6.11. The upper end of a pendulum is attached to a linear spring of stiffness k, where the spring is constrained so as to move in the vertical direction (Fig. 6.5). Derive the equations of motion for the vertical displacement a and the angular displacement by means of Hamilton's principle. Assume that a is measured from the equilibrium position and that is arbitrarily large. L m 6.15. Derive Newton's and Lagrange's equation of motion for the system of Problem 6.11, discuss differences and show how Newton's equations can be reduced to Lagrange's equations.