The Variational Principles of Mechanics [antikvár]

INTRODUCTION 1. The variational approach to mechanics. Ever since Newton laid the solid foundation of dynamics by formulating the laws of motion, the science of mechanics dieveloped along two main lines. One branch, which we shall call "vectorial mechanics, starts directly from Newton's laws of motion. It aims at recognizing all the forces acting on any given particle, its motion being uniquely determined by the known forces acting on it at every instant. The analysis and synthesis of forces and moments is thus the basic concern of vectorial mechanics. While in Newton's mechanics the action of a force is measured by the momentum produced by that force, the great philosopher and universalist Leibniz, a contemporary of Newton, advocated another quantity, the vis viva (living force), as the proper gauge for the djmamical action of a force. This vis viva of Leibniz coincides—^apart from the unessential factor 2—with the quantity we call today "kinetic energy." Thus Leibniz replaced the "momentum" of Newton by the "kinetic energy." At the same time he replaced the "force" of Newton by the "work of the force." This "work of the force" was later replaced by a still more basic quantity, the "work function." Leibniz is thus the originator of that second branch of mechanics, usually called "analytical mechanics,which bases the entire study of equilibrium and motion on two fundamental scalar quantities, the "kinetic energy" and the "work function," the latter frequently replaceable by the "potential energy." Since motion is by its very nature a directed phenomenon, it seems puzzling that two scalar quantities should be sufficient to determine the motion. The energy theorem, which states that the sum of the kinetic and potential energies remains unchanged *This use of the term does not necessarily imply that vectorial methods are used. 'See note on terminology at end of chap. i, section 1, xxi