A Neutral Delay Differential Equation Formulation of Newtonian Mechanics: Application to the Ideal Spring

A Neutral Delay Differential Equation Formulation of Newtonian Mechanics: Application to the Ideal Spring | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Neutral Delay Differential Equation Formulation of Newtonian Mechanics: Application to the Ideal Spring M. A. Elfouly This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7908926/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Classical mechanics treats causality as instantaneous, a force is presumed to set acceleration at the very same moment. In many real systems, however, momentum is exchanged over a finite time because signals propagate through material, internal microstructures relax, or control loops introduce latency. To close this gap, we recast Newton’s second law in causal terms by averaging momentum change over a short yet finite window. The resulting neutral delay formulation places delay in the update of the momentum rate rather than imposing it on the position state. This construction preserves standard symmetries, respects momentum conservation under pairwise interactions, and smoothly recovers the classical law as the window shrinks to zero. We develop the associated stability picture and an explicit bridge between instantaneous and delayed dynamics, identifying a practical operating domain in which the neutral model remains stable and free of spurious oscillations. Section five subjects the framework to a stringent spring–mass benchmark spanning lumped, transitional, and distributed regimes. The neutral model reproduces the hallmarks of finite-speed transport—earlier phase accumulation, a flat positive group delay, and contractive transients without resonant growth—while a phase-first identification procedure estimates the effective delay directly from measured frequency response. Overall, the neutral delay formulation is not an ad hoc add-on but a physics-consistent generalization of Newton’s second law: it aligns modeling with how measurements are actually made, provides clear rules for model selection and parameter identification, and lays a robust foundation for extensions to nonlinear dynamics, frictional interfaces, and systems with distributed memory. Theoretical Physics Applied Mathematics Newtonian Mechanics Neutral Newton law Neutral Delay Differential Equation Delay Differential Equation Finite-Speed Propagation Hopf bifurcation Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7908926","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":532515582,"identity":"67d8e814-d35c-4cb1-97f1-a8c9877bc41e","order_by":0,"name":"M. A. 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