# 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 $(M,L)$ of a configuration space $M$ and a smooth function $L=L(q,v,t)$ called Lagrangian. By convention, $L=T-V,$ where $T$ and $V$ are the kinetic and potential energy of the system, respectively. Here $q\in M,$ and $v$ is the velocity vector at $q$ $(v$ is tangential to $M).$ (For those familiar with tangent bundles, $L:TM\times \mathbb {R} _{t}\to \mathbb {R} ,$ and $v\in T_{q}M).$ Given the time instants $t_{1}$ and $t_{2},$ Lagrangian mechanics postulates that a smooth path $x_{0}:[t_{1},t_{2}]\to M$ describes the time evolution of the given system if and only if $x_{0}$ is a stationary point of the action functional

${\cal {S}}[x]\,{\stackrel {\text{def}}{=}}\,\int _{t_{1}}^{t_{2}}L(x(t),{\dot {x}}(t),t)\,dt.$ If $M$ is an open subset of $\mathbb {R} ^{n}$ and $t_{1},$ $t_{2}$ are finite, then the smooth path $x_{0}$ is a stationary point of ${\cal {S}}$ if all its directional derivatives at $x_{0}$ vanish, i.e., for every smooth $\delt$ :[t_{1},t_{2}]\to \mathbb {R} ^{n},} $\delta {\cal {S}}\ {\stackrel {\text{def}}{=}}\ {\frac {d}{d\varepsilon }}{\Biggl |}_{\varepsilon =0}{\cal {S}}\left[x_{0}+\varepsilon \delta \right]=0.$ The function $\delta (t)$ on the right-hand side is called perturbation or virtual displacement. The directional derivative $\delta {\cal {S}}$ 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.