Advanced Mechanics. Mathematical Introduction

TL;DR

This paper presents a covariant formulation of classical non-relativistic mechanics using fibre bundles, unifying Newtonian, Lagrangian, and Hamiltonian systems with their conservation laws.

Contribution

It introduces a fibre bundle framework for non-relativistic mechanics, emphasizing covariance under reference frame transformations and formalizing key concepts like forces and motions.

Findings

Formulation of mechanics using fibre bundle connections.

Unified description of Newtonian, Lagrangian, and Hamiltonian systems.

Derivation of conservation laws within this geometric framework.

Abstract

Classical non-relativistic mechanics in a general setting of time-dependent transformations and reference frame changes is formulated in the terms of fibre bundles over the time-axis R. Connections on fibre bundles are the main ingredient in this formulation of mechanics which thus is covariant under reference frame transformations. The basic notions of a non-relativistic reference frame, a relative velocity, a free motion equation, a relative acceleration, an external force are formulated. Newtonian, Lagrangian, Hamiltonian mechanical systems and the relations between them are defined. Lagrangian and Hamiltonian conservation laws are considered.