Classical Dynamics of Particles and Systems · Problem 6-1

Hamilton's Principle—Lagrangian and Hamiltonian Dynamics — Marion & Thornton

Physics — Mechanics · calculus-of-variations-path-length

Problem

Consider the line connecting

(x_1, y_1) = (0, 0)

and

(x_2, y_2) = (1, 1)

. Show explicitly that the function

y(x) = x

produces a minimum path length by using the varied function

y(\alpha, x) = x + \alpha \sin \pi(1 - x)

. Use the first few terms in the expansion of the resulting elliptic integral to show the equivalent of Equation 6.4.