0020*p3
means that either p4 or p3 or both are non-vanishing.
2 2 2 2 2
{a22 *k1*w123 + 2*a22 *k1*w125 + a22 *k1*w127 + a22 *k10*w116 + a22 *k10*w118
2 2 2 2
+ a22 *k20*w037 + a22 *k20*w040 + a22 *k20*w042 + 2*a22 *k20*w052
2 2 2
+ 2*a22 *k20*w084 + a22 *k20*w086 + a22 *k20*w108 + 2*a22*b33*k1*w066
+ 2*a22*b33*k1*w068 + a22*b33*k10*w063 + a22*b33*k20*w026 + a22*b33*k20*w045
+ a22*b33*k20*w058 + 2*a22*c33*k1*w040 + 2*a22*c33*k1*w042 + a22*c33*k10*w037
- a22*c33*k20*w029 - a22*c33*k20*w034 - a22*c33*k20*w073 - 2*a22*c33*k20*w078
2 2
- a22*c33*k20*w103 + b33 *k1*w037 + 2*b33*c33*k1*w026 - 2*c33 *k1*w029
2 2 2 2
- 2*c33 *k1*w034 - c33 *k1*w073 - 2*c33 *k1*w078 - c33 *k1*w103,
a22,
a22 - a33,
a33}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2 2
HAM=a22*u1 + a22*u2 + a33*u3 + b33*u3*v3 + c33*v3 + m3*v3 + n3*u3
has apart from the Hamiltonian and Casimirs the following first integrals:
2 4 2 2 2 2 4 2
INT=a22 *u1 + 2*a22 *u1 *u2 + a22 *u2 + 2*a22*b33*u1 *u3*v3
2 2 2 2 2
+ 2*a22*b33*u2 *u3*v3 + 2*a22*c33*u1 *v3 + 2*a22*c33*u2 *v3
2 2 2 2 2 3
+ 2*a22*m3*u1 *v3 + 2*a22*m3*u2 *v3 + b33 *u3 *v3 + 2*b33*c33*u3*v3
2 2 4 2 2 2 2 2 2 2 4
+ 2*b33*m3*u3*v3 - c33 *v1 - 2*c33 *v1 *v2 - 2*c33 *v1 *v3 - c33 *v2
2 2 2 3 2 2 2 2
- 2*c33 *v2 *v3 + 2*c33*m3*v3 - m3 *v1 - m3 *v2
2 2 2 2 2 2
=(b33 *u3 + 2*c33*m3*v3)*v3 - (v1 + v2 )*m3
2 2 2 2 2 2 2 2 2 2
- (v1 + v2 + 2*v3 )*(v1 + v2 )*c33 + (u1 + u2 ) *a22
2 2 2
+ 2*(c33*v3 + m3)*b33*u3*v3 + 2*(c33*v3 + m3 + b33*u3)*(u1 + u2 )*a22*v3
2 2 2 2
INT=u3 *(a22*u1 + a22*u2 + b33*u3*v3 + c33*v3 + m3*v3)
2 2 2
=((c33*v3 + m3 + b33*u3)*v3 + (u1 + u2 )*a22)*u3
2 2 2 2 2 2 2 2 2 2
INT=a22*u1 *v2 + a22*u1 *v3 + a22*u2 *v1 + 2*a22*u2 *v2 + a22*u2 *v3
2 2 2 2
+ 2*a22*u2*u3*v2*v3 + a22*u3 *v3 + b33*u3*v1 *v3 + b33*u3*v2 *v3
3 4 2 2 2 2 4
+ b33*u3*v3 - c33*v1 - 2*c33*v1 *v2 - c33*v1 *v3 - c33*v2
2 2 2 2 3
- c33*v2 *v3 + m3*v1 *v3 + m3*v2 *v3 + m3*v3
2 2 2 2 2 2 2
=((2*u2*v2 + u3*v3)*u3*v3 + (v2 + v3 )*u1 + (2*v2 + v3 + v1 )*u2 )*a22
2 2 2 2 2
+ (b33*u3*v3 - c33*v1 - c33*v2 + m3*v3)*(v1 + v2 + v3 )
2 2 2
INT=u3*(a22*u1 + a22*u2 + b33*u3*v3 + c33*v3 + m3*v3)
2 2
=((c33*v3 + m3 + b33*u3)*v3 + (u1 + u2 )*a22)*u3
3
INT=u3
3
=u3
2 2 2
INT=u3*(v1 + v2 + v3 )
2 2 2
=(v2 + v3 + v1 )*u3
2 2 2 2
INT=a22*u1 + a22*u2 + b33*u3*v3 - c33*v1 - c33*v2 + m3*v3
2 2 2 2
=(b33*u3 + m3)*v3 - (v1 + v2 )*c33 + (u1 + u2 )*a22
INT=u3
=u3
And again in machine readable form:
HAM=a22*u1**2 + a22*u2**2 + a33*u3**2 + b33*u3*v3 + c33*v3**2 + m3*v3 + n3*u3$
INT=a22**2*u1**4 + 2*a22**2*u1**2*u2**2 + a22**2*u2**4 + 2*a22*b33*u1**2*u3*v3 +
2*a22*b33*u2**2*u3*v3 + 2*a22*c33*u1**2*v3**2 + 2*a22*c33*u2**2*v3**2 + 2*a22*
m3*u1**2*v3 + 2*a22*m3*u2**2*v3 + b33**2*u3**2*v3**2 + 2*b33*c33*u3*v3**3 + 2*
b33*m3*u3*v3**2 - c33**2*v1**4 - 2*c33**2*v1**2*v2**2 - 2*c33**2*v1**2*v3**2 -
c33**2*v2**4 - 2*c33**2*v2**2*v3**2 + 2*c33*m3*v3**3 - m3**2*v1**2 - m3**2*v2**2
$
INT=u3**2*(a22*u1**2 + a22*u2**2 + b33*u3*v3 + c33*v3**2 + m3*v3)$
INT=a22*u1**2*v2**2 + a22*u1**2*v3**2 + a22*u2**2*v1**2 + 2*a22*u2**2*v2**2 +
a22*u2**2*v3**2 + 2*a22*u2*u3*v2*v3 + a22*u3**2*v3**2 + b33*u3*v1**2*v3 + b33*u3
*v2**2*v3 + b33*u3*v3**3 - c33*v1**4 - 2*c33*v1**2*v2**2 - c33*v1**2*v3**2 - c33
*v2**4 - c33*v2**2*v3**2 + m3*v1**2*v3 + m3*v2**2*v3 + m3*v3**3$
INT=u3*(a22*u1**2 + a22*u2**2 + b33*u3*v3 + c33*v3**2 + m3*v3)$
INT=u3**3$
INT=u3*(v1**2 + v2**2 + v3**2)$
INT=a22*u1**2 + a22*u2**2 + b33*u3*v3 - c33*v1**2 - c33*v2**2 + m3*v3$
INT=u3$