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Lagrangian field theory is a formalism in classical field theory. It is the field theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used for discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom. This article uses for the Lagrangian density, and ''L'' for the Lagrangian. The Lagrangian mechanics formalism was generalized further to handle field theory. In field theory, the independent variable is replaced by an event in spacetime (''x'', ''y'', ''z'', ''t''), or more generally still by a point ''s'' on a manifold. The dependent variables (''q'') are replaced by the value of a field at that point in spacetime ''φ''(''x'', ''y'', ''z'', ''t'') so that the equations of motion are obtained by means of an action principle, written as: : where the ''action'', , is a functional of the dependent variables φ''i''(''s'') with their derivatives and ''s'' itself : and where ''s'' = denotes the set of ''n'' independent variables of the system, indexed by ''α'' = 1, 2, 3,..., ''n''. Notice ''L'' is used in the case of one independent variable (''t'') and is used in the case of multiple independent variables (usually four: ''x, y, z, t''). ==Definitions== In Lagrangian field theory, the Lagrangian as a function of generalized coordinates is replaced by a Lagrangian density, a function of the fields in the system and their derivatives, and possibly the space and time coordinates themselves. In field theory, the independent variable ''t'' is replaced by an event in spacetime (''x'', ''y'', ''z'', ''t'') or still more generally by a point ''s'' on a manifold. Often, a "Lagrangian density" is simply referred to as a "Lagrangian". 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lagrangian (field theory)」の詳細全文を読む スポンサード リンク
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