The Euler-Lagrange Equations and Characteristics

Abstract

In the preceding chapter, we considered the problem of minimizing the functional

$$J\left( y \right) = \int\limits_0^t {L\left[ {x\left( t \right), y\left( t \right)} \right] dt}$$

(1)

subject to relations of the form

$$frac{{d{{x}_{i}}}}{{dt}} = {{g}_{i}}\left( {x, y} \right), {{x}_{i}}\left( 0 \right) = {{c}_{i}}, i = l, 2, ..., n,$$

(2)

where x and y represent the n-tuples (x₁, x₂, …, x_n), (y₁, y₂, …, y_n).