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homework6 - Homework Assignment 6 Physics 302, Classical...

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Homework Assignment 6 Physics 302, Classical Mechanics Fall 2010 A. V. Kotwal Handed out: Friday, October 15, 2010 Due in class on: Friday, October 22, 2010 Problems 1. The Lagrangian for a system of one degree of freedom can be written as L = m 2 ( ˙ q 2 sin 2 ωt + ˙ qqω sin 2 ωt + q 2 ω 2 ) What is the corresponding Hamiltonian? Is it conserved? Introduce a new coordinate defined by Q = q sin ωt Find the Lagrangian in terms of the new coordinate, and the corresponding new Hamiltonian. Is the new Hamiltonian conserved? 2. An object is bouncing vertically and perfectly elastically in an accelerating elevator. If the time dependence of the acceleration a ( t ) is slow enough to satisfy the adiabatic assumption, find the maximum heights h max ( t ) that the object reaches on its bounces [ h max ( t ) is measured relative to the floor of the elevator]. 3. A particle with mass m moves in one dimension under the influence of a potential V ( x ) = - k | x | , where k > 0. For energies that are negative, the motion is bounded and oscillatory.
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homework6 - Homework Assignment 6 Physics 302, Classical...

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