Condições de transversalidades na mecânica

Abstract

In this work we show a deduction of Euler-Lagrange equation and transversality conditions in a geometric view. To do that we use a functional F, measuring some physical aspect of our system, as a function over the set of all possible solutions φ(t) = (t, q(t), q̇(t)) that describe the development of the system. To generate this set, given one curve φ(t), we compose it with two one-parameter groups of functions, φ{Q, Ɛ} at left and φ^{ −1}{I, Ɛ} at right, and after that generate a family of curves φƐ (t) = φ{Q, Ɛ} ◦ q ◦ φ^{−1}_(I, Ɛ} (t). The first composition generate vertical variations and the second one generate horizontal variations. To choose the solution φ(t) = (t, q(t), q̇(t)) that can be a candidate to optimal solution of our problem we use the variational principle F'[φ] = 0.