Solution 5 to problem e3c2new
Expressions |
Parameters |
Inequalities |
Relevance |
Back to problem e3c2new
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
n1=0
n2=0
m3=0
r6=0
r5=0
r4=0
r3=0
r2=0
r1=0
q20=0
q19=0
8*k1*m2*n3
q17=------------
2
a33
8*k1*m1*n3
q16=------------
2
a33
2 2
4*k1*m1 + 4*k1*m2
q15=---------------------
2
a33
q14=0
q13=0
q12=0
q11=0
2 2
4*k1*m1 + 4*k1*m2
q10=---------------------
2
a33
q9=0
q8=0
q7=0
q5=0
q4=0
2
- 4*k1*n3
q3=-------------
2
a33
q2=0
2
- 4*k1*n3
q1=-------------
2
a33
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
p25=0
p24=0
- 4*k1*m2
p23=------------
a33
- 8*k1*m1
p22=------------
a33
4*k1*m2
p21=---------
a33
p20=0
p19=0
p18=0
p17=0
p16=0
p15=0
p14=0
4*k1*m1
p13=---------
a33
- 8*k1*m2
p12=------------
a33
- 4*k1*m1
p11=------------
a33
p10=0
p9=0
p8=0
- 4*k1*n3
p7=------------
a33
p6=0
- 4*k1*n3
p5=------------
a33
p4=0
p3=0
p2=0
p1=0
k125=0
k124=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
k101=0
k100=0
k99=0
k98=0
k97=0
k96=0
k95=0
k94=0
k93=0
k92=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
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
k38=0
k37=0
k36=0
k35=0
k34=0
k33=0
k32=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
k14=0
k13=0
k12=0
k11=0
k10=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:
m2, k1, n3, m1, a33
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.
{k1,a33,m1,n3}
Relevance for the application:
The system of equations related to the Hamiltonian HAM:
2 2 2
a33*u1 + a33*u2 + 2*a33*u3 + 2*m1*v1 + 2*m2*v2 + 2*n3*u3
HAM=-------------------------------------------------------------
2
has apart from the Hamiltonian and Casimirs the following only first integral:
2 4 2 2 2 2 4 2
INT=a33 *u1 + 2*a33 *u1 *u2 + a33 *u2 - 4*a33*m1*u1 *v1 - 8*a33*m1*u1*u2*v2
2 2 2
+ 4*a33*m1*u2 *v1 + 4*a33*m2*u1 *v2 - 8*a33*m2*u1*u2*v1 - 4*a33*m2*u2 *v2
2 2 2 2 2 2
- 4*a33*n3*u1 *u3 - 4*a33*n3*u2 *u3 + 4*m1 *v1 + 4*m1 *v2 + 8*m1*n3*u1*v3
2 2 2 2 2 2 2 2
+ 4*m2 *v1 + 4*m2 *v2 + 8*m2*n3*u2*v3 - 4*n3 *u1 - 4*n3 *u2
2 2 2 2
=4*((m1 + m2 )*(v1 + v2 ) + 2*(m1*u1 + m2*u2)*n3*v3)
2 2 2 2 2 2
+ ((u1 + u2 )*a33 - 4*n3 )*(u1 + u2 ) + 4*(
2 2 2 2
(u1 *v2 - 2*u1*u2*v1 - u2 *v2)*m2 - (u1 + u2 )*n3*u3
2 2
- (u1 *v1 + 2*u1*u2*v2 - u2 *v1)*m1)*a33
And again in machine readable form:
HAM=(a33*u1**2 + a33*u2**2 + 2*a33*u3**2 + 2*m1*v1 + 2*m2*v2 + 2*n3*u3)/2$
INT=a33**2*u1**4 + 2*a33**2*u1**2*u2**2 + a33**2*u2**4 - 4*a33*m1*u1**2*v1 - 8*
a33*m1*u1*u2*v2 + 4*a33*m1*u2**2*v1 + 4*a33*m2*u1**2*v2 - 8*a33*m2*u1*u2*v1 - 4*
a33*m2*u2**2*v2 - 4*a33*n3*u1**2*u3 - 4*a33*n3*u2**2*u3 + 4*m1**2*v1**2 + 4*m1**
2*v2**2 + 8*m1*n3*u1*v3 + 4*m2**2*v1**2 + 4*m2**2*v2**2 + 8*m2*n3*u2*v3 - 4*n3**
2*u1**2 - 4*n3**2*u2**2$