The **Averaged Lagrangian** is a concept often used in the context of dynamical systems, particularly in the fields of mechanics and control theory. It is associated with the method of averaging, which is a mathematical technique used to simplify the analysis of systems with periodic or oscillatory behavior.
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The averaged Lagrangian is the phase average of a Lagrangian evaluated on a periodic wave profile. Variation with respect to profile parameters gives the local dispersion relation, while variation of the slow phase gives a modulation conservation law.