Lagrangian Mechanics and Hamilton's Principle

Lagrangian mechanics reformulated classical mechanics by focusing on scalar quantities (energy) rather than vectors (forces).

The Lagrangian

The Lagrangian $L$ of a system is defined as the difference between its kinetic energy $T$ and potential energy $V$:

$L(q, \dot{q}, t) = T - V$

Where $q$ represents the generalized coordinates, and $\dot{q}$ represents the generalized velocities.

Hamilton's Principle of Least Action

Hamilton's principle states that the path taken by a system between times $t_1$ and $t_2$ is the one that extremizes (usually minimizes) the action $S$:

$S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt$

By applying the calculus of variations, we arrive at the Euler-Lagrange Equations:

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