Notes for AKT-140117/0:26:27

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]\displaystyle{ q }[/math] be the coordinate of a particular configuration space and [math]\displaystyle{ \dot{q} }[/math] be its time derivative. Then, the action is described as

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

where [math]\displaystyle{ t_i,t_f }[/math] are the initial time and final time, respectively. The integrand [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ =\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]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \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.