A particle moves in a plane under the influence of a force f= -Ara-1 directed to- ward the origin; A and a (> 0) are constants. Choose appropriate generalized co- ordinates, and let the potential energy be zero at the origin. Find the Lagrangian equations of motion. Is the angular momentum about the origin conserved? Is the total energy conserved?

Using Lagrangian formalism, solve the following problem: A solid cylinder is released from rest to roll without slipping down a ramp of slope ϕ. a) What is the best choice of generalized coordinates to be used?

Please obtain the same result as in the book.

The dynamics of a particle moving one-dimensionally in a potential V (x) is governed by the Hamiltonian Ho = p²/2m + V (x), where p = is the momentuin operator. Let E, n = of Ho. Now consider a new Hamiltonian H given parameter. Given A, m and E, find the eigenvalues of H. -ih d/dx 1, 2, 3, ... , be the eigenvalues Ho + Ap/m, where A is a %3|

Consider the question of finding the points on the curve xy² = 2 closest to the origin. (a) State what function is being minimized for this problem and what the constraint is. Label each. (b) Use Lagrange multipliers to find a system of equations for finding the closest point. Write this system of equations without any vectors. Include the constraint as one of the equations. (c) Solve the system of equations from part (b) to find the points closest on the curve xy² = 2 closest to the origin.