Lagrangian system

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

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

A Lagrangian density $L$ (or, simply, a Lagrangian) of order $r$ is defined as an $n$ -form, $n = dim X$ , on the $r$ -order jet manifold $J r Y$ of $Y$ .

A Lagrangian $L$ can be introduced as an element of the variational bicomplex of the differential graded algebra $O * \infty (Y)$ of exterior forms on jet manifolds of $Y \to X$ . The coboundary operator of this bicomplex contains the variational operator $δ$ which, acting on $L$ , defines the associated Euler–Lagrange operator $δL$ .

Given bundle coordinates $x λ, y i$ on a fiber bundle $Y$ and the adapted coordinates $x λ, y i, y i Λ$ , $(Λ = (λ 1, ..., λ k)$ , $|Λ| = k \leq r$ ) on jet manifolds $J r Y$ , a Lagrangian $L$ and its Euler–Lagrange operator read

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