Dynamics: A different outlook

TL;DR

This paper offers a comprehensive and unconventional overview of classical mechanics, integrating vector, differential geometric, Lagrangian, and Hamiltonian perspectives, including extensions to curved space-time, suitable for advanced students.

Contribution

It uniquely combines traditional mechanics with differential geometry and extends Lagrangian mechanics to curved space-time, providing a unified and detailed pedagogical approach.

Findings

Unified vectorial formulations of Newtonian mechanics

Extension of Lagrangian mechanics to curved space-time

Detailed examples and differential geometric insights

Abstract

The present article deals with general mechanics in an unconventional manner. At first, Newtonian mechanics for a point particle has been described in vectorial picture, considering Cartesian, polar and tangent-normal formulations in a unified manner. An extensive differential geometric notions have been used when motion on curved surfaces has been considered. Both the Lagrangian and the Hamiltonian formulations have been discussed with various examples. The relevant part of the calculus of variation has been presented in the appendix. Also, a detailed study about the nature of the Lagrangian has been shown in the appendix and it has been found that the usual Lagrangian can be extended to curved space-time using the notion of differential geometry. The article may be suitable both for undergraduate and postgraduate students.