Why is Euler equation similar to the gradient of Hamilton-Jacobi equation?

Max

The Hamilton-Jacobi equation determines the generating function S(r,t)
(Hamilton's principal function) for the canonical transformation,
which requires that the transformed Hamiltonian be identically zero.

If we consider an ensemble of such a dynamical system, hydrodynamics
gives us continuity equation and Euler equation for density ro(r, t)
and velocity v(r, t). Taking the gradient of Hamilton-Jacobi equation,
we would found a correspondence with the Euler equation, if one
substitutes gradient of generating function Grad S(r, t) with the
velocity distribution v(r, t).

This is quite amazing! Hamilton-Jacobi equation describes a single
dynamical system, while Euler equation describes an ENSEMBLE of such a
dynamical system.

Why is there such a correspondence?

-Max

a student

Max wrote:
> The Hamilton-Jacobi equation determines the generating function S(r,t)
> (Hamilton's principal function) for the canonical transformation,
> which requires that the transformed Hamiltonian be identically zero.
>
> If we consider an ensemble of such a dynamical system, hydrodynamics
> gives us continuity equation and Euler equation for density ro(r, t)
> and velocity v(r, t). Taking the gradient of Hamilton-Jacobi equation,
> we would found a correspondence with the Euler equation, if one
> substitutes gradient of generating function Grad S(r, t) with the
> velocity distribution v(r, t).
>
> This is quite amazing! Hamilton-Jacobi equation describes a single
> dynamical system, while Euler equation describes an ENSEMBLE of such a
> dynamical system.
>
> Why is there such a correspondence?

I can only give a partial answer. Note first that the H-J equation
describes more than one system - it describes a whole family of
trajectories (with momentum grad S associated with position x), and
hence is ripe for the notion of bringing in a probability measure
P(x,t) over the possible positions. There is only the question of
whether this can be done consistently. One would then have a natural
correspondence between "H-J ensembles" and hydrodynamics, where one
identifies P(x,t) with the normalised mass density rho(x,t).

The answer to the question is yes: for a particle of mass m moving in
potential V(x) one can choose the Lagrangian
L[P, S] = int dx P [ |grad S|^2/(2m) + V + del S/del t ].
Varying with respect to P and S gives the H-J equation and continuity
equation respectively.

There is a quantum generalisation of this too, and a corresponding
hydrodynamical interpretation of quantum mechanics.