Every differential form on a jet space can be expressed as a sum of wedge products of 1-forms on M and contact 1-forms.  A differential form omega is called a bi-form of degree (r, s) if it is a sum of wedge products of r 1-forms on M and s contact 1-forms.  Alternatively, omega is of type (r, s) if omega(X_1, X_2, ... X_(r + s)) = 0 whenever more than r of the vector fields X_i are total vector fields or more than s of the vector fields X_i are vertical vector fields on J^k(E) ->M.  The non-negative integer r is called the horizontal degree of omega.  The non-negative integer s is called the vertical degree of omega.

Let E -> M and F -> N be two fiber bundles with dim(M) = m and dim(N) = n and let Phi: J^k(E) -> J^l(F) be a transformation.  If Phi is the prolongation of a projectable transformation from E to F and omega is a differential bi-form of type (r, s) on J^l(F), then the pullback Phi^*(omega) is a differential bi-form of type (r, s) on J^k(E)-- that is, Phi^* is bi-degree preserving.  In this special case where Phi is the prolongation of a projectable transformation, the pullback of a differential bi-form can be computed using the Pullback command found in the DifferentialGeometry package.

Suppose now that Phi: J^k(E) -> J^l(F) is the prolongation of a point transformation, a contact transformation, a differential substitution or a generalized differential substitution. (See AssignTransformationType for the definitions of these different types of transformations.)  Then if omega is a differential bi-form of type (r, s) on J^l(F) the pullback Phi^*(omega) = eta_0 + eta_0 + eta_0 + ... where eta_i is a differential bi-form of type (r - i, s + i) on J^k(E).  In these cases the command ProjectedPullback(Phi, omega) returns the type (r, s) bi-form eta_0.

Use ProjectedPullback to transform a Lagrangian bi-form to a new Lagrangian bi-form using any of the transformations listed in the previous bullet.

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