Euler–Lagrange equation. History[edit] The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem.
This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.[2] Calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points.
If there are no constraints, the solution is obviously a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Lagrangian mechanics. Lagrangian mechanics is a re-formulation of classical mechanics using the principle of stationary action (also called the principle of least action).[1] Lagrangian mechanics applies to systems whether or not they conserve energy or momentum, and it provides conditions under which energy, momentum or both are conserved.[2] It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788.
The use of generalized coordinates may considerably simplify a system's analysis. For example, consider a small frictionless bead traveling in a groove.