set2 - a = 2 and a =-2 b Sketch the phase portraits of the...

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Department of Physics Prof. R. Bundschuh The Ohio State University Second Problem Set for Physics 664 (Theoretical Mechanics) Spring quarter 2004 Important dates: May 31 no class, Jun 9 9:30am-11:18am Fnal exam Due date: Thursday, Apr 15 4. Overdamped oscillations 10 points If a harmonic oscillator with friction is in the overdamped regime it performs the motion x ( t ) = e - βt [ A 1 e ω 2 t + A 2 e - ω 2 t ] where ω 2 q β 2 - ω 2 0 and A 1 and A 2 are real constants determined by the initial conditions. a) Calculate the velocity ˙ x ( t ). b) Determine A 1 and A 2 as functions of the position x 0 and velocity v 0 of the motion at t = 0. c) Discuss the geometric shape of the phase path of the motion in the case A 1 = 0. d) Discuss the asymptotic behavior of the phase paths of the motion in the case A 1 6 = 0 for t → ∞ . 5. Phase portrait 12 points A particle of mass m moves along the x axis under the in±uence of the potential V ( x ) = - 1 x 4 + ax 2 + 2 where a is a dimensionless constant. a) Sketch the potential for

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Unformatted text preview: a = 2 and a =-2. b) Sketch the phase portraits of the system for the cases a = 2 and a =-2. c) Indicate stable and instable equilibrium points and the seperatrices in these phase por-traits if they exist. Annotate the seperatices with the energies they correspond to. 1 6. Fourier expansion 12 points We want to consider the periodic driving force F ( t ) = -1 t ∈ i 2 π ω n-π ω , 2 π ω n h for some integer n +1 t ∈ i 2 π ω n, 2 π ω n + π ω h for some integer n . To this end we have to Fnd the ±ourier expansion of this function. a) Sketch F ( t ). b) Obtain the ±ourier expansion F ( t ) = 1 2 a + ∞ X n =1 ( a n cos nωt + b n sin nωt ) of F ( t ). c) Plot the sums of the Frst two terms, Frst three terms, and Frst four terms in the interval [-2 π/ω, 2 π/ω ] for ω = 1rad /s . 2...
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This note was uploaded on 07/17/2008 for the course PHYS 664 taught by Professor Bundschuh during the Fall '04 term at Ohio State.

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set2 - a = 2 and a =-2 b Sketch the phase portraits of the...

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