A function L on J^k(R^n, R^m) defines the action integral or fundamental integral I[s] = int L(j^k(s)(x)) dx^1...dx^n for a k-th order multiple integral problem in the calculus of variations.  Here s is a mapping from R^n to R^m and j^k(s) : R^n -> J^k(R^n, R^m) is the prolongation of s to jet space.  The Euler-Lagrange equations are the system of m, 2k-th order partial differential equations for the extremals of the action integral I[s]. See Gelfand and Fomin for an excellent introduction to the calculus of variations.

For the first calling sequence, EulerLagrange(L) returns a list of m functions on J^2k(R^n, R^m), the a-th function being the Euler-Lagrange expression E_a(L) derived from the variation of the a-th dependent variable u^a.

Let E -> M be a fiber bundle with dim(M) = n.  The differential forms on the jet spaces of E can be bi-graded by their horizontal and contact degree.  A differential form of horizontal degree n and vertical degree 0 is called a Lagrangian form or Lagrangian bi-form.  In terms of local coordinates (x^i, u^a) on E, a Lagrangian bi-form lambda can be expressed as lambda = L Dx^1...Dx^n, where L is a locally defined function on J^k(E).  The associated Euler-Lagrange form is a differential bi-form of horizontal degree n and vertical degree 1.  It is defined in terms of the usual Euler-Lagrange expressions E_a(L) by E(L) = E_a(L) C^a Dx^1...Dx^n, where C^a = du^a - u^a_i dx^i are the contact 1-forms on J^1(E).  For geometrical aspects of the calculus of variations, the representation of the Euler-Lagrange equations as the components of a differential bi-form is very useful.

The second calling sequence EulerLagrange(lambda) returns the differential bi-form E_a(L) C^a Dx^1...Dx^n.

The third calling sequence EulerLagrange(omega) returns a list of m differential bi-forms of vertical degree 1 less than the vertical degree of omega.  Here the partial derivatives with respect to the jets of dependent variables u^a_I in the usual formula for the Euler-Lagrange operator acting on functions are replaced by interior products from the coordinate vector fields D_{u^a_I}.

The command EulerLagrange is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form EulerLagrange(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-EulerLagrange(...).