The physical foundations of geometric mechanics

Andrew D. Lewis

doi:10.3934/jgm.2017019

Article Contents

2017, Volume 9, Issue 4: 487-574. Doi: 10.3934/jgm.2017019

This issue Previous Article On the relationship between the energy shaping and the Lyapunov constraint based methods Next Article

The physical foundations of geometric mechanics

Andrew D. Lewis

Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada

Received: November 2015

Revised: May 2017

Published: November 2017

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Abstract

The principles of geometric mechanics are extended to the physical elements of mechanics, including space and time, rigid bodies, constraints, forces, and dynamics. What is arrived at is a comprehensive and rigorous presentation of basic mechanics, starting with precise formulations of the physical axioms. A few components of the presentation are novel. One is a mathematical presentation of force and torque, providing certain well-known, but seldom clearly exposited, fundamental theorems about force and torque. The classical principles of Virtual Work and Lagrange-d'Alembert are also given clear mathematical statements in various guises and contexts. Another novel facet of the presentation is its derivation of the Euler-Lagrange equations. Standard derivations of the Euler-Lagrange equations from the equations of motion for Newtonian mechanics are typically done for interconnections of particles. Here this is carried out in a coordinate-free rmner for rigid bodies, giving for the first time a direct geometric path from the Newton-Euler equations to the Euler-Lagrange equations in the rigid body setting.

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Mathematics Subject Classification: Primary: 70E55; Secondary: 70G45, 70G65, 70G75, 70H03.

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Figure 1. A rigid transformation with spatial and body frames

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Figure 2. Rod with tip constrained to move in a plane

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Figure 3. Central torque-force on a rigid body in a configuration

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