Lagrangian mechanics

In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his 1788 work, Mécanique analytique.[1]

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${\displaystyle {\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})}$ Second law of motion
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Lagrangian mechanics describes a mechanical system as a pair ${\textstyle (M,L)}$ consisting of a configuration space ${\textstyle M}$ and a smooth function ${\textstyle L}$ within that space called a Lagrangian. For many systems, ${\textstyle L=T-V,}$ where ${\textstyle T}$ and ${\displaystyle V}$ are the kinetic and potential energy of the system, respectively.[2]

The stationary action principle requires that the action functional of the system derived from ${\textstyle L}$ must remain at a stationary point (a maximum, minimum, or saddle) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.[3]