Question

6.10 * In general the integrand/(y, y', x) whose integral we wish to minimize depends on y, y, and x. There is a considerable simplification if f happens to be independent of y, that is, f-f(y, x). (This happened in both Examples 6.1 and 6.2, though in the latter the roles of x and y were interchanged.) Prove that when this happens, the Euler-Lagrange equation (6.13) reduces to the statement that df/dy = const. (6.42) Since this is a first-order differential equation for y(x), while the Euler-Lagrange equation is generally second order, this is an important simplification and the result (6.42) is sometimes called a first integral of the Euler-Lagrange equation. In Lagrangian mechanics we'll see that this simplification arises when a component of momentum is conserved.

Answer 1