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.
The configuration space of such a Lagrangian system is a fiber bundle
A Lagrangian L can be introduced as an element of the variational bicomplex of the differential graded algebra O∗∞(Y) of exterior forms on jet manifolds of Y → X.
Given bundle coordinates xλ, yi on a fiber bundle Y and the adapted coordinates xλ, yi, yiΛ, (Λ = (λ1, ...,λk), |Λ| = k ≤ r) on jet manifolds JrY, a Lagrangian L and its Euler–Lagrange operator read where denote the total derivatives.
[1] In a different way, Lagrangians, Euler–Lagrange operators and Euler–Lagrange equations are introduced in the framework of the calculus of variations.