Entry and leaving arcs of turnpikes: their exact computation in the calculus of variations

Abstract

We settle the question of how to compute the entry and leaving arcs for turnpikes in autonomous variational problems, in the one-dimensional case using the phase space of the vector field associated to the Euler equation, and the initial/final and/or the transversality condition. The results hinge on the realization that extremals are the contours of a well-known function and that that the transversality condition is (generically) a curve. An approximation algorithm is presented and an example included for completeness.