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Lagrangian

 Definition/Summary The Lagrangian is a function that summarizes equations of motion. It appears in the action, a quantity whose extremum (minimum or maximum) yields the classical equations of motion by use of the Euler-Lagrange equation. In quantum mechanics, the action, and thus the Lagrangian, appears in the path integral, and the eikonal or geometric-optics limit of the path integral is finding the extremum of the action. Though originally developed for Newtonian-mechanics particle systems, Lagrangians have been used for relativistic particle systems, classical fields, and quantum-mechanical systems. Lagrangians are often more convenient to work with than the original equations of motion, because one can easily change variables in them to make a problem more easily tractable. Lagrangians are also easier for testing for symmetries, since unlike equations of motion, they are unchanged by symmetries.

 Equations Action principle: the action I for time t and system coordinate q(t) is the integral of the Lagrangian L: $I = \int L(q(t), \dot q(t), t) dt$ The extremum of the action yields the Euler-Lagrange equation, which gives: $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q(t)}\right) - \frac{\partial L}{\partial q(t)} = 0$ with appropriate terms for any higher derivatives which may be present. It is easily generalized to multiple independent variables $x_i$ and multiple dependent variables $q_a(x)$: $\sum_i \frac{\partial}{\partial x_i}\left(\frac{\partial L}{\partial (\partial q_a(x) / \partial x_i)}\right) - \frac{\partial L}{\partial q_a(x)} = 0$

 Scientists Joseph Louis Lagrange (1736-1813)

 Extended explanation Let us start with a Newtonian equation of motion for a particle with position q and mass m moving in a potential V(q,t): $m \frac{d^2 q}{dt^2} = - \frac{\partial V}{\partial q}$ It can easily be derived from this Lagrangian with the Euler-Lagrange equations: $L = T - V$ where the kinetic energy has its familiar Newtonian value: $T = \frac12 m \left( \frac{dq}{dt} \right)^2$