Reading:

Read B & O Chapter 3 -- 3.6; KB 361 lecture notes 16 -- 18.

Problems for Review:

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1. A particle of mass m moves in one dimension such that it has the Lagrangian

   

   where V is some function of x. Find the Euler-Lagrange equation for x(t). Can you think of a simpler Lagrangian with classical motions that solve this same Euler-Lagrange equation? Is there a conserved energy?

2. Using a Lagrangian approach, find the equilibrium angle and the frequency of small oscillations about equilibrium, for a simple pendulum placed in a railroad car that has a constant acceleration a in the x-direction.
3. Consider an incline plane system, where a mass m slides down a frictionless inclined plane of mass M which itself may slide along a frictionless table. Both objects start from rest.
   1. As objects in the xy-plane, how many coordinates are required to describe the two masses' positions? How many constraints restrict these coordinates? What are those constraints? How many degrees of freedom does the system have? What is a proper set of generalized coordinates (eliminating the constraints)?
   2. After eliminating the constraints, write down the Lagrangian and all Euler-Lagrange equations for this system.
   3. Show that in the limit where , the Euler-Lagrange equations agree with our expectations for motion on a fixed incline plane.

4. B & O 3-13a.

Problems to Solve:

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1. B & O 3-2.
2. B & O 3-6.
3. 1. B & O 3-7a.
   2. In the case where and we force , the system is just a single simple pendulum. Show that the Euler-Lagrange equations above give the expected result in this case.
4. B & 0 3-12.

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