Transcription of Chapter 4. Lagrangian Dynamics
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Chapter 4. Lagrangian Dynamics
56 Chapter 4. Lagrangian Dynamics (Most of the material presented in this Chapter is taken from Thornton and Marion, Chap. 7) Important Notes on Notation In this Chapter , unless otherwise stated, the following notation conventions will be used: 1. Einstein s summation convention. Whenever an index appears twice (an only twice), then a summation over this index is implied. For example, xixi xixii =xi2i . ( ) 2. The index i is reserved for Cartesian coordinates. For example, xi, for i=1,2,3, represents either x,y, or z depending on the value of i. Similarly, pi can represent px,py, or pz. This does not mean that any other indices cannot be used for Cartesian coordinates, but that the index i will only be used for Cartesian coordinates. 3. When dealing with systems containing multiple particles, the index will be used to identify quantities associated with a given particle when using Cartesian coordinates.
61 Figure 4-1 – A simple pendulum of mass m and length . Solution. In Cartesian coordinates the kinetic and potential energies, and the Lagrangian are T= 1 2 mx 2+ 1 2 my 2 U=mgy L=T−U= 1 2 mx 2+ 1 2 my 2−mgy. (4.20) We can now transform the coordinates with the following relations
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