Non-Relativistic and Relativistic Lagrangian Pairing in Fluid Mechanics Inspired by Quantum Theory

Ismail-Sutton, Sara; Scholle, Markus; Gaskell, Philip H.

Abstract

The pairing of non-relativistic and relativistic Lagrangians within the context of fluid mechanics, advancing methodologies for constructing Poincare-invariant Lagrangians, is explored. Through leveraging symmetries and Noether’s theorem in an inverse framework, three primary cases are investigated: potential flow, barotropic flow expressed in terms of Clebsch variables, and an extended Clebsch Lagrangian incorporating thermodynamic effects. To ensure physical correctness, the eigenvalue relation of the energy–momentum tensor, together with velocity normalisation, are applied as key criteria. The findings confirm that the relativistic Lagrangians successfully reduce to their non-relativistic counterparts in the limit c→∞. These results demonstrate a systematic approach that enhances the relationship between symmetries and variational formulations, providing the advantage of deriving Lagrangians that unify non-relativistic and relativistic theories.

Citation

Ismail-Sutton, S., Scholle, M., & Gaskell, P. H. (2025). Non-Relativistic and Relativistic Lagrangian Pairing in Fluid Mechanics Inspired by Quantum Theory. Symmetry, 17(3), Article 315. https://doi.org/10.3390/sym17030315

• Journal Article Type: Article
• Acceptance Date: Feb 18, 2025
• Online Publication Date: Feb 20, 2025
• Publication Date: 2025-03
• Deposit Date: Apr 14, 2025
• Publicly Available Date: Apr 14, 2025
• Journal: Symmetry
• Electronic ISSN: 2073-8994
• Publisher: MDPI
• Peer Reviewed: Peer Reviewed
• Volume: 17
• Issue: 3
• Article Number: 315
• DOI: https://doi.org/10.3390/sym17030315
• Keywords: fluid mechanics, Hamilton’s principle, Galilean invariance, Lagrange formalism, inverse variational problems, Poincaré invariance
• Public URL: https://durham-repository.worktribe.com/output/3714846