Editing Euler–Lagrange equation (section)

{{Short description|Second-order partial differential equation describing motion of mechanical system}}
In the [[calculus of variations]] and [[classical mechanics]], the '''Euler–Lagrange equations'''<ref>{{cite book|first=Charles|last=Fox|title=An introduction to the calculus of variations|publisher=Courier Dover Publications|year=1987|isbn=978-0-486-65499-7}}</ref> are a system of second-order [[ordinary differential equation]]s whose solutions are [[stationary point]]s of the given [[action (physics)|action functional]]. The equations were discovered in the 1750s by Swiss mathematician [[Leonhard Euler]] and Italian mathematician [[Joseph-Louis Lagrange]].

Because a differentiable functional is stationary at its local [[maxima and minima|extrema]], the Euler–Lagrange equation is useful for solving [[optimization (mathematics)|optimization]] problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to [[Fermat's theorem (stationary points)|Fermat's theorem]] in [[calculus]], stating that at any point where a differentiable function attains a local extremum its [[derivative (mathematics)|derivative]] is zero. 
In [[Lagrangian mechanics]], according to [[Hamilton's principle]] of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the [[action (physics)#Action (functional)|action]] of the system. In this context Euler equations are usually called '''Lagrange equations'''. In [[classical mechanics]],<ref name="Goldstein"> {{cite book|author1-link=Herbert Goldstein|author2-link=Charles P. Poole|first1=H.|last1=Goldstein|first2=C.P.|last2=Poole|first3=J.|last3=Safko|title=Classical Mechanics|publisher=Addison Wesley|year=2014|edition=3rd}}</ref> it is equivalent to [[Newton's laws of motion]]; indeed, the Euler-Lagrange equations will produce the same equations as Newton's Laws.  This is particularly useful when analyzing systems whose force vectors are particularly complicated. It has the advantage that it takes the same form in any system of [[generalized coordinate]]s, and it is better suited to generalizations. In [[classical field theory]] there is an [[classical field theory#Lagrangian dynamics|analogous equation]] to calculate the dynamics of a [[field (physics)|field]].