[Paper Review] On the Hamiltonian Description of Fluid Mechanics

I. Αντωνίου, G. P. Pronko|ArXiv.org|Jun 14, 2001

Fluid Dynamics and Turbulent Flows5 references3 citations

TL;DR

This paper develops a Hamiltonian formulation of fluid mechanics using Lagrangian particle trajectories, deriving canonical variables that clarify the origin of Clebsch variables and linking circulation conservation (Tompson's theorem) to invariance under volume-preserving diffeomorphisms. The formalism establishes new conservation laws and extends naturally to ideal plasma, providing a foundation for quantization of quantum fluids.

ABSTRACT

We suggest the Hamiltonian approach for fluid mechanics based on the dynamics, formulated in terms of Lagrangian variables. The construction of the canonical variables of the fluid sheds a light of the origin of Clebsh variables, introduced in the previous century. The developed formalism permits to relate the circulation conservation (Tompson theorem) with the invariance of the theory with respect to special diffiomorphisms and establish also the new conservation laws. We discuss also the difference of the Eulerian and Lagrangian description, pointing out the incompleteness of the first. The constructed formalism is also applicable for ideal plasma. We conclude with several remarks on the quantization of the fluid.

Motivation & Objective

• To develop a Hamiltonian description of fluid mechanics based on Lagrangian particle trajectories, avoiding reliance on Eulerian field variables.
• To clarify the mathematical origin of Clebsch variables, traditionally introduced empirically in fluid dynamics.
• To establish a connection between circulation conservation (Tompson's theorem) and invariance under volume-preserving diffeomorphisms.
• To demonstrate the formalism's applicability to ideal plasma, including electromagnetic interactions via vector potentials.
• To lay the groundwork for quantization of fluid systems by constructing canonical variables and Poisson brackets.

Proposed method

• Formulates fluid dynamics using particle trajectories $\vec{x}(\xi_i, t)$ as fundamental configurational variables, with $\xi_i$ labeling initial positions.
• Derives the Lagrangian $L = \int d^3\xi \frac{m}{2} \dot{\vec{x}}^2(\xi_i, t)$, leading to equations of motion $m\ddot{\vec{x}} = 0$ in the absence of forces.
• Introduces canonical conjugate momenta $\vec{\pi}(\vec{x}, t)$ and $\vec{\Pi}(\vec{x}, t)$ for fluid and plasma components via functional derivatives.
• Constructs the Hamiltonian in terms of canonical variables using the Legendre transform, with Poisson brackets defined on phase space.
• Applies Noether's theorem to volume-preserving diffeomorphisms to derive conserved circulations $V_\Lambda = \oint_\Lambda dx_j \frac{l_j(x)}{\rho(x)}$.
• Extends the formalism to plasma by incorporating electromagnetic potentials $\vec{A}(x)$, modifying the momentum variables to include $-e\vec{A}$ and $+e\vec{A}$ terms.

Experimental results

Research questions

• RQ1 How can a Hamiltonian formulation of fluid mechanics be consistently constructed using Lagrangian particle trajectories?
• RQ2 What is the canonical origin of Clebsch variables in fluid dynamics, and how do they emerge from the phase space structure?
• RQ3 How is circulation conservation (Tompson's theorem) related to symmetry under volume-preserving diffeomorphisms?
• RQ4 Can the Hamiltonian formalism be extended to include electromagnetic effects in ideal plasma?
• RQ5 What are the implications for quantization of the fluid, particularly in constructing a quantum fluid theory outside the Fock representation?

Key findings

• The canonical variables $\vec{\pi}(x)$ and $\vec{\Pi}(x)$ are derived from the Lagrangian via functional derivatives, enabling a full Hamiltonian formulation.
• Clebsch variables emerge naturally as components of the conjugate momenta in the phase space, resolving their historical ambiguity.
• Circulation conservation is shown to be a Noether current associated with invariance under volume-preserving diffeomorphisms of particle trajectories.
• The formalism yields new conserved quantities, such as $V^{el}_\Lambda = \oint_\Lambda dx_j \frac{l_j(x)}{\rho_{el}(x)}$ for electrons and $V^{ion}_\Lambda$ for ions.
• For plasma, the conserved circulation $V_\Lambda = V^{el}_\Lambda + V^{ion}_\Lambda = \oint_\Lambda dx_j (m v_{el} + M v_{ion})_j$ is conserved even in the presence of electromagnetic fields.
• The phase space of plasma is constructed with canonical variables $\vec{A}_\perp(x), \vec{P}^{em}_\perp(x); \vec{\xi}(x), \vec{\pi}(x); \vec{\Xi}(x), \vec{\Pi}(x)$, with well-defined Poisson brackets and a Hamiltonian that includes electromagnetic coupling.