Lagrangian mechanics

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${\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}$ Second law of motion
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In physics, Lagrangian mechanics is an alternate formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760^[1] culminating in his 1788 grand opus, Mécanique analytique.^[2] Lagrange’s approach greatly simplifies the analysis of many problems in mechanics, and it had crucial influence on other branches of physics, including relativity and quantum field theory.

Lagrangian mechanics describes a mechanical system as a pair (M, L) consisting of a configuration space M and a smooth function ${\textstyle L}$ within that space called a Lagrangian. For many systems, L = T − V, where T and V are the kinetic and potential energy of the system, respectively.^[3]

The stationary action principle requires that the action functional of the system derived from L must remain at a stationary point (specifically, a maximum, minimum, or saddle point) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.^[4]