Timeline for Why should we expect the brachistochrone problem to be solvable?
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| when toggle format | what | by | license | comment | |
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| 51 mins ago | answer | added | Qmechanic♦ | timeline score: 1 | |
| 7 hours ago | history | became hot network question | |||
| 8 hours ago | comment | added | peek-a-boo | Re last paragraph: But we do use compactness several times when discussing minimizers of functionals (particularly in the Sobolev space context). Namely we use Banach-Alaoglu and also the Rellich compactness theorems. See Evans’ PDE book chapter 8 for explicit assumptions on the class of functionals considered. | |
| 10 hours ago | comment | added | Cleonis | I copy a paragraph from my existing answer: The Brachistochrone problem straddles two categories of application of calculus of variations. It's a Statics problem in the sense that the solution is a static shape: which shape has the property that sliding along it happens in the fastest time? But to get at the solution Dynamics must be applied: given a curve, how fast will an object slide down the curve? It's not a Statics problem: something is moving. It's not a Dynamics problem: for any value of the gravitaitonal acceleration the solution has the same shape. | |
| 11 hours ago | comment | added | Buzz♦ | Compactness seems like something far stronger than necessary. A completeness property on the metric space of paths (say, continuous and of bounded variation) ought to be enough. | |
| 13 hours ago | answer | added | Cleonis | timeline score: 3 | |
| 15 hours ago | answer | added | controlgroup | timeline score: 6 | |
| 15 hours ago | comment | added | controlgroup | "Given the terrible failure of compactness in all reasonable function spaces why do so many natural/interesting calculus of variations problems turn out to be solvable?" -- you might consider that we, as mere mortal humans, might happen to be interested in problems that are reasonably simple and thus are more likely to have solutions. | |
| S 15 hours ago | review | First questions | |||
| 14 hours ago | |||||
| S 15 hours ago | history | asked | Noah Schweber | CC BY-SA 4.0 |