From: Alexander Poquet Subject: Calculus of Variations Date: 13 Dec 1999 22:45:44 GMT Newsgroups: sci.math Hi, I am looking for a good on-line introduction to the calculus of variations, preferably in a non-physics oriented fashion (though I'll take what I can get). From a physics standpoint, I can 'use' some basic methods (hamilton's first function, etc) but I have some questions about what motivates certain things. As an example: I have a particle in N-space that is assumed to follow the path which minimizes hamilton's first function. defining the 'action' of a path 's' which may not be the true path: / t_2 / t_2 S = | KE - PE dt = | (m/2)[ds/dt]^2 - V(s) dt / t_1 / t_1 s is parametrically defined and the 'square' of its derivative is defined in terms of the vector inner product. V(s) is the potential energy at point s(t). Presumably, I can define 's' in terms of the true path x varied by some function n(t): / t_2 S = | (m/2)[dx/dt + dn/dt]^2 - V(x + n) dt / t_1 Which motivates the expansion: / t_2 S = | (m/2)[dx/dt]^2 + m[dx/dt . dn/dt] + (m/2)[dn/dt]^2 - / t_1 V(x) - [n(t) . grad V(x)] - O(t) dt Where O(t) represents the other terms in the taylor expansion of V. Now, we are told that we are only minimizing the first order (i dont understand why, but i intuit that this analogous to only setting the first derivative equal to zero to find extrema in classic calculus optimization problems -- if someone could clear this up for me, i would be grateful.) Breaking up the integral, / t_2 S = | (m/2)[dx/dt]^2 - V(x) dt + / t_1 / t_2 | m[dx/dt . dn/dt] - [n(t) . grad V(x)] dt / t_1 / t_2 | O(t) dt / t_1 Now we are told we are only interested in the second integral. Let me see if I understand why -- the first integral is clearly the action of the true path, which we have assumed to be at the minimum. The second integral is the first order variation of that path in terms of n(t); the third integral contains the higher order variations due to n(t). Our goal, we are told, is to write the integrand as [f(x,x',n',t) . n(t)]. This is easily accomplished with integration by parts a la inner product: | t_2 dS = m[dx/dt]n(t)| + | t_1 / t_2 | (-m[d^2x/dt^2] . n(t)) - [n(t) . grad V(x)] dt / t_1 I have written dS to represent the variation of S, because tty doesn't do greek lowercase deltas very well. Given our initial boundary conditions, mainly that our ideal path and varied path start and end at the same points -- that is, n(t_1) = n(t_2) = zero vector, we can throw out non-integral term. At this point we do something I'm not sure I understand; we set the first order variation to zero: / t_2 dS = | (-m[d^2x/dt^2] - grad V(x)) . n(t) dt = 0 / t_1 We then reason that, if the integral itself is zero, it follows that the integrand is zero. If t_2 and t_1 are arbitrary, it is fairly easy to see that this must be so. However, we then state that the integrand is always zero, for an arbitrary n(t). I dont understand this. It seems like the varied curve, s(t), will not necessarily have dS = 0, and that the whole point of the operation is to make figure out what s(t) is if s(t) = x(t). Some clarification would be cool. Of course, as a student I just swallow this and go ahead and say: therefore, if n(t) can be anything, then in order for the integrand to consistantly be zero, the coefficient of n(t) must be zero: -m[d^2x/dt^2] - grad V(x) = 0 which gives - grad V(x) = m[d^2x/dt^2] which is equivalent, of course to F = ma which is clearly the right answer, so my computation is right. But I'm not sure I understand the steps and this wishy-washy physics approach of 'use math as a tool and dont ask questions' is disconcerting. From a less physical standpoint, I was interested in proving that a circle maximizes enclosed area for a given perimeter. There are easier ways to do this, I know! But I wanted to apply the technique. The most straightforward way seemed to be to use greens theorem: / // A = (|) F . ds = || curl F . k dA / C // R Where C is the boudary of the region R in the plane and s is what I want to know. Choosing F to make the k component of its curl equal 1 (and thus the enclosed area), and then converting the line integral to a reimann integral (using the definition of the differential), I obtain an integral to minimize. I used F = < 0, x >: / t_2 A = | x(t)[dy/dt] dt / t_1 Which I can then vary by introducing n_x and n_y in an analogous fashion. This technique yields a minimum: s(t) = 0 vector. I never introduced the other (important!) condition -- the constant perimeter, namely: / t_2 P = | sqrt[(ds/dt)^2] dt = C / t_1 where the square has again been used to imply the vector inner product. I'm not sure how to introduce it; I don't know how I'd get a square root in my area integral, though I can choose whatever field I want as long as its curl is one. I guess I don't need a root, because if the integral of square root is constant the integral of the square is constant. Anyway I only played with that a little. This has probably been done before, but I'd appreciate any tips. Thanks for sticking with this :) alexander ============================================================================== From: foltinek@math.utexas.edu (Kevin Foltinek) Subject: Re: Calculus of Variations Date: 16 Dec 1999 11:59:12 -0600 Newsgroups: sci.math In article <38557718$0$206@nntp1.ba.best.com> Alexander Poquet writes: > Hi, I am looking for a good on-line introduction > to the calculus of variations, preferably in a non-physics > oriented fashion (though I'll take what I can get). I don't know off-hand of any on-line discussions of CoV but you could probably do a search for it or Lagrangian or mechanics or dynamics. There are a lot of actual books about the subject, many of them inexpensive (Dover), though a lot of them say pretty much the same thing. I don't know if I'd recommend any one of them over any others as an introduction. An approach which is slightly different (more rigorous) than the one you outlined is the following (in some generality). Let Y(u,t) be a smooth function, defined for all real u and for all t1<=t<=t2. Given any value for u, define a curve x_u by x_u(t)=Y(u,t). Let L(t,x,v) be an arbitrary (smooth) function of the real variable t and the position x and the velocity v. Given a curve x(t), the velocity will be v(t) = x'(t). The "action" of a path x(t) defined for t1<=t<=t2 is S[x] = \int_{t1}^{t2} L(t,x(t),x'(t)) dt . The problem is (without trying to make this precise): find the path x(t) between two fixed points x1 and x2, defined for t1<=t<=t2, such that S[x] is minimal. We can look at S[x_u] for any value of u; if, for some value of u, x_u is a minimizer (or other critical point) of S[x_u], then d/du S[x_u] = 0. So you can compute d/du S[x_u], integrate by parts, and obtain something similar to what you have already seen. (Hint: your n(t) will be replaced by (Y_u = \partial Y/ \partial u).) This approach is more general and more rigorous than the Taylor-type expansion. Eventually (using the given boundary data) you end up (when 0=dS/du) with something that looks like: 0 = \int_{t1}^{t2} g(t,x_u(t),x'_u(t)) Y_u dt . Note that Y_u is an arbitrary function (subject to the boundary conditions). If g(...) is non-zero at some value of t, then it will be non-zero and of constant sign in an interval containing t; we can choose Y_u to have the same sign in that interval and zero everywhere else, so we integrate something which is non-negative, and positive on a non-empty interval, therefore the integral is positive, a contradiction. (In the case when x is a vector, the above g will be a vector, as will Y_u; then if g is non-zero at some point, it is non-zero on some interval, and we choose Y_u to point in the same direction as g on the interval, zero elsewhere, giving a non-negative integrand.) Therefore g(...) is zero everywhere. The fact that Y_u is arbitrary is easy in the case of unconstrained paths joining two points. When you add local constraints to the problem, things become quite a bit messier (particularly the "non-holonomic" constraints). But that's a whole different story. Your last question involved a constrained optimization problem: minimize an action subject to a constraint of the form \int_{t1}^{t2} k(t,x,x') dt = C . There you have to use Lagrange multipliers (just as in the constrained optimization problems from vector calculus). I don't remember off the top of my head but I think that instead of (L_x - (L_x')') = 0 you get (L_x - (L_x')') = \lambda * (k_x - (k_x')') . By the way, there are all sorts of nasty technical details that can be considered, and to make anything precise, probably should be considered. For example, "what is a path?". (How smooth?) Also, the point where I said "if, for some value of u, x_u is a minimizer": there are cases where the true minimizer is not a member of a smooth family of paths. (Example: minimize the surface area of a surface of revolution, with specified end radii.) Hope this helps some. Kevin.