Abstract
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.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this chapter
Cite this chapter
Kriz, I., Pultr, A. (2013). Calculus of Variations and the Geodesic Equation. In: Introduction to Mathematical Analysis. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0636-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0636-7_14
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0635-0
Online ISBN: 978-3-0348-0636-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)