Lagrangian Formalism

The Lagrangian formalism of classical mechanics is a framework which can be used to predict how a system of point particles evolves over time. It is based on an empirically derived principle stating that for each set of generalized coordinates and their generalized velocities there is a special function such that the system evolves in such a way so as to either minimize or maximize a particular integral.

Definition: Lagrangian

Let $q_1,\dots ,q_s$ and ${\dot{q}}_1,\dots ,{\dot{q}}_s$ are any generalized coordinates and their generalized velocities for a system $\mathcal{S}$ of point particles.

A Lagrangian of $\mathcal{S}$ is function

$$\mathcal{L}(q_1,\dots ,q_s,{\dot{q}}_1,\dots ,{\dot{q}}_s,t)$$

of $q_1,\dots ,q_s$, ${\dot{q}}_1,\dots ,{\dot{q}}_s$ and the time $t$.

Definition: Action

Let $t_1$ and $t_2$ be two moments in time.

The action of a path $\mathbf{q}:[t_1;t_2]\to {\mathbb{R}}^s$ in configuration space is defined via the following integral:

$$S[\mathbf{q}]\stackrel{\text{def}}{=}{\int }_{t_1}^{t_2}\mathcal{L}(q_1(t),\dots ,q_s(t),{\dot{q}}_1(t),\dots ,{\dot{q}}_s(t),t)\mathrm{dt}$$

The Lagrangian formalism is based on the principle that for each set of generalized coordinates and their generalized velocities there is a Lagrangian, usually called the Lagrangian of the system, which can be used to accurately predict the evolution of the system through time.

Axiom: Principle of Stationary Action

For each set of generalized coordinates and their generalized velocities for a system of point particles there exists a Lagrangian $\mathcal{L}$ such that, between any moments $t_1$ and $t_2$, the path of the system in configuration space is the one whose action is either the lowest or highest possible out of all paths.