Timeline for mechanics: convergence to an equilibrium point
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 30, 2011 at 9:01 | vote | accept | Damien S. | ||
| Dec 30, 2011 at 9:01 | |||||
| Dec 29, 2011 at 21:03 | comment | added | fedja | In general, you get attracted to some connected closed set where $V$ is constant and $\nabla V=0$ but that's all. | |
| Dec 29, 2011 at 21:00 | comment | added | fedja | $\int |x'|^2dt<+\infty$, $x''$ is bounded. Hence $x'\to 0$. Hence $V$ tends to some limit. That much is always true. On the line, if the limit of $x$ fails to exist, there exists a point from which you depart and go fixed distance in both directions and to which you return infinitely many times with arbitrarily low velocity. Moreover, this point is a (non-strict) local minimum (if not, the potential nearby is less and once the velocity drops low enough, the return is impossible). But you cannot go far from a local minimum if you do not have much kinetic energy. | |
| Dec 29, 2011 at 20:32 | comment | added | Damien S. | Thanks for this nice example ! I will work it out. How do you prove simply that it impossible in 1D ? | |
| Dec 29, 2011 at 19:59 | history | answered | fedja | CC BY-SA 3.0 |