Elements of Analytic Mechanics

Abstract

As you may have noticed, the previous chapters used a vector approach to mechanics based on Newton’s laws, which describe motion using vector quantities such as force, speed, and acceleration. These quantities characterize the motion of a body, which is idealized as a “material point” or “particle,” understood as the only point to which mass is attached. In contrast, analytical mechanics uses scalar properties of motion that represent the system as a whole—usually its total kinetic energy and potential energy—rather than the Newton vector forces of individual particles. A scalar is a quantity, and a vector is a quantity and direction. These equations of motion are derived from a scalar quantity. At the same time, such artificial techniques as the principle of possible displacements, generalized coordinates, and generalized forces are used. In various fields of modern technology, complex problems arise, for the solution of which it is desirable to have a universal analytical apparatus based on the general principles of mechanics. The development of such an apparatus, the presentation of the general principles of mechanics, the derivation of the differential equations of motion from them, and the study of the equations themselves and methods of their integration constitute the main content of analytical mechanics.