Advanced Theoretical Mechanics

Advanced Theoretical Mechanics

A Course of Mathematics for Engineers and Scientists

1st Edition - January 1, 1966

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  • Authors: Brian H. Chirgwin, Charles Plumpton
  • eBook ISBN: 9781483137407

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Advanced Theoretical Mechanics deals with advanced theoretical mechanics in three dimensions, making use of concepts and methods such as matrices, vectors, tensors, and transformation methods. The definition of a vector via the transformation law obeyed by its components is emphasized, and matrix methods are used to handle sets of components. Special attention is given to the definition of angular velocity and the proof that it can be represented by a vector. This book is comprised of 11 chapters and begins with an introduction to kinematics in three dimensions. Lagrange's equations and analytical dynamics are then presented, along with the simpler problems of three-dimensional dynamics, often with the help of rotating axes. Stability and small oscillations are also considered. The subsequent chapters focus on the dynamics of a particle and the motion of a system of particles; gyroscopic motion, free rotation, and steady motion; oscillations of a dynamical system with a finite number of degrees of freedom; and the vibrations of strings. The final chapter is devoted to analytical dynamics, paying particular attention to Hamilton's principle and equations of motion as well as the Hamilton-Jacobi equation. This monograph is intended for engineers and scientists as well as students of mathematics, physics, and engineering.

Table of Contents

  • Preface

    Chapter I. Kinematics in Three Dimensions


    The Transformation Law for Vectors

    Finite Rotations

    Successive Rotations: Euler's Angles

    Angular Velocity

    Relative Motion

    Moving Frames of Reference

    The Acceleration of a Particle

    The General Motion of a Rigid Body

    Angular Velocities about Non-Intersecting Axes

    Chapter II. Sets of Forces: Equilibrium



    Equivalent Sets of Forces

    The Principle of Virtual Work

    Other Sets of Line Vectors

    Chapter III. The Dynamics of a Particle

    General Principles

    A Particle with One Degree of Freedom

    The Use of Rotating and Accelerated Axes

    The Spherical Pendulum

    Motion on a Surface of Revolution

    Motion Relative to the Rotating Earth

    The Motion of a Charged Particle

    Chapter IV. The Motion of a System of Particles

    Description of the System

    The Dynamical Variables

    Conservation Laws

    The Inertia Matrix

    Principal Axes of Inertia

    Dynamical Variables for Rigid Systems

    The Motion of a Sphere

    Chapter V. Gyroscopic Motion, Free Rotation and Steady Motion


    Rotation under No Forces of Bodies with Kinetic Symmetry

    The Steady Motion of a Gyroscope or Top

    The General Motion of a Top

    Euler's Dynamical Equations

    Free Rotation

    More General Motions

    Chapter VI. Lagrange's Equations

    Generalized Methods

    The Dynamical Variables

    Generalized Forces

    Classification of Constraints

    Application of the Principle of Virtual Work

    Conservation Laws

    Ignoration of Coordinates

    The Motion of a Charged Particle

    Chapter VII. Stability of Motion


    Steady Motion with Two Degrees of Freedom

    The Stability of Free Rotation of a Rigid Body

    The Stability of a Top

    The Gyro-Compass

    The Stability of a Rolling Wheel

    Chapter VIII. Impulsive Motion

    Elementary Discussion

    Generalized Methods

    General Theorems

    Chapter IX. The Oscillations of a Dynamical System with a Finite Number of Degrees of Freedom - Normal Modes


    Systems with Two Degrees of Freedom

    Stability of Equilibrium: Free Oscillations of a System with n Degrees of Freedom

    The Oscillations of a Linearly Constrained System—Rayleigh's Principle

    A Reciprocal Theorem

    Chapter X. The Vibrations of Strings

    The Fundamental Concepts of Wave Motion

    Transverse Vibrations

    Normal Modes

    Forced Vibrations and Damping

    Reflection and Transmission at a Discontinuity

    Longitudinal Vibrations

    Application of Rayleigh's Principle

    Miscellaneous Problems

    Chapter XI. Analytical Dynamics


    Hamilton's Principle

    The Principle of Least Action

    Hamilton's Equations of Motion

    Transformation Theory: Contact Transformations

    Infinitesimal Contact Transformations

    The Hamilton-Jacobi Equation

    Answers to the Exercises


Product details

  • No. of pages: 518
  • Language: English
  • Copyright: © Pergamon 2013
  • Published: January 1, 1966
  • Imprint: Pergamon
  • eBook ISBN: 9781483137407

About the Authors

Brian H. Chirgwin

Charles Plumpton

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