Is the Modified Hamilton-Jacobi Equation a Wave Equation?

  • Thread starter Marin
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In summary, the conversation discusses the modified Hamilton-Jacobi equation, which includes a term for the wave velocity u. This u is interpreted as the velocity of 'action waves' in phase space. The conversation also mentions that this equation is a special nonlinear case of the popular wave equation. However, there is some confusion about the meaning of the squares in both equations. Further discussion and explanation is requested.
  • #1
Marin
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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
 
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  • #2
Hmmm, haven't fould anything so far..

Any ideas left?
 

FAQ: Is the Modified Hamilton-Jacobi Equation a Wave Equation?

What is the Hamilton-Jacobi Equation?

The Hamilton-Jacobi Equation (HJE) is a partial differential equation in classical mechanics that describes the evolution of a system over time. It is also known as the Hamiltonian-Jacobi equation or the H-J equation.

What is the significance of the Hamilton-Jacobi Equation?

The HJE is significant because it allows for the determination of the complete solution to a classical mechanical system, including the position and momentum of all particles, from just the initial conditions.

How is the Hamilton-Jacobi Equation used in physics?

The HJE is used to find the equations of motion for a classical mechanical system by solving for the action function, which is a mathematical function that describes the system's motion. It is also used in the study of fluid dynamics and quantum mechanics.

What are the assumptions made in the Hamilton-Jacobi Equation?

The HJE makes several assumptions, including that the system is deterministic, meaning that the future state of the system is completely determined by its current state. It also assumes that the system is conservative, meaning that energy is conserved and that the system is time-independent.

Can the Hamilton-Jacobi Equation be applied to non-classical systems?

Yes, the HJE can also be applied to non-classical systems, such as quantum systems, by using the Hamilton-Jacobi formalism. However, in these cases, the HJE may have different interpretations and may not fully describe the system's behavior.

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