Lagrange Mechanics Practice Problem

A roller coaster of mass m moves along a frictionless track that lies in the plane ( horizontal and y vertically up). The height of the track above the ground is given by . (a) Using as your generalized coordinate, write down the Lagrangian, the generalized momentum , and the Hamiltonian (as a function of and ). (b) Find Hamilton’s equations and show that they agree with what you would get from the Newtonian approach. [Hint: You know from Section 4.7 that Newton’s second law takes the form , where s is the distance measured along the track. Rewrite this as an equation fo and show that you get the same result from Hamilton’s equations.]