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.
Newtonian mechanics. This is, however, a simple problem that can easily (and probably more quickly) be solved directly from the Newtonian formalism. But, the benefits of using the Lagrangian approach become obvious if we consider more complicated problems. For example, we try to determine the equations of motion of a particle of mass
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