Suppose we have a mechanical system whose dynamics
are governed by a Lagrangian function together with Hamilton's variational
principle. How does one go about constructing an
averaged Lagrangian for
the system? In what sense will this averaged Lagrangian give the averaged
dynamics of the system? We provide a new, concrete procedure for constructing
an averaged Lagrangian. The equations of motion resulting from this new
Lagrangian will retain geometric structures inherent in the original mechanical system. We compare our method to other methods for constructing averaged
Lagrangians: methods developed by Andrews and McIntyre, Gjaja and Holm, and
Marsden and Shkoller. In doing so, we describe the first steps towards a
theoretical foundation for all dynamical methods/theories of averaging.
