# Definition:Lagrangian

## Definition

Let $P$ be a physical system composed of $n \in \N$ particles.

Let the real variable $t$ be the time of $P$.

$\forall i \le n$ let $\map {{\mathbf x}_i} t$ be the position of the $\cal i$th particle.

Suppose, the action $S$ of $P$ is of the following form:

$\displaystyle S = \int_{t_1}^{t_2} L \rd t$

where $L$ is a mapping of (possibly) $t$, $\map {{\mathbf x}_i} t$ and their derivatives.

Then $L$ is the Lagrangian of $P$.

## Source of Name

This entry was named for Joseph Louis Lagrange.