Liouville equation with Dirac delta as probability density

[andresB]

I would like to know the solution to Liouville equation
∂ρ/∂t=-{ρ,H}

given the initial condition

ρ(t=0)=δ(q,p)

where δ(q,p) is a dirac delta centered in some point (q,p) in phase space.

I have the feeling, but I'm not sure, that the solution is of the form

ρ(t)=δ(q(t),p(t))

where q(t) and p(t) are the trajectories from Hamilton equations.

Any help?

 

[Orodruin]

Did you try inserting your suspicion into the Liouville equation to see if it does solve it?

Note that your suspicion is actually
$$
\rho(q,p,t) = \delta(q-q(t),p-p(t)),
$$
where ##q(t)## and ##p(t)## are just functions of time (that satisfy the Hamilton equations of motion), when expressed properly.

 

[andresB]

Did you try inserting your suspicion into the Liouville equation to see if it does solve it?

I tried, but it is confusing to me due to the nature of Dirac delta. Liouville equation read
##∂δ/∂t= -(∂δ/∂q)(∂H/∂p) + (∂δ/∂p)(∂H/∂q)## (1)

Equality should be understood in the distribution sense. Using a test function ##F(q,p)##, I get for the left hand side of (1)

##∫F(q,p) ∂δ/∂t dqdp= ∂/∂t∫F(q,p)δ(q-q(t),p-p(t) dqdp= ∂F(q(t),p(t))/∂t##

But I'm unsure how to prove the equality to the right hand side of (1).

 

[Orodruin]

You do not need to use test functions. Just apply the standard rules of derivatives that hold also for distributions. Take it step by step and show your work.