TABLE OF CONTENTS During the mid-19th century, a family of fundamental developments was introduced, led by Lagrange, Hamilton, and Jacobi. These results provided a unifying perspective on analytical mechanics and also stimulated fundamental advances in allied mathematical subfields such as variational calculus, differential equations, and topology. The most central developments are embodied in elegant and powerful methods for deriving differential equations of motion by taking gradients of scalar functions they introduced (e.g., the Lagrangian and the Hamiltonian, closely related to the mechanical kinetic and potential energies of the system), relationships of mechanical system motion to variational principles (e.g., d'Alembert's Principle and Hamilton's Principle), and efficient methods for accommodating constraints and constraint forces. Collectively, these insights amounted to a revolution in analysis of dynamical systems, even given that their starting point was the summation of the monumental works of Newton, Gauss, and Euler. This chapter and the following one provides the most fundamental aspects of these classical developments; we start with Newtonian=Eulerian principles and utilize a system of particles as a conceptual representation for a large class of systems. We introduce virtual and related variational arguments leading to d'Alembert's principle, Lagrange's equations, and Hamilton's principle. Finally, we generalize these particle mechanics results to establish the corresponding developments applicable to systems idealized as collections of particles, rigid bodies, and distributed parameter systems. Examples are utilized throughout this discussion to illustrate the ideas and provide some insights into their utility.
Consider the familiar problem of a particle moving relative to an...
