So while calculus of variation has similar goals as calculus and uses similar ideas it is substantially more technical. However, in calculus of variation we usually have that critical points are characterized as the solution set of a PDE. ![]() In calculus, typically the set of critical points is relatively "small". Let me briefly comment on the critical points. One can the pass to different topologies where one does have compactness (the Banach–Alaoglu theorem is often quite useful), however, then one needs to ensure that the function in question is also continuous with respect to that new topology (typically this is not really true, but we can get away by showing that it is lower semicontinuous in that new topology). For example closed balls are no longer compact Is it true that the unit ball is compact in a normed linear space iff the space is finite-dimensional?. Fundamental theory of the calculus of variations variable end points the parametric problem the isoperimetric problem constraint inequalities introduction. In infinite dimensions the first point becomes quite difficult as compact sets are kind of "small". ![]() For example the function is continuous and goes to infinity for $\vert x \vert \rightarrow \infty$.ģ.) We know that global extrema are critical points.Ĥ.) We check all the critical points and the smallest one is the global minimum. This is usually done using some sort of compactness. Typically we proceed in the following way:ġ.) Show that some extremum must exist. One of the things we care about in calculus is to find extrema of functions. The most basic one is that linear functions no longer need to be continuous in infinite dimensions! See here for an example Discontinuous linear functional. While the definition of integration, differentiation and so on are quite similar to the one in calculus, there are always some technical difficulties that pop up in infinite dimensions. ![]() This variational approach is used in classical physics as well and is called Hamilton's principle.Īs mentioned in my comment above, calculus is usually done for functions from $\mathbb^k$. Or you might want to have a look at chapter 10 in Lieb & Loss "Analysis", where they explain how the calculus of variation is useful in quantum mechanics. This answer consists of me waving my hands, if you want to get a better understanding I would advise you to have a glimpse at any textbook in calculus of variation, like the book by Struwe.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |