Variational Principles in Classical Mechanics

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Douglas Cline, University of Manchester

Pub Date: 2017

ISBN 13: 978-0-9988372-4-6

Publisher: Independent

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Table of Contents

  • Contents
  • Preface
  • Prologue
  • 1 A brief history of classical mechanics
  • 2 Review of Newtonian mechanics
  • 3 Linear oscillators
  • 4 Nonlinear systems and chaos
  • 5 Calculus of variations
  • 6 Lagrangian dynamics
  • 7 Symmetries, Invariance and the Hamiltonian
  • 8 Hamiltonian mechanics
  • 9 Conservative two-body central forces
  • 10 Non-inertial reference frames
  • 11 Rigid-body rotation
  • 12 Coupled linear oscillators
  • 13 Hamilton’s principle of least action
  • 14 Advanced Hamiltonian mechanics
  • 15 Analytical formulations for continuous systems
  • 16 Relativistic mechanics
  • 17 The transition to quantum physics
  • 18 Epilogue
  • Appendices

About the Book

Two dramatically different philosophical approaches to classical mechanics were developed during the 17th - 18th centuries. Newton developed his vectorial formulation that uses time-dependent differential equations of motion to relate vector observables like force and rate of change of momentum. Euler, Lagrange, Hamilton, and Jacobi, developed powerful alternative variational formulations based on the assumption that nature follows the principle of least action. These powerful variational formulations have become the preeminent philosophical approach used in modern science, as well as having applications to other fields such as economics and engineering.

This book introduces variational principles, and illustrates the intellectual beauty, the remarkable power, and the broad scope, of applying variational principles to classical mechanics. A brief review of Newtonian mechanics compares and contrasts the relative merits of the intuitive Newtonian vectorial formulation, with the more powerful analytical variational formulations. Applications presented cover a wide variety of topics, as well as extensions to accommodate relativistic mechanics, and quantum theory.

About the Contributors


Douglas Cline received his BSc 1st Class Honours in Physics, (1957) and his PhD in Physics (1963) both from the University of Manchester. He joined the University of Rochester in 1963 as a Research Associate, and was promoted to Assistant Professor (1965), Associate Professor(1970), and Professor (1977). At the University of Rochester Nuclear Structure Research Laboratory he served as Associate Director (1977-88) and Director (1988-1999). He has held visiting appointments at Laval University, (1965), Niels Bohr Institute in Copenhagen (1973), Lawrence Berkeley Laboratory (1975-76), Australian National University (1978), and the University of Uppsala (1981). He is a Fellow of the American Physical Society (1981), and a recipient of the Lawrence Berkeley Laboratory Gammasphere Dedication Award (1995), the Award for Excellence in Teaching from the Department of Physics and Astronomy (2007, 2009), and the 2013 Marian Smoluchowski Medal from the Polish Physical Society.