A note on the foundations of mechanics

TL;DR

This paper reviews the foundational principles of mechanics with enhanced clarity, clarifying concepts like constrained systems, time, and relativistic forces, and explores the classical roots of quantum equations.

Contribution

It offers a clearer conceptual and mathematical presentation of mechanics foundations, linking classical solutions to quantum equations and clarifying fundamental principles.

Findings

Improved understanding of constrained systems and time.

Classical solutions relate to Schrödinger and Klein-Gordon equations.

Clarification of the nature of relativistic forces.

Abstract

This short note is intended to review the foundations of mechanics, trying to present them with the greatest mathematical and conceptual clarity. It was attempted to remove most of inessential, even parasitic issues, which can hide the true nature of basic principles. The pursuit of that goal results in an improved understanding of some topics such as constrained systems, the nature of time or the relativistic forces. The Sr\"odinger and Klein-Gordon equations appear as conditions fulfilled by certain types of classical solutions of the field equations although the meaning of quantum equations is not, even remotely, exhausted by these cases. A part of this note comes from previous works.