# Advanced Theoretical Mechanics

## 1st Edition

### A Course of Mathematics for Engineers and Scientists

**Authors:**Brian H. Chirgwin Charles Plumpton

**eBook ISBN:**9781483137407

**Imprint:**Pergamon

**Published Date:**1st January 1966

**Page Count:**518

## Description

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

Introduction

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

Introduction

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

Introduction

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

Introduction

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

Introduction

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

Introduction

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

Index

## Details

- No. of pages:
- 518

- Language:
- English

- Copyright:
- © Pergamon 1966

- Published:
- 1st January 1966

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483137407