11midterm - m moving in the presence of a spherically...

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ADVANCED DYNAMICS — PHY–4241/5227 Midterm Exam March 21, 2011 Each problem counts 25 points. PROBLEM 1 Write the principle of least action and the Euler-Lagrange equation(s) of motion for a 1-dimensional Lagrangian of the form: L = 1 2 m ˙ x 2 - V ( x ) . Is the resulting equation consistent with Newton’s second law? PROBLEM 2 The Lagrangian of the 1D harmonic oscillator is L = 1 2 m ˙ x 2 - 1 2 k x 2 . 1. Use the definition of the generalized momentum to find the momentum p . 2. Write down the Hamiltonian H ( p,x ). 3. Write down Hamilton’s equations. 4. Show that Hamilton’s equations give Newton’s force law. PROBLEM 3 Consider the 3-dimensional Lagrangian for a particle of mass
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Unformatted text preview: m moving in the presence of a spherically symmetric potential V ( r )= V ( r ). That is, L = 1 2 m r 2 + r 2 2 + r 2 sin 2 2 -V ( r ) . Identify as many conserved quantities ( i.e., constants of the motion) as you can from the mere structure of the Lagrangian. PROBLEM 4 Consider a point mass m on the surface of a sphere of radius R under the inuence of gravity-g z (spherical pendulum). 1. Write down the Lagrange function using spherical coordinates. 2. Find the Euler-Lagrange equations. 3. Calculate the special solutions for = constant . Describe this motion....
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