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arXiv · 2607.18312

Vakonomic Fluids

Ritoban Roy-Chowdhury·Mohammad Sina Nabizadeh·Oliver Gross·Anthony Gruber·Albert Chern
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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.

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Ritoban Roy-Chowdhury, Mohammad Sina Nabizadeh, Oliver Gross, Anthony Gruber, Albert Chern. 2026-07-17. Vakonomic Fluids. https://arxiv.org/abs/2607.18312

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