Definition:Lagrangian

Definition

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

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

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

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

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

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

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

Also known as

A Lagrangian is also known as a Lagrangian function.

Also see

• Definition:Standard Lagrangian

• Results about Lagrangians can be found here.

Source of Name

This entry was named for Joseph Louis Lagrange.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Lagrangian function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Lagrangian function