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Unformatted text preview: lecture 6 Topics: F = ma → EulerLagrange equations Hamilton’s principle Functions of functions Calculus of variations Functional derivatives Finding functional derivatives Back to Hamilton’s principle More degrees of freedom The Lagrangian and the action Quantum mechanics and the classical trajectory Appendix: On the functional Taylor series F = ma → EulerLagrange equations This week, we will introduce some difficult, but beautiful material  Lagrangian mechanics. Essen tially, having spent the last couple of weeks getting a better understanding of classical mechanics in terms of second order differential equations, we are now going to introduce a completely different formulation of mechanics based on the calculus of an infinite number of variables! This proba bly sounds really scary. But DON’T PANIC! I am introducing this for several reasons. I think that you will find it fascinating, and I hope that it will give you a different philosophical slant on classical mechanics. It may also give you a hint about how the quantum mechanical nature of the world may determine the shape of the classical mechanics that we are used to. The mathematics will sound unfamiliar to almost all of you. But in the end, we will not actually be doing anything with it that you do not already know how to do. Finally, it is really useful — not just theoretically but also for solving real problems. The result of the painful analysis I am about to subject you is actually pretty simple. The statement is this. To solve many classical problems (and we will discuss later why this works in the most interesting cases) you construct a function (called the Lagrangian) which is the kinetic energy T minus the potential energy V as a function of the coordinates and their time derivatives (the velocities). L ‡ q, ˙ q · T ‡ q, ˙ q · V ‡ q, ˙ q · (1) ( T and V may not depend on both q and ˙ q , but that will depend on the problem). Then for each coordinate, you get an equation of motion as follows: d dt ˆ ∂ ∂ ˙ q L ‡ q, ˙ q · ! = ∂ ∂q L ‡ q, ˙ q · (2) This gives you a set of straightforward rules that will allow you to write down the second order differential equations for the time evolution of a physical system in a single step, without identifing all the separate forces acting on the system. So I hope that even if you never fully appreciate all the deep philosophical implications of the Lagrangian approach, you will love it as a laborsaving device. 1 I could of course just write down (1) and skip all the difficult math that leads up to it, because the hard stuff won’t really be important in the way we use these results most of the time. I think it is good to go through it once, but really what I am really most interested in is the philosophical implications of all this. The deeper question that we are heading to is Why are energy and momentum conserved? Next week, I will try to convince you that these conservation laws follow directly from fundamental symmetries of the world...
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 Spring '09
 JohnRobertson

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