Abstract
The principal of least action is one of the fundamental ideas in physics. The path of the shortest time of a particle in the presence of gravity is an example of this principal. In this paper some methods are introduced to teach the optimal path in introductory physics courses. The optimal path (path of the shortest time) is calculated for a few families of paths. Finally a numerical method according to Snell’s law in a discrete medium is used to find the general optimal path and is compared with the brachistochrone path.
Highlights
The principle of least action is one of the fundamental ideas in physics
In the numerical method the shooting method is used in order to find the path that passes through the end points
The brachistochrone path is introduced in this paper and all optimal paths, including the numerical Snell’s path are compared to the brachistochrone curve
Summary
The principle of least action is one of the fundamental ideas in physics. The calculus of variations is a powerful mathematical tool used to understand and identify the path for the shortest time (Boas, 2006; Taylor, 2005; Thornton & Marion, 2003). These are common examples that are used for a variety of purposes in any introductory physics course as sample problems in a lecture, homework problems, and demonstrations in a class or as part of an experiment in a lab Because these are familiar examples and students know their relevant kinematic equations, we will use them to introduce the idea of the optimal path (the path for the shortest time). The derivative of with respect to for this case is complicated and to find the value of the minimum time and corresponding we need to use a numerical method to solve = 0. In section we will use a parabolic example and use numerical techniques to find time as a function of a parameter, the minimum time, and the corresponding optimal path
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