The Principle of Least Action in Dynamics
N. S. Manton
DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK. [email protected]
25 April 2013
Abstract The mathematical laws of nature can be formulated as variational principles. This paper surveys the physics of light rays using Fermat’s Principle and the laws of motion of bodies using the Principle of Least Action. The aim is to show that this approach is suitable as a topic for enriching 6th-form mathematics and physics. The key ideas, calculations and results are presented in some detail.
1 Introduction
In everyday life, some of our activities are directed towards optimising something. We try to do things with minimal effort, or as quickly as possible. Here are a couple of examples: We may plan a road journey to minimise the time taken, and that may mean taking a longer route to travel along a section of motorway. Figure 1 is a schematic road map between towns A and B. Speed on the ordinary roads is 50 mph, on the motorway it is 70 mph. Show that the quickest journey time is along the route AFGB.