Action Functional for Ideal Fluids

• Action functional for ideal fluids is a variational principle defined via Lagrangians to derive conservation laws and equations of motion.
• It employs canonical Clebsch parametrizations and Lagrange multipliers to yield continuity, Euler, and stress-energy tensor formulations.
• This framework underpins symmetry, conservation, and field-theoretic approaches, informing models in gravity, turbulence, and multiscale dynamics.

An action functional for an ideal (perfect) fluid is a variational principle that encodes the equations of motion, conservation laws, and symmetries of fluid dynamics in terms of a Lagrangian or Hamiltonian constructed from fundamental variables such as particle-number currents, velocity fields, and thermodynamic potentials. In both relativistic and non-relativistic settings, the fluid action functional is central to derivations of the continuity equation, Euler equation, and the stress-energy tensor. Canonical approaches, Clebsch parametrizations, and extensions to fluids with anomalies or multiscale structure reveal deep geometric and physical content.

1. Fundamental Formulations and Variables

The action for an ideal fluid is conventionally posed in four-dimensional spacetime $(M_4, g_{\mu\nu})$ using the particle-number current $J^\mu$, entropy density ratio $S$, and Clebsch potentials $\{\theta,\alpha,\beta,\gamma\}$, which form the canonical fluid 1-form $\Pi_\mu$ (Wiegmann, 29 Mar 2024): $$\Pi_\mu = \partial_\mu \theta + \alpha\,\partial_\mu \beta + \gamma\,\partial_\mu S$$ The particle density is $n = \sqrt{-g_{\mu\nu}J^\mu J^\nu}$ and the energy density $\varepsilon = \varepsilon(n, S)$. The action reads: $$S_0 = \int_{M_4} [ J^\mu\,\Pi_\mu - \varepsilon(n, S) ]\, d^4x$$ This captures the kinetic and thermodynamic structure, reduces on-shell to the familiar “pressure action”: $$S_0 = -\int_{M_4} P(n, S) \sqrt{-g} \,d^4x + \text{total derivative}$$ where $P$ is the pressure from the Legendre transform.

Variational principles are built from these ingredients or, equivalently, in terms of matter/entropy densities, 4-velocity $u^\mu$, and particle currents, as elaborated in (Mendoza et al., 2020, Paston, 2017).

2. Variational Principle and Equations of Motion

Independent variation of the fluid action with respect to $J^\mu$, $S$, and Clebsch potentials yields:

• Continuity equation ($\nabla_\mu J^\mu=0$): via $\delta\theta$.
• Kelvin’s theorem constraints: via $\delta\alpha$, $\delta\beta$; these enforce advection of the label fields.
• Entropy advection: via $\delta S$; $J^\mu\partial_\mu S=0$ for adiabatic flows, with $T = \partial\varepsilon/\partial S$ as conjugate.
• Euler equation: via $\delta J^\mu$; produces

$n\,u^\nu\nabla_\nu(w\,u_\mu) + \partial_\mu P = 0$

with $w = \partial\varepsilon/\partial n$. The combination of continuity and Euler equations is encoded as $\nabla_\nu T^{\nu\mu} = 0$, $T^{\nu\mu} = n^\nu p^\mu + P g^{\nu\mu}$ (Wiegmann, 29 Mar 2024, Mendoza et al., 2020).

Lagrange multipliers serve to enforce conservation laws and normalization constraints; e.g., for the matter action in general relativity: $$S_m = \int d^4x \sqrt{-g} [ -\varepsilon(\rho,s) + \phi\,\nabla_\mu(\rho u^\mu) + \theta\,u^\mu\nabla_\mu s + \alpha (g_{\mu\nu}u^\mu u^\nu-1)]$$ (Mendoza et al., 2020).

3. Clebsch and Polynomial Parametrizations

The Clebsch representation offers a general parametrization for the 4-velocity: $$u_\mu = \partial_\mu\phi + \alpha\,\partial_\mu\beta$$ enabling the description of both irrotational and vortical flows within the variational framework. This parametrization is central to the relabeling symmetry and Kelvin’s theorem (Wiegmann, 29 Mar 2024, Paston, 2017). Several equivalent forms of the action exist, including:

Formulation	Key Variables	Notable Features
Canonical Clebsch	$J^\mu, \theta,\alpha,\beta,\gamma$	Fully general vorticity
Scalar-potential (“radical”)	$j^\mu, \lambda$	Potential/dust flows
Rational/polynomial	$J^\mu, a$ or $n,\lambda$	Quadratic action
Lagrange-multiplier	$n, \lambda$	Mimetic gravity links

All forms are classically equivalent via auxiliary field elimination and share the physical content: continuity, Euler/relativistic geodesic equations, and dust/perfect-fluid stress-energy tensor (Paston, 2017).

