The Euler-Lagrange equation is a fundamental equation in the field of analytical mechanics, specifically within the framework of Lagrangian mechanics.
It provides a method to derive the equations of motion for a system based on a function called the Lagrangian, L, which represents the difference between the kinetic energy (T) and potential energy (V) of the system, $L = T−V$.
The equation emerges from the principle of stationary action, which states that the path taken by a system between two states, over a certain time, is the one for which the action integral (the integral of the Lagrangian over time) is stationary (usually a minimum).
In mathematical terms, for a system described by generalized coordinates qᵢ and their time derivatives q˙ᵢ, the Euler-Lagrange equation is expressed as:
This equation determines how the coordinates qᵢ evolve over time, effectively describing the dynamics of the system under conservative forces. It is used widely across many fields of physics, including classical mechanics, quantum mechanics, and field theory, to derive the equations governing the motion of systems from molecular to astronomical scales.
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