Demystifying the Lagrangian of classical mechanics

[1907.07069] Demystifying the Lagrangian of classical mechanics Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Physics > Classical Physics arXiv:1907.07069 (physics) [Submitted on 16 Jul 2019 ( v1 ), last revised 12 Feb 2022 (this version, v4)] Title: Demystifying the Lagrangian of classical mechanics Authors: Gerd Wagner , Matthew W. Guthrie View a PDF of the paper titled Demystifying the Lagrangian of classical mechanics, by Gerd Wagner and Matthew W. Guthrie View PDF HTML (experimental) Abstract: The Lagrangian formulation of classical mechanics is widely applicable in solving a vast array of physics problems encountered in the undergraduate and graduate physics curriculum. Unfortunately, many treatments of the topic lack explanations of the most basic details that make Lagrangian mechanics so practical. In this paper, we detail the steps taken to arrive at the principle of stationary action, the Euler-Lagrange equations, and the Lagrangian of classical mechanics. These steps are: 1) we derive the Lagrange formalism purely mathematically from the problem of the minimal distance between two points in a plane, introducing the variational principle and deriving the Euler-Lagrange equation; 2) we transform Newton's second law into an Euler-Lagrange equation, proving that the Lagrangian is kinetic minus potential energy; 3) we explain why it is important to reformulate Newton's law. To do so, we prove that the Euler-Lagrange equation is astonishingly the same in any set of coordinates. We demonstrate that because of this feature the role that coordinates play in classical mechanics is much simpler and clearer in the Lagrangian as compared to the Newtonian formulation. This is important because the choice of coordinates is not relevant to physical reality, rather they are arbitrarily chosen to provide a convenient way of analyzing a physical system. Comments: Updated to reflect submission to journal Subjects: Classical Physics (physics.class-ph) Cite as: arXiv:1907.07069 [physics.class-ph] (or arXiv:1907.07069v4 [physics.class-ph] for this version) https://doi.org/10.48550/arXiv.1907.07069 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Demystifying Classical Particles, Fields, and Relativity Through the Lagrangian Formalism. World Scientific, May 2026 Related DOI : https://doi.org/10.1142/14762 Focus to learn more DOI(s) linking to related resources Submission history From: Matthew Guthrie [ view email ] [v1] Tue, 16 Jul 2019 15:29:20 UTC (12 KB) [v2] Wed, 12 Feb 2020 01:14:28 UTC (11 KB) [v3] Fri, 10 Dec 2021 18:37:57 UTC (430 KB) [v4] Sat, 12 Feb 2022 20:30:24 UTC (20 KB) Full-text links: Access Paper: View a PDF of the paper titled Demystifying the Lagrangian of classical mechanics, by Gerd Wagner and Matthew W. Guthrie View PDF HTML (experimental) TeX Source view license Current browse context: physics.class-ph < prev | next > new | recent | 2019-07 Change to browse by: physics References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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