Calculus of Variations and the Geodesic Equation

The aim of this chapter is to give a glimpse of the main principle of the calculus of variations which, in its most basic problem, concerns minimizing certain types of linear functions on the space of continuously differentiable curves in $${\mathbb{R}}^{n}$$ with fixed beginning point and end point. For further study in this subject, we recommend [7]. We derive the Euler-Lagrange equation which can be used to axiomatize a large part of classical mechanics. We then consider in more detail the possibly most fundamental example of the calculus of variations, namely the problem of finding the shortest curve connecting two points in an open set in $${\mathbb{R}}^{n}$$ with an arbitrary given (smoothly varying) inner product on its tangent space. The Euler-Lagrange equation in this case is known as the geodesic equation. The smoothly varying inner product captures the idea of curved space. Thus, solving the geodesic equation here goes a long way toward motivating the basic techniques of Riemannian geometry, which we will develop in the next chapter.