Parametric Variational Integrals

Abstract

In this chapter we shall treat the theory of one-dimensional variational problems in parametric form. Problems of this kind are concerned with integrals of the form

$$F(x) = \int_a^b {F(x(t))} ,\dot x(t))dt$$

(1)

, whose integrand F(x, υ)is positively homogeneous of first degree with respect to υ. Such integrals are invariant with respect to transformations of the parameter t, and therefore they play an important role in geometry. A very important example of integrals of the type (1) is furnished by the weighted arc length

$$S(x): = \int_a^b {\omega (x(t))} \left| {\dot x(t)} \right|dt$$

(2)

, which has the Lagrangian F(x, υ) = ω(x)|υ|. Many celebrated questions in differential geometry and mechanics lead to variational problems for parametric integrals of the form (2), and because of Fermat’s principle also the theory of light rays in isotropic media is governed by the integral (2), whereas the geometrical optics of general anisotropic media is just the theory of extremals of the integral (1).