# Definition:Lagrangian

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## 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 $i$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.