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euler-lagrange_equations [2015/03/29 18:59]
nikolaj
euler-lagrange_equations [2015/03/29 19:00]
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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