f11midterm

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

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## This note was uploaded on 12/11/2011 for the course PHY 4936 taught by Professor Berg during the Fall '11 term at FSU.

Ask a homework question - tutors are online