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A Lagrangian is a function or density whose stationary integral encodes an extremization problem. In Lagrangian mechanics its action gives the equations of motion; in geometry it describes geodesics and minimal surfaces.

In Lagrangian, the action integrates such a density over spacetime.

A Lagrangian density is integrated over spacetime to give the action $S=\int d^{d}x\mathcal{L}$ in Lagrangian.