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euler-lagrange_equations [2015/03/29 18:58]
nikolaj
euler-lagrange_equations [2015/03/30 00:36]
nikolaj
Line 32: Line 32:
 $S(t_i,​t_{i+1})[q]=\int_{t_i}^{t_{i+1}}L(q,​q',​t)\,​{\mathrm dt}.$ $S(t_i,​t_{i+1})[q]=\int_{t_i}^{t_{i+1}}L(q,​q',​t)\,​{\mathrm dt}.$
  
-is minimal at $q_\text{sol}$,​ with respect to "small and smooth"​ variations $q\mapsto q+\delta q$ that leave the endpoints fixed ($\delta q(t_1)=\delta q(t_2)=0$). To remove the quotes here, one needs a little of variational calculus.+is minimal at $q_\text{sol}$,​ with respect to "small and smooth"​ variations $q\mapsto q+\delta q$ that leave the endpoints fixed ($\delta q(t_1)=\delta q(t_2)=0$). ​ 
 +To remove the quotes here, one needs a little of variational calculus.
  
-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 $q(t_1)$. ​
  
-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).
  
-A nice aspect about this correspondence is that the action requirement makes the following apparent: If we re express $q$ in terms of other coordinates ${\hat q}$, then $L(q,​q',​t)$ rewrites as ${\hat L}({\hat q},{\hat q}',​t)$ with some new ${\hat L}$. Now optimizing $q$ w.r.t. to the action for $L$ is literally the same as optimizing the ${\hat q}$ w.r.t. to the action for ${\hat L}$. The derivation of the Euler-Lagrange equations doesn'​t care for details about the Lagrangian, though, so both systems lead to a problem of the form "​$\frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial v} = \frac{\partial L}{\partial x}$", merely for different Lagrangians. So it turns that when re-expressing coordinates,​ we don't actually have to think about how the differential equation transformes,​ we just have switch to the new induced Lagrangian in terms of the new coordinates and automatically know what we have to solve is the new Euler-Lagrange equations.+$\bullet$ ​A nice aspect about this correspondence is that the action requirement makes the following apparent: If we re express $q$ in terms of other coordinates ${\hat q}$, then $L(q,​q',​t)$ rewrites as ${\hat L}({\hat q},{\hat q}',​t)$ with some new ${\hat L}$. Now optimizing $q$ w.r.t. to the action for $L$ is literally the same as optimizing the ${\hat q}$ w.r.t. to the action for ${\hat L}$. The derivation of the Euler-Lagrange equations doesn'​t care for details about the Lagrangian, though, so both systems lead to a problem of the form "​$\frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial v} = \frac{\partial L}{\partial x}$", merely for different Lagrangians. So it turns that when re-expressing coordinates,​ we don't actually have to think about how the differential equation transformes,​ we just have switch to the new induced Lagrangian in terms of the new coordinates and automatically know what we have to solve is the new Euler-Lagrange equations.
  
 === Reference === === Reference ===
 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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