Problem Set-Lagrangians and Hamiltonians 1. A box of mass m is on an inclined plane as shown in the below picture. Ignore friction. a. Write the Lagrangian in terms of x and ‫ݔ‬ሶ . (Hint: Write the height of the box off the ground in terms of x and ߠ.) b. Write the Hamiltonian in terms of p and x. c. Find the differential equation of motion x(t) from both the Lagrangian and the Hamiltonian. Your result should be the same using both methods. 2. A ball of mass m is suspended from the ceiling by a spring (spring constant k). It does not swing side to side. See below picture. y a. Write the Langrangian in terms of y and ‫ݕ‬ሶ . b. Write the Hamiltonian in terms of p and y. c. Find the differential equation of motion y(t) from both the Lagrangian and the Hamiltonian. Your result should be the same using both methods. ଵ 4. For the 2D harmonic oscillator with mass m in potential ܸ = ଶ ݇ሺ‫ ݔ‬ଶ + ‫ ݕ‬ଶ ሻ a. Write the Lagrangian in terms of x, ‫ݔ‬ሶ , y, and ‫ݕ‬ሶ . b. Write the Hamiltonian in terms of px, py, x, y. c. Find the differential equation of motion x(t) from both the Lagrangian and the Hamiltonian. Your result should be the same using both methods. ଵ 5. For the 2D harmonic oscillator with mass m in potential ܸ = ଶ ݇‫ ݎ‬ଶ . a. Write the Lagrangian in terms of r, ‫ݎ‬ሶ , θ, and ߠሶ . b. Write the Hamiltonian in terms of pr, pθ, r, θ. c. Find the differential equation of motion x(t) from both the Lagrangian and the Hamiltonian. Your result should be the same using both methods. 6. A pendulum is attached to a block of mass M. The block is moving vertically. The length of the pendulum is ℓ and the bob has mass m. (Hint: Look at Section 10.3) a. Write the kinetic and potential energies of the system in terms of Cartesian coordinates. b. Write equations of constraint. c. Use the constraints to transform the coordinates. How many degrees of freedom are there? d. Explain whether the new expressions for kinetic and potential energy make sense conceptually. e. Write the Lagrangian in terms of generalized coordinates.