[Paper Review] Canonical formulation for nonrelativistic Euler fluids and Schwinger type conditions

Rabin Banerjee, Arpan Krishna Mitra|arXiv (Cornell University)|Oct 1, 2015

Cosmology and Gravitation Theories4 references3 citations

TL;DR

This paper presents a novel canonical Lagrangian formulation for nonrelativistic nonisentropic Euler fluids using Noether's energy-momentum tensor to derive a generalized Clebsch parametrization for fluid velocity. The method yields a noncanonical Hamiltonian algebra directly from the symplectic structure and reveals new Schwinger-type relations among stress tensor components, which ensure conservation and generalize relativistic field-theoretic structures to nonrelativistic hydrodynamics.

ABSTRACT

We present a new approach, based on Noether's energy-momentum tensor, to construct the lagrangian for nonrelativistic nonisentropic Euler fluids. An advantage of this approach is that it naturally provides a generalised Clebsh decomposition for the fluid velocity. This is used to develop a hamiltonian formulation inolving a noncanonical algebra. This algebra is very simply obtained from the symplectic structure. It is used to show that the components of the Noether's energy-momentum tensor satisfy certain Schwinger-type relations. These relations, which are reminiscent of corresponding relations in relativistic field theory, are new.

Motivation & Objective

• To develop a systematic, Eulerian-variable-based Lagrangian formulation for nonrelativistic nonisentropic fluids, avoiding ad-hoc assumptions in prior approaches.
• To resolve the obstruction in transitioning from Hamiltonian to Lagrangian formalism due to noncanonical brackets and Casimirs in fluid dynamics.
• To derive a generalized Clebsch decomposition that includes entropy for nonisentropic fluids, providing a physically motivated parametrization of velocity.
• To establish a noncanonical Hamiltonian algebra directly from the symplectic structure, ensuring consistency with hydrodynamic equations.
• To uncover and prove the existence of Schwinger-type algebraic relations among components of the stress tensor in nonrelativistic fluid theory, previously unknown in this context.

Proposed method

• Utilize Noether's definition of the energy-momentum tensor to construct the Lagrangian density, ensuring consistency with known Hamiltonian and current densities.
• Introduce a conjugate variable θ to the fluid density ρ, leading to a Lagrangian of the form 𝒟 = ρ∂₀θ − (½ρv² + V(ρ)), with velocity v expressed via θ.
• Derive a generalized Clebsch parametrization for velocity involving entropy S, yielding two distinct forms (45, 46), one of which is newly identified.
• Construct the symplectic structure from the Lagrangian, from which noncanonical Poisson brackets between ρ, vᵢ, and their conjugate variables are derived directly.
• Use the derived algebra to show that components of the stress tensor satisfy Schwinger-type relations analogous to those in relativistic quantum field theory.
• Verify consistency by showing that the integrated form of the Schwinger-type relation reproduces the time component of the stress tensor conservation law.

Experimental results

Research questions

• RQ1 How can a consistent Lagrangian formulation be constructed for nonrelativistic nonisentropic fluids without relying on ad-hoc conservation laws or canonical transformations?
• RQ2 What is the natural generalization of the Clebsch parametrization that includes entropy in nonisentropic fluids, and how does it affect the Hamiltonian structure?
• RQ3 Can noncanonical Poisson brackets in fluid dynamics be systematically derived from the symplectic structure of an Eulerian Lagrangian, rather than by inspection or transformation?
• RQ4 Do Schwinger-type algebraic relations—previously known in relativistic field theory—also emerge in nonrelativistic fluid dynamics, and what is their physical significance?
• RQ5 How does the proposed formalism ensure consistency between the Hamiltonian and Lagrangian formulations despite the presence of Casimirs and noninvertible symplectic structures?

Key findings

• The paper derives a generalized Clebsch parametrization for nonisentropic fluids involving entropy, with one form being a new result not previously reported in the literature.
• The symplectic structure of the Lagrangian directly yields noncanonical Poisson brackets between fluid variables, providing a systematic derivation independent of canonical transformations.
• The components of the stress tensor satisfy new Schwinger-type relations, which are algebraic identities reminiscent of those in relativistic quantum field theory but now established in nonrelativistic fluid dynamics.
• These Schwinger-type relations imply the conservation of the stress tensor, and their integrated form matches the time component of the standard stress-energy conservation law.
• The method resolves the long-standing issue of Lagrangian formulation in noncanonical fluid systems by avoiding the need for Legendre transforms when Casimirs obstruct inversion.
• The formalism is robust under gauge-like transformations of the Clebsch potentials, preserving both the velocity field and the Hamiltonian algebra, thus ensuring physical consistency.