Donghao.ouyang: Created page with "Just some generalization for the least action principle and Euler-Lagrange equation for the classical cases. In the calculus of variation, we have developed a tool for descri..."

2018-05-23T16:35:49Z

Created page with "Just some generalization for the least action principle and Euler-Lagrange equation for the classical cases. In the calculus of variation, we have developed a tool for descri..."

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Just some generalization for the least action principle and Euler-Lagrange equation for the classical cases. In the calculus of variation, we have developed a tool for describing various physics situation. In general, let <math>q</math> be the coordinate of a particular configuration space and <math>\dot{q}</math> be its time derivative. Then, the action is described as

<math>\mathcal{L}=\int_{t_i}^{t_f}dt L\left(q,\dot{q}\right),</math>

where <math>t_i,t_f</math> are the initial time and final time, respectively. The integrand <math>L</math> is known as the '''Lagrangian''' and is assumed to be time-independent for convenience. The idea here is to find the path that minimize the action <math>\mathcal{L}</math>. Now, we introduction the idea of variation, which can be viewed as an infinitesimal shift from the original path; however, it does not change the terminal points. Since the path we are interested is the path that minimizes the action, then the variation of the action should be 0 and that is

<math>0=\delta \mathcal{L}=\delta \int_{t_i}^{t_f}dt L\left(q,\dot{q}\right)=\int_{t_i}^{t_f}dt \delta L\left(q,\dot{q}\right)</math>
<math>=\int_{t_i}^{t_f}dt \left(\frac{\partial L}{\partial q}\delta q+\frac{\partial L}{\partial\dot{q}}\delta \dot{q}\right)=\int_{t_i}^{t_f}dt \left(\frac{\partial L}{\partial q}\delta q+\frac{\partial L}{\partial\dot{q}}\frac{d}{dt}\left(\delta q\right)\right).</math>

Then, by the integration by parts, we have that

<math>0=\int_{t_i}^{t_f}dt \left(\frac{\partial L}{\partial q}-\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)\right)\delta q+\int_{t_i}^{t_f}\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\delta q\right)=\int_{t_i}^{t_f}dt \left(\frac{\partial L}{\partial q}-\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)\right)\delta q+\frac{\partial L}{\partial\dot{q}}\delta q\big|_{t_i}^{t_f}=\int_{t_i}^{t_f}dt \left(\frac{\partial L}{\partial q}-\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)\right)\delta q,</math>

since the boundary term does not vary so that <math>\delta q\left(t_i\right)=\delta q\left(t_f\right)=0</math>. Thus, we arrive at the point where the classical particle must obey the path where the equation

<math>\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)-\frac{\partial L}{\partial q}=0.</math>

This equation is known as the Euler-Lagrange equation.

Notes for AKT-140117/0:26:27 - Revision history

Donghao.ouyang: Created page with "Just some generalization for the least action principle and Euler-Lagrange equation for the classical cases. In the calculus of variation, we have developed a tool for descri..."