In calculus, we are introduced to stationary points, points on a function  f(x) that, when x increases or decreases by an infinitesimal amount, f(x) does not change at all. In other words, f’(x)=0. These points occur at local maximums and minimums of functions. In what has been named the Calculus of Variations, Lagrange’s equations can be derived if a quantity called action, denoted S, is minimized along a particle’s path. The action of a system is simply the integral of the Lagrangian over time:

If S is at a minimum (or any stationary point), then the instantaneous rate of change of S is zero. Any such minute change or variation along S is given by . As long as there is no variation at time t1 and t2 (the path can vary throughout but the beginning and end points must remain the same), by manipulating the integrand L, and integrating by parts, the following equation can be derived:

which is the Euler-Lagrange equation, the equation of motion for the generalized coordinate and velocity corresponding to a degree of freedom, for conservative conditions. Lagrange’s equations hold for all points along the particle’s path.

The principle of stationary action, , has implemented to facilitate several significant insights in more modern physics.

Introduction - Core Terminology and Concepts - Euler-Lagrange Equation - Example of Lagrangian Mechanics

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