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f11midterm - θ = constant Describe this motion PROBLEM...

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Intermediate Mechanics II — PHY 4936 Midterm Exam October 7, 2011 PROBLEM 1 (33 points) The potential of the 1D harmonic oscillator is U ( x ) = k x 2 / 2. 1. Write down the Lagrangian. 2. Write down the Euler-Lagrange equation. 3. Solve the Euler-Lagrange equation. Express integration constants through time zero initial conditions x 0 and ˙ x 0 . PROBLEM 2 (34 points) Consider a point mass m on the surface of a sphere of radius R under the influence of gravity - g ˆ z (spherical pendulum). 1. Write down the Lagrange function using spherical coordinates. 2. Identify the conservation laws. 3. Find the special solutions for
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Unformatted text preview: θ = constant . Describe this motion. PROBLEM 3 (33 points) Assume a Lagrangian L = L ( { q i } , { ˙ q i } , t ) where q i , i = 1 , . . . , n are generalized coor-dinates, ˙ q i , i = 1 , . . . , n are generalized velocities and t is the time. 1. Write down the principle of least action. 2. Derive the Euler-Lagrange equations from the principle of least action. 3. Assume that the Lagrangian is invariant under translation q k → q k = q k + ± k of one or more generalized coordinates q k . Find the corresponding conserved quantities....
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This note was uploaded on 12/11/2011 for the course PHY 4936 taught by Professor Berg during the Fall '11 term at FSU.

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