Lagrangian functions

Logistics play a vital role in the prosperity of today’s cities, but current urban
logistics delivery practices have proven problematic and to be causing various negative
effects in cities. This study proposes an alternative method for delivering cargo with the
leasing of a network of logistics hubs within urban areas for designated daily time intervals
and handcart last-mile deliveries. The objective of the study is the development of a
mathematical programming model for identifying the optimal number and locations of
hubs for serving demand with the minimum cost, as well as the optimal times during the
day for leasing the facilities, while also allocating hubs to customers. The problem is
effectively solved by applying a Lagrangian relaxation and subgradient optimization
approach. Numerical examples and a sensitivity analysis provide evidence of the
robustness of the model and its ability to be effectively applied to address real problems.

A second-order Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lower-order derivatives play a key role in forcing certain types of dynamics. However, the application of these techniques requires an analytic restriction on the Lagrangian that it satisfy a twist property. In this dissertation we approach this problem from the point of view of curve shortening in an effort to remove the twist condition. In classical curve shortening a family of curves evolves with a velocity which is normal to the curve and proportional to its curvature. The evolution of curves with decreasing action is more general, and in the first part of this dissertation we develop some results for curve shortening flows which shorten lengths with respect to a Finsler metric rather than a Riemannian metric. The second part of this dissertation focuses on analytic methods to accommodate the fact that the Finsler metric for second-order Lagrangian system has singularities. We prove the existence of simple periodic solutions for a general class of systems without requiring the twist condition. Further; our results provide a frame work in which to try to further extend the topological forcing theorems to systems without the twist condition.