CHAPTER III - THE CALCULUS OF VARIATIONS

Abstract

This chapter discusses the calculus of variations. One of the central problems in the classical analysis of functions of one or several variables was that of discovering the extrema of differentiable functions. Extremum problems play an important part in functional spaces. For example, the problem of determining the minimal surface of revolution bounded by two circles with a common axis is interpreted as the problem of finding the extremum of the function the argument y = y(x) of which is itself an element of a normed linear space. The calculus of variations has as its aim a generalization of the structures of classical analysis that will make possible the solution of similar extremum problems. More widely conceived, the calculus of variations is the analysis of infinitesimals in infinite-dimensional spaces. Hamilton's principle possesses a characteristic feature that its formulation omits any assumption of finiteness in respect of the number of degrees of freedom. It can, therefore, be applied even too mechanical systems with infinitely many degrees of freedom, and in particular, to problems with a continuous mass distribution, provided the potential and kinetic energies can be calculated for these systems.