Question

parabolic shape and rotates with a constant angular velocity ? set up Haniltons equations of motion. 6. A mass m is suspended by a massless spring of spring constant k and unstretched length b. The suspension point has a constant upward acceleration ao. Gravity is acting vertically downward. Find the Lagrangian and Hamiltonian functions and obtain Hamilton's equa- tions of motion. What is the period of motion?

Solve problem 6

Answer 1

Given that kinetic energy of the system is T = 1/a mx^2 potential energy of the system is v = 1/2 kx^2 the Lagrangian of the system is L = T - V l = 1/2 mx^-2 - 1/2 kx^2 Hamiltonian of the system is H = x_p_x - L p_x = partial differential L/partial differential x = partial differential/partial differential x (1/2 mx^2 - 1/2 x^2 t) = mx x = p_x/m H = p_x/m (p_x) - 1/2 mx^2 + 1/2 kx^2 H = m x^2 - 1/2 m x^2 + 1/2 k x^2 H = 1/2 m x^2 + 1/2 k x^2 \r\n Hamiltonian equation of motion are x = partial differential H/partial differential partial differential/partial differential p_x (1/2 m (p_x/m)^2 + 1/2 k x^2) partial differential/partial differential p_x (1/2 m p^2_a/m^2 + 1/2 kx^2) 1/2m 2 p_x x = p_x/m p_x = -partial differential H/partial differential x (1/2 mx^2 + 1/2 kx^2) = -1/2 k^2 x p_x = -kx from the Lagrangian