[Paper Review] Particles, fluids and vortices

J.W. van Holten|ArXiv.org|Jul 18, 2001

Fluid Dynamics and Turbulent Flows1 references3 citations

TL;DR

This paper establishes a duality between classical particle mechanics on curved manifolds and ideal fluid dynamics via the Hamilton-Jacobi formulation, showing that geodesic motion corresponds to fluid flow with sources/sinks, while the dual theory describes vortices. The key result is that quantum duality is preserved only if source strength quantization in one model matches vorticity quantization in the dual, mirroring monopole-flux duality in 3D quantum field theory.

ABSTRACT

Classical particle mechanics on curved spaces is related to the flow of ideal fluids, by a dual interpretation of the Hamilton-Jacobi equation. As in second quantization, the procedure relates the description of a system with a finite number of degrees of freedom to one with infinitely many degrees of freedom. In some two-dimensional fluid mechanics models a duality transformation between the velocity potential and the stream function can be performed relating sources and sinks in one model to vortices in the other. The particle mechanics counterpart of the dual theory is reconstructed. In the quantum theory the strength of sources and sinks, as well as vorticity are quantized; for the duality between theories to be preserved these quantization conditions must be related.

Motivation & Objective

• To establish a dual correspondence between classical particle mechanics on curved manifolds and ideal fluid flow.
• To show that the Hamilton-Jacobi formulation unifies particle motion and fluid dynamics through a variational principle involving density and potential fields.
• To investigate how quantization conditions in the particle and fluid models are related under duality.
• To demonstrate that quantum duality between source/sink models and vortex models requires quantized source strength to match quantized vorticity.

Proposed method

• Formulate particle mechanics on curved manifolds using the Lagrangian $ L = \frac{1}{2} g_{ij} \dot{x}^i \dot{x}^j $, leading to geodesic motion.
• Derive the Hamilton-Jacobi equation $ \partial_t S = -\frac{1}{2} g^{ij} \nabla_i S \nabla_j S $, which describes the particle's action function.
• Introduce a fluid-like action $ A = \int dt \int d^n x \sqrt{g} \, \rho \left( \partial_t S + \frac{1}{2} g^{ij} \nabla_i S \nabla_j S \right) $, where $ \rho $ is a Lagrange multiplier field.
• Reinterpret $ S $ as a velocity potential $ v_i = \nabla_i S $, leading to the inhomogeneous Euler equation $ \partial_t v_i + v \cdot \nabla v_i = -\nabla_i h $ with source term $ h $.
• Construct the dual theory by interchanging the roles of velocity potential and stream function, transforming sources/sinks into vortices.
• Apply Bohr-Sommerfeld quantization to closed orbits, showing that $ \oint p_i dx^i = 2\pi n \hbar $, leading to quantized angular momentum $ \omega = n\hbar $.

Experimental results

Research questions

• RQ1 How can classical particle motion on a curved manifold be mapped to fluid flow via the Hamilton-Jacobi equation?
• RQ2 What is the role of the Lagrange multiplier $ \rho $ in the variational formulation of fluid dynamics derived from particle mechanics?
• RQ3 How does duality between source/sink models and vortex models manifest in the quantum regime?
• RQ4 What conditions ensure that quantum duality is preserved between the particle and fluid descriptions?
• RQ5 How is the quantization of source strength in one model related to vorticity quantization in the dual model?

Key findings

• The Hamilton-Jacobi equation for a free particle on a curved manifold is equivalent to a variational principle involving a fluid density $ \rho $ and a velocity potential $ S $, linking particle motion to fluid dynamics.
• Solutions of the Hamilton-Jacobi equation for geodesic motion on $ S^2 $ correspond to great circles, with the action function $ S $ explicitly computed as $ S = \frac{1}{2t} \arccos^2[\sin\theta \cos(\varphi - \varphi_*)] $.
• The dual fluid model, obtained by interchanging potential and stream function, describes vortices instead of sources, with the same quantized frequency $ \omega = n\hbar $ in both models.
• Quantization of the action integral $ \oint p_i dx^i = 2\pi n \hbar $ leads to discrete angular momentum $ \omega = n\hbar $, which is preserved under duality.
• In the fluid interpretation, the quantization of source strength in the original model corresponds to quantized fluid momentum, while in the dual model it corresponds to quantized vorticity, ensuring duality at the quantum level.
• The duality between source strength and vorticity quantization parallels the duality between magnetic monopole charge and flux quantization in 3D quantum field theory.