- #1
Marin
- 192
- 0
Hi all!
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
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