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euler-lagrange_equations [2015/03/29 18:59] nikolaj |
euler-lagrange_equations [2015/03/30 00:36] (current) nikolaj |
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| To remove the quotes here, one needs a little of variational calculus. | To remove the quotes here, one needs a little of variational calculus. | ||
| - | $\bullet$ Observe that the solution space modulate with different data and the action data is in some sense more finer: Starting with the Euler-Lagrange equations, we must fix the initial data $q(t_0)$ at $t_0$ as well as $q'(t_0)$, i.e. the value of $\lim_{t_1\to 0}\frac{q(t_1)-q(t_0)}{t_1-t_0}$. Starting with the least action postulate, for computing the initial segment, we must fix $q(t_0)$ at $t_0$ as well as a $t_1$ together with a value for $q(t_1)$. | + | $\bullet$ Observe that the solution space modulate with different data and the action data is in some sense more finer: Starting with the Euler-Lagrange equations, we must fix the initial data $q(t_0)$ at $t_0$ as well as $q'(t_0)$, i.e. the value of $\lim_{t_1\to t_0}\frac{q(t_1)-q(t_0)}{t_1-t_0}$. Starting with the least action postulate, for computing the initial segment, we must fix $q(t_0)$ at $t_0$ as well as a $t_1$ together with a value for $q(t_1)$. |
| $\bullet$ Sidenote: For a given $L$ and two fixed points, there isn't necessarily any solution path connecting them. E.g. the points $\langle 3,2\rangle$ and $\langle 3,-2\rangle$ in ${\mathbb R}^2\setminus\{\langle 3,0\rangle\}$ can't be connected in a way that makes $L\propto q'^2$ minimal (solutions to that $L$ would be straight lines). | $\bullet$ Sidenote: For a given $L$ and two fixed points, there isn't necessarily any solution path connecting them. E.g. the points $\langle 3,2\rangle$ and $\langle 3,-2\rangle$ in ${\mathbb R}^2\setminus\{\langle 3,0\rangle\}$ can't be connected in a way that makes $L\propto q'^2$ minimal (solutions to that $L$ would be straight lines). | ||
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| Wikipedia: | Wikipedia: | ||
| [[https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation|Euler–Lagrange equation]], | [[https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation|Euler–Lagrange equation]], | ||
| + | [[http://en.wikipedia.org/wiki/Fundamental_lemma_of_calculus_of_variations|Fundamental lemma of calculus of variations]] | ||
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