The Calculus of Variations for One-dimensional Problems

The goal of this chapter is to study one-dimensional variational problems where one tries to minimize an integral functionalIntegral functional of the form \(\displaystyle I(\mathbf {u})=\int _J F(x, \mathbf {u}(x), \mathbf {u}'(x))\,dx \) among a certain class of functions (paths) \(\mathcal {A}\) . Here J is a finite interval in \(\mathbb {R}\) , and the integrand \(\displaystyle F(x, \mathbf {u}, \mathbf {v}):J\times \mathbb {R}^n\times \mathbb {R}^n\to \mathbb {R} \) is a function whose properties are the main object of our concern.