Lagrangian Mechanics

Lagrangian mechanics is based on the Principle of Stationary Action (also known as the principle of least action). It uses the Lagrangian, $L$, a function that represents the difference between the kinetic energy, $T$, and potential energy, $V$, of a system:

$$L=T-V$$

The dynamics of the system are derived by finding the path for which the action, $S$, defined as the integral of the Lagrangian over time, is stationary (i.e., does not change for small variations of the path). This leads to the Euler-Lagrange equations:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_i}\right)-\frac{\partial L}{\partial q_i}=0$$

where $q_i$ are the generalized coordinates of the system, and ${\dot{q}}_i$ are their time derivatives.