Transcription of The Lagrangian Method - Harvard University
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The Lagrangian Method - Harvard University
chapter 6 The Lagrangian MethodCopyright 2007 by David Morin, version)In this chapter , we re going to learn about a whole new way of looking at things. Considerthe system of a mass on the end of a spring. We can analyze this, of course, by usingF=mato write downm x= kx. The solutions to this equation are sinusoidal functions, as we wellknow. We can, however, figure things out by using another Method which doesn t explicitlyuseF=ma. In many (in fact, probably most) physical situations, this new Method is farsuperior to usingF=ma. You will soon discover this for yourself when you tackle theproblems and exercises for this chapter .
VI-4 CHAPTER 6. THE LAGRANGIAN METHOD 6.2 The principle of stationary action Consider the quantity, S · Z t 2 t1 L(x;x;t_ )dt: (6.14) S is called the action.It is a quantity with the dimensions of (Energy)£(Time). S depends on L, and L in turn depends on the function x(t) via eq. (6.1).4 Given any function x(t), we can produce the quantity S.We’ll just deal with one coordinate, x, …
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