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

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
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

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 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.