From: "Michael Weiss" Subject: Re: Lagrangian formalism in general relativity Date: Thu, 18 May 2000 03:08:12 GMT Newsgroups: sci.physics.research Summary: [missing] Oz grapples with action: |I thought the action was essentially the sum of the potential energy |terms (eg gravitational, electromagnetic etc) less the kinetic energy |(keeping newtonian here). Since any increase in the total PE will result |in a decrease in the KE then in essence this merely states (local) |conservation of energy. 'Fraid it's not that simple. PE + KE = constant. KE - PE is a different beast. Of course you could use conservation of energy to rewrite KE-PE. Since PE + KE = total energy, PE = total energy - KE, and so KE - PE = KE - (total energy - KE) = 2KE - total energy. I don't know that 2KE - total energy is any more intuitive than KE - PE. But read on. Also be warned: KE - PE is the formula for the lagrangian of a classical point particle in Newtonian mechanics (moving in a conservative force field without velocity-dependent terms, just to dot all the p's and q's). In other cases, the lagrangian is different. When I tried to grok analytical mechanics one fine summer, I finally found a measure of insight from reading some of Hamilton's original papers. The "royal road" (if there is one) passes through Hamiltonian *optics*. Herewith some chatty remarks. I hope they help. Start with Fermat's principle of least time. Fermat said that light tries to get from point A to point B in as little *time* as possible. Fermat showed how to derive Snell's law of refraction from his principle; the derivation looks like that old calculus chestnut, where a decathlon champion has to get from point A on the beach to point B in the water as fast as possible, first running and then swimming. As you know, once upon a time, Newton and Huyghens had two different theories of light. Huyghen's light waves *slow down* when passing from air to water, while Newton's light corpuscles *speed up*. (They get a "kick" as they enter the water, and that increase in the vertical velocity component is Newton's explanation for refraction.) Both Newton's corpuscles and Huyghens' account for ray optics well enough. But Fermat's derivation works out only if you assume that light *slows down*. Can we adapt it to Newton's theory? We can indeed. According to Newton, the index of refraction is v_water / v_air; according to Huyghens, it's v_air / v_water. Fermat said that time is minimized. In other words, he found the minimum of integral_{A to B} dt = integral_{A to B} (1/v) dx (This was one of the earliest problems in the calculus of variations, incidentally.) Since Newton's formula for the index of refraction is just the reciprocal of Huyghens, someone (Maupertuis, if I remember) had the bright idea of replacing (1/v) with v in Fermat's principle. In other words, to find the path of a light corpuscle, minimize: integral_{A to B} v dx = integral_{A to B} v^2 dt. OK, so light *isn't* really Newtonian corpuscles. But the derivation works fine for any kind of little particles obeying Newtonian mechanics. So to find the path of a Newtonian particle going from point A to point B, it looks like we should minimize the integral of v^2 dt. Well, v^2 should make you think of the kinetic energy. The lagrangian seems to be emerging from the mist. But what about the PE term? Let's think a little more about that integral. You need to know v at all points along any contemplated path. Otherwise you can't do the integral! Maupertuis computed v using the conservation of energy. For the refraction problem, he assumed a discontinuous change in the PE at the air/water interface. But you can use any nice-enough potential function. We need to be a bit clearer about the term *path*. Do we mean just the curve in space, or a function from time to space? The function tells you not only where the particle was, but at what time. Let's use "path" for the time-to-space function from now on, and "curve" just for the track the path traces out. So Maupertuis was restricting attention to paths that obey conservation of energy. In other words, he placed three constraints on the motion of the particle, even before he started minimizing integrals: 1. The path starts at A. 2. It ends at B. 3. PE + KE = E_total all along the path. Someone else (Euler? Lagrange?) decided to *drop* that last constraint. Well, you can't just drop it and still minimize v^2 dt, and expect to get the right answer! But remember our alternate form for the lagrangian: (2KE - E_total) dt = (mv^2 - E_total) dt Maupertuis was holding E_total fixed (both along a path, and between paths). So as we vary the curve, the E_total term won't affect which path gives the minimum. Euler said, in effect: let's consider *all* paths that: 1. Start at A, at time t_0. 2. End at B, at time t_1. E_total will vary from path to path, and can even vary for different points along the *same* path. No matter. Euler discovered that the path that minimizes (mv^2 - E_total) *automatically* does so using a constant value for E_total. Enough chatting for one post.