4. Symmetry Properties and Conservation Laws

The ideal-fluid action exhibits:

• Spacetime diffeomorphism invariance: yields energy, momentum, and angular momentum conservation ($\nabla_\nu T^{\nu\mu}=0$).
• Particle relabeling (Clebsch gauge): invariance under internal variable transformations, underpinning Kelvin’s circulation theorem.
• Gauge invariance: $\theta \to \theta +$ constant for $U(1)$ symmetry, enforces particle conservation.

Noether’s theorem quantifies these invariants as integrated currents associated with continuous symmetries (Wiegmann, 29 Mar 2024, Lingam et al., 2014, Cotter et al., 2014).

5. Boundary Conditions and Hamiltonian Structure

Natural boundary conditions in the Hamilton–Piola or Eulerian actions enforce impermeability (vanishing normal velocity) or null variation at the domain boundary and initial/final times (Elgindi, 20 Nov 2025, Auffray et al., 2013). The Hamiltonian reduction is explicit in the ADM split for gravity–fluid systems and in finite-dimensional truncations (“jetlet” hierarchy) for multiscale structure (Cotter et al., 2014, Kluson, 30 Nov 2025).

The canonical structure is preserved in advanced settings, such as Born–Infeld inspired gravity, where the fluid action is coupled to determinantal gravitational sectors and the canonical form emerges after elimination of non-dynamical auxiliaries (Kluson, 30 Nov 2025).

6. Generalizations: Extended Models and Topological Terms

Action functionals extend naturally to incompressible or multiscale cases. For incompressible flows, the action includes a Lagrange multiplier enforcing $\nabla\cdot v=0$ (Elgindi, 20 Nov 2025): $$S[v,p] = \int_{t_1}^{t_2}\int_\Omega \left( \tfrac12 |v|^2 - p\,\nabla\cdot v \right)\,d^3x\,dt$$ Higher-order expansions (jetlet hierarchy) furnish weak, multiscale solutions and encode energy cascade phenomena; action functionals incorporate Taylor jets and their evolution via Lagrange multipliers, realizing finite-dimensional canonical Hamiltonian systems (Cotter et al., 2014).

No closed or Wess–Zumino–type topological term appears in the ideal (anomaly-free) fluid action; the helicity integral $H = \int_{M_3} (\Pi \wedge d\Pi)$ is conserved but excluded from the local action, only appearing with anomaly-inducing deformations (Wiegmann, 29 Mar 2024).

7. Field-Theoretic and Geometric Approaches

For 2D ideal fluids, field-theoretic models—such as non-Abelian Chern–Simons gauge theory coupled to a matter spinor—provide an action extremum principle for coherent states and relaxation phenomena. Self-dual states minimize the energy and satisfy a sinh–Poisson equation, with the action: $$S[\psi, A] = \int d^2x\,dt\,\left\{ -\kappa\,\epsilon^{\mu\nu\rho}\mathrm{Tr}(A_\mu\partial_\nu A_\rho + \tfrac23 A_\mu[A_\nu, A_\rho]) + i\,\mathrm{Tr}(\psi^\dagger D_0 \psi) - \frac{1}{2m}\mathrm{Tr}[(D_k\psi)^\dagger (D^k\psi)] - \frac{g}{4\kappa}\mathrm{Tr}([\psi, \psi^\dagger]^2) \right\}$$ (Spineanu et al., 2013). These models relate fluid vorticity to topological invariants and soliton-like stationary flows.

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In summary, the action functional for ideal fluid dynamics serves as a universal generator for conservation laws, field equations, and symmetry constraints across classical, relativistic, and field-theoretic regimes. The variational construction admits multiple parametrizations (Clebsch, polynomial, Lagrange-multiplier) and supports extensions to gauge, anomaly, gravity, and multiscale domains. This framework underpins both theoretical analyses and computational strategies in fluid and plasma physics, general relativity, and topological hydrodynamics (Wiegmann, 29 Mar 2024, Mendoza et al., 2020, Paston, 2017, Elgindi, 20 Nov 2025, Cotter et al., 2014, Spineanu et al., 2013, Kluson, 30 Nov 2025, Auffray et al., 2013, Lingam et al., 2014).