Timeline for Invariance of the l.h.s. of Euler-Lagrange equation
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 4, 2013 at 12:09 | history | edited | Robert Bryant | CC BY-SA 3.0 |
cleaned up notation a bit
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| Apr 2, 2013 at 22:17 | comment | added | Robert Bryant | @alvarezpaiva: Actually, it's the more natural covector, i.e., the thing you apply to a variation vector field and then integrate over the domain to compute the derivative of the functional in that direction. @Sergei: Of course, $\omega_L$ is the pullback to $TM$ of the symplectic form on $T^\ast M$ and $E_L$ is the pullback of $H$ under the Legendre mapping, so it's really the same thing, just in a different language. In practice, computing $H$ as a function on $T^\ast M$ for $L$ not quadratic in the velocity variables can sometimes be a challenge, so staying on $TM$ can bring advantages. | |
| Apr 2, 2013 at 21:07 | comment | added | alvarezpaiva | More generally, isn't this the "mean curvature vector"? | |
| Apr 2, 2013 at 20:02 | vote | accept | Sergei Ivanov | ||
| Apr 2, 2013 at 20:02 | comment | added | Sergei Ivanov | Thank you. It seems that this description gets better when translated to $T^*M$ by Legendre transform. If $s(t)\in T^*M$ is the Legendre transform of $\gamma'(t)$, the analog of $\beta$ is a 1-form $\omega(s'(t),\cdot)+dH(\cdot)\in T^*_{s(t)}T*M$, where $\omega$ is the symplectic form and $H$ is the Hamiltonian. And it projects down to $M$ because vanishes on the fiber of $T^*M$. | |
| Apr 2, 2013 at 18:49 | history | edited | Robert Bryant | CC BY-SA 3.0 |
simplified notation for the variation 1-form
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| Apr 2, 2013 at 18:29 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed some grammar and typos
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| Apr 2, 2013 at 17:19 | history | answered | Robert Bryant | CC BY-SA 3.0 |