Timeline for Period of Oscillation of a potential $ U = A|x|^n$ problem [closed]
Current License: CC BY-SA 4.0
14 events
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| Nov 1, 2021 at 22:17 | comment | added | Lightcone | There is a neat way to deal with this kind of problem: the action variable. In the phase space $(x,p)$, the particle orbits around a closed trajectory $p^2/2m + A|x|^n = E$ clockwise and encloses an area $J \propto \sqrt{2mE}\cdot (E/A)^{1/n} \propto E^{1/2+1/n}$. Then the formula $\tau = \partial J/\partial E$ implies that $\tau \propto E^{1/n-1/2}$. Finding the precise expression for the proportional constant is an easy exercise. Tried to explain more details by posting as an answer, but failed since it is closed. You may refer to: physics.stackexchange.com/a/389454/100521 | |
| Mar 18, 2021 at 19:27 | history | closed |
G. Smith ZeroTheHero Jon Custer Buzz♦ Chemomechanics |
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| Mar 18, 2021 at 18:43 | vote | accept | Adán González | ||
| Mar 18, 2021 at 17:57 | answer | added | Eli | timeline score: 2 | |
| Mar 17, 2021 at 18:16 | review | Close votes | |||
| Mar 18, 2021 at 19:27 | |||||
| Mar 17, 2021 at 17:25 | answer | added | ytlu | timeline score: 5 | |
| Mar 17, 2021 at 17:07 | history | edited | Adán González | CC BY-SA 4.0 |
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| Mar 17, 2021 at 17:06 | comment | added | Adán González | @ytlu I made the assumption that $ x'(0) = 0 $ | |
| Mar 17, 2021 at 17:05 | comment | added | ytlu | The Lplace transform of derivative shoudl involve the intital conditions, the intial position and intial velocity. | |
| Mar 17, 2021 at 17:04 | comment | added | Andrew Steane | ... or (same idea another way) start at $x = 0$ with non-zero velocity, and find out how long it takes for the velocity to reach zero. | |
| Mar 17, 2021 at 17:02 | comment | added | Andrew Steane | Just an idea: from energy considerations one can deduce that the particle will oscillate, but not in a simple harmonic fashion. So to get the period one way is to take an initial condition of at rest at some $x > 0$, and then don't solve for the whole trajectory, but just aim to find out how long it takes to reach the location $x=0$. That will be one quarter of the period. | |
| Mar 17, 2021 at 16:58 | history | edited | Adán González | CC BY-SA 4.0 |
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| Mar 17, 2021 at 16:51 | history | edited | Adán González | CC BY-SA 4.0 |
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| Mar 17, 2021 at 16:43 | history | asked | Adán González | CC BY-SA 4.0 |