The Action Principle

Define the Lagrangian as the following combination of the kinetic energy \(T\) and the potential energy \(V\):

\[ L = T - V \]

\(L\) is a function of the coordinates \(x\) and the velocity \(\dot{x}\), which are treated as independent variables. For a point particle

\[ L(x,\dot{x}) = \frac{1}{2}m \sum_{i=1}^D (\dot{x}^i)^2 - V(x) \]

Taking derivatives

\begin{align*} \frac{\partial L}{\partial x^i} &= - \frac{\partial V}{\partial x^i} \\ \frac{\partial L}{\partial \dot{x}^i} &= m \dot{x}^i \end{align*}

Using Newton’s equations, it follows that

\[ \boxed{0 = \frac{\partial L}{\partial x^i} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}^i}\right)} \]

These are the Euler-Lagrange equations.