Timeline for Hamilton-Jacobi equation with a second-order Lagrangian
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 6, 2023 at 15:48 | history | edited | Qmechanic♦ | CC BY-SA 4.0 |
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| Nov 1, 2016 at 0:07 | vote | accept | Milou | ||
| Oct 14, 2016 at 13:38 | answer | added | Qmechanic♦ | timeline score: 4 | |
| Oct 14, 2016 at 13:38 | history | edited | Qmechanic♦ | CC BY-SA 3.0 |
added 196 characters in body; edited tags; edited title
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| Oct 2, 2016 at 16:40 | comment | added | Milou | What I've written in my comment is absolutely wrong :) I'm sorry! So I've decided to use the Ostogradsky Hamiltonian. But I still don't know if the Hamilton-jacobi equation is still true anyway. | |
| Sep 30, 2016 at 17:55 | comment | added | ACuriousMind♦ | The point is - what space is that new $H$ living on? In the first-order setting, we switch from the Lagrangian tangent bundle with coordinates $(q,\dot{q})$ to the Hamiltonian phase space/cotangent bundle with coordinates $(q,p)$. But here you can't just Legendre transform the $\dot{q}$, you need to somehow handle the $\ddot{q}$ dependence. There are difference schemes for that, leading to different notions of the final Hamiltonian, so it's not exactly clear what you want to do here. | |
| Sep 30, 2016 at 17:47 | comment | added | Milou | @ACuriousMind am I allowed to write this? I've never worked with a higher-order L. | |
| Sep 30, 2016 at 1:39 | comment | added | Milou | umm if it's true to consider that ∂S/∂t=−H, I would define my Hamiltonian from \delta S (wich is equal to L \delta t), so H will be =∑j p\dot{q}+∑j ∂S/∂\dot{q} \ddot{q}−L (all will j index) | |
| Sep 30, 2016 at 1:26 | comment | added | ACuriousMind♦ | Which definition of the "Hamiltonian" are you using for higher-order Lagrangians? There are several (e.g. that by Ostrogradsky) that are similar, but not the same. | |
| Sep 30, 2016 at 1:13 | history | edited | Milou |
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| Sep 29, 2016 at 20:11 | history | asked | Milou | CC BY-SA 3.0 |