Variational calculus and applications
Keywords:Calculus of variations, Functionals, Lagrangian, Optimization
This paper consists of a brief review and introduction to the main concepts of Classic Variational Calculus. Starting from the
definitions of the concepts of first and second variation of a functional, we present a mathematically rigorous treatment for the
Variational Calculus, establishing necessary and sufficient conditions for obtaining extrema. In this context, the notion of conjugate
points is introduced, which is fundamental for the classification of weak extrema. Some simple and enlightening examples are dealt
with throughout the paper. Strong extrema are characterized and sufficient conditions for their occurrence are given. The paper
concludes with a brief application to Lagrange mechanics, showing the existence of actions whose stationary points are saddle
points instead of minima.
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