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  • Print publication year: 1999
  • Online publication date: April 2013

I - The Calculus of variations

from Appendices


IT is a general feature of equations of classical physics that they can be derived from variational principles. Two early examples are Fermat's principle in optics (1657) and Maupertuis’ principle in mechanics (1744). The equations of elasticity, hydrodynamics and electrodynamics can also be represented in this way.

However, when one deals with field equations, involving as a rule four or more independent variables x, y, z, t, …, one makes little use, owing to the great complexity of partial differential equations, of the property that the solution expresses stationary values of certain integrals. The only essential advantage of the variational approach in such cases is connected with the derivation of conservation laws — e.g. for energy. The situation is quite different in problems involving one independent variable (time in mechanics, or length of a ray in geometrical optics). Then one deals with a set of ordinary differential equations and it turns out that a study of the behaviour of the solution is greatly facilitated by a variational approach. This approach is in fact a straightforward generalization of ordinary geometrical optics in every detail. Its modern representation owes much to David Hilbert, on whose unpublished lectures, given at Gottingen in about 1903, we base the considerations of the following sections. The theory is presented here for a three-dimensional space (x, y, z) only, but can easily be extended to more dimensions.