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I was studying the HJ-formalism of classical mechanics when I came upon a modified HJE:

[tex](\nabla S)^2=\frac{1}{u^2}(\frac{\partial S}{\partial t})^2[/tex]

where

[tex]u=\frac{dr}{dt}[/tex]

and [tex]dr=(dx,dy,dz)[/tex] is the position vector.

(I read the derivation and it's ok)

Now, u is interpreted to be the wave velocity of the so called 'action waves' in phase space.

However, my book (Nolting, Volume 2) states that this is a wave equation, or at least a special nonlinear case of the popular wave equation

[tex]\nabla^2S=\frac{1}{u^2}\frac{\partial^2}{\partial t^2}S[/tex]

which is somehow unclear to me, as the squares in both equations mean different things.

A similar statement is also made in Wikipedia:

http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation

(cf. Eiconal apprpximation and relationship to the Schrödinger equation)

I hope someone of you can explain this to me :)

best regards,

marin

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# Hamilton-Jacobi-Equation (HJE)

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