Lagrangian system

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In mathematics, a Lagrangian system is a pair (Y, L), consisting of a smooth fiber bundle Y → X and a Lagrangian density L, which yields the Euler–Lagrange differential operator acting on sections of Y → X.

In classical mechanics, many dynamical systems are Lagrangian systems. The configuration space of such a Lagrangian system is a fiber bundle ${\displaystyle Q\rightarrow \mathbb {R} }$ over the time axis ${\displaystyle \mathbb {R} }$. In particular, ${\displaystyle Q=\mathbb {R} \times M}$ if a reference frame is fixed. In classical field theory, all field systems are the Lagrangian ones.