A Maple Package for the Calculus of Variations based on Jet Manifolds

This contribution presents a computer algebra package for Lagrangian systems with p>1 independent and q>1 dependent variables. The Lagrangian may depend on the partial derivatives up to the order n>0 of the dependent variables with respect to the independent ones. In the case of one independent variable, p=1, the package derives the equations of motion in the form of a system of q ordinary differential equations of order 2n, for p>1 the result is a system of q partial differential equation up to the order 2n. In addition the package determines all the required boundary conditions in the case of p<3 and n<2. Since the presented method uses the concept of jet manifolds, a short introduction to the notation of jet theory is provided. Two examples - the Timoshenko beam and the Kirchhoff plate - demonstrate the main features of the presented computer algebra based approach.