Vakonomic Fluids

2026-07-17

AI Summary

This work addresses the challenge that traditional discrete methods fail to preserve variational structures while handling the nonholonomic constraints introduced by Koopman representations, leading to violations of conservation laws in long-term simulations of the incompressible Euler equations. For the first time, the vakonomic variational principle is incorporated into fluid discretization, leveraging a geodesic framework on the group of volume-preserving diffeomorphisms. By combining low-rank Clebsch variable momentum maps with discrete Lie group geometric structures, the proposed method constructs a discrete system satisfying sub-Riemannian geodesic properties. It rigorously preserves the Lie–Poisson structure and relabeling symmetry, conserves Casimir invariants and Kelvin’s circulation theorem to machine precision, and demonstrates exceptional stability and physical fidelity even at low resolutions, significantly enhancing numerical robustness and realism.

Abstract

We introduce a novel discretization of the incompressible Euler equations based on their interpretation as geodesic equations on the Lie group of volume-preserving diffeomorphisms. It is well known that encoding diffeomorphisms and their infinitesimal generators through a discretized Koopman representation places a nonholonomic constraint on discrete velocities, for which there is no consensus on a variational treatment. We show that taking the vakonomic perspective, as opposed to the usual perspective of Lagrange--d'Alembert, yields discrete fluid trajectories that remain geodesics on a (sub-)Riemannian manifold. In particular, the resulting vakonomic dynamics are Lie--Poisson and their solutions admit a discrete relabeling symmetry, leading to machine-precision satisfaction of Casimir invariants along with a discrete analogue of Kelvin's Circulation Theorem. Using an efficient momentum map representation based on low-rank Clebsch variables, we show that these vakonomic fluids behave stably and consistently even at low grid resolutions, leading to increased robustness and physical realism in the long term.

Problem

• vakonomic
• nonholonomic constraint
• incompressible Euler equations
• discrete geodesics
• Casimir invariants

Innovation

• vakonomic dynamics
• discrete geodesics
• Lie–Poisson structure
• Casimir invariants
• Clebsch variables