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.

Joseph-Louis Lagrange (1736–1813)

Lagrangian mechanics defines a mechanical system to be a pair of a configuration space and a smooth function called Lagrangian. By convention, where and are the kinetic and potential energy of the system, respectively. Here and is the velocity vector at is tangential to (For those familiar with tangent bundles, and

Given the time instants and Lagrangian mechanics postulates that a smooth path describes the time evolution of the given system if and only if is a stationary point of the action functional

If is an open subset of and are finite, then the smooth path is a stationary point of if all its directional derivatives at vanish, i.e., for every smooth

The function on the right-hand side is called perturbation or virtual displacement. The directional derivative 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.