# Lagrangian mechanics

Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle.

Lagrangian mechanics defines a mechanical system to be a pair ${\displaystyle (M,L)}$ of a configuration space ${\displaystyle M}$ and a smooth function ${\displaystyle L=L(q,v,t)}$ called Lagrangian. By convention, ${\displaystyle L=T-V,}$ where ${\displaystyle T}$ and ${\displaystyle V}$ are the kinetic and potential energy of the system, respectively. Here ${\displaystyle q\in M,}$ and ${\displaystyle v}$ is the velocity vector at ${\displaystyle q}$ ${\displaystyle (v}$ is tangential to ${\displaystyle M).}$ (For those familiar with tangent bundles, ${\displaystyle L:TM\times \mathbb {R} _{t}\to \mathbb {R} ,}$ and ${\displaystyle v\in T_{q}M).}$

Given the time instants ${\displaystyle t_{1}}$ and ${\displaystyle t_{2},}$ Lagrangian mechanics postulates that a smooth path ${\displaystyle x_{0}:[t_{1},t_{2}]\to M}$ describes the time evolution of the given system if and only if ${\displaystyle x_{0}}$ is a stationary point of the action functional

${\displaystyle {\cal {S}}[x]\,{\stackrel {\text{def}}{=}}\,\int _{t_{1}}^{t_{2}}L(x(t),{\dot {x}}(t),t)\,dt.}$

If ${\displaystyle M}$ is an open subset of ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle t_{1},}$ ${\displaystyle t_{2}}$ are finite, then the smooth path ${\displaystyle x_{0}}$ is a stationary point of ${\displaystyle {\cal {S}}}$ if all its directional derivatives at ${\displaystyle x_{0}}$ vanish, i.e., for every smooth ${\displaystyle \delta$ :[t_{1},t_{2}]\to \mathbb {R} ^{n},}

${\displaystyle \delta {\cal {S}}\ {\stackrel {\text{def}}{=}}\ {\frac {d}{d\varepsilon }}{\Biggl |}_{\varepsilon =0}{\cal {S}}\left[x_{0}+\varepsilon \delta \right]=0.}$

The function ${\displaystyle \delta (t)}$ on the right-hand side is called perturbation or virtual displacement. The directional derivative ${\displaystyle \delta {\cal {S}}}$ on the left is known as variation in physics and Gateaux derivative in mathematics.

Lagrangian mechanics has been extended to allow for non-conservative forces.