# Hamilton-Jacobi equation and particle-wave motion

• Vicol
In summary, the Hamilton-Jacobi equation is a formulation of classical mechanics that can also treat motion of particles as wave motion. It involves the use of Hamilton's principal function and Hamilton's characteristic function, which change in time like a wavefront. This is similar to the wave-particle duality seen in quantum mechanics. In the classical limit, the Schrodinger equation can be reduced to the Hamilton-Jacobi equation. The Hamilton-Jacobi equation allows for any choice of function at t = 0, similar to the wavefunction in quantum mechanics. For this equation, a real solution is taken for S, leading to a discussion on page 8 of the referenced source about dropping terms in the equation.
Vicol
I've seen somwhere a claim that Hamilton-Jacobi euqation is the only formulation of classical mechanics which can treat motion of particle as wave motion. There was something about hamilton prinicpal function, hamilton characteristic function and one of these change in time like wavefront or something like this :P Sorry for not beeing clear, I don't understand the math behind but the claim sounds really interesting. Mainly beacuse QM does the same - there is a wave associate with particle.

Does anyone have good source of knowledge for this topic or can explain this?

Vicol said:
there is a wave associate with particle.

in the classical limit the schrodinger equation goes to H - J equation ...

we see that in the classical limit h→ 0 the Schrodinger equation is just the Hamilton-Jacobi equation.

The H-J equation was a partial differential equation that could be solved with any choice of function at t = 0.

This function acts like the wavefunction that we encounter in quantum mechanics.

For the H-J equation we will take a real solution S, and thus we will indeed be dropping...
pl. see the discussion in ref. below p-8

<http://www.physics.ohio-state.edu/~mathur/821hj.pdf>

## What is the Hamilton-Jacobi equation and how is it related to particle-wave motion?

The Hamilton-Jacobi equation is a partial differential equation that describes the dynamics of a system in classical mechanics. It is closely related to the principle of least action, which states that the path taken by a particle between two points is the one that minimizes the action integral. This principle also applies to wave propagation, making the Hamilton-Jacobi equation a useful tool for studying both particle and wave motion.

## How is the Hamilton-Jacobi equation derived?

The Hamilton-Jacobi equation is derived from the Hamiltonian formalism, which is a mathematical framework for analyzing the dynamics of a system. It involves expressing the system's energy in terms of generalized coordinates and momenta, and then using the Hamiltonian equations to obtain the equations of motion. The Hamilton-Jacobi equation is then obtained by introducing a new variable, the action, which simplifies the Hamiltonian equations.

## What is the significance of the Hamilton-Jacobi equation in quantum mechanics?

In quantum mechanics, the Hamilton-Jacobi equation is used to derive the Schrödinger equation, which is the fundamental equation that describes the evolution of a quantum system. This connection between classical and quantum mechanics highlights the importance of the Hamilton-Jacobi equation in understanding the behavior of particles and waves at the microscopic level.

## Can the Hamilton-Jacobi equation be solved analytically?

In general, the Hamilton-Jacobi equation is difficult to solve analytically for complex systems. However, there are some special cases where analytical solutions can be found, such as for systems with certain symmetries. In most cases, numerical methods are used to solve the Hamilton-Jacobi equation and obtain solutions.

## What are some applications of the Hamilton-Jacobi equation in physics and engineering?

The Hamilton-Jacobi equation has a wide range of applications in physics and engineering, including classical mechanics, quantum mechanics, optics, and control theory. It is used to study various physical systems, such as celestial mechanics, fluid dynamics, and particle accelerators. It is also used in engineering applications, such as optimal control and trajectory planning for robotic systems.

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