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Unformatted text preview: Ph-507. Homework 8 (due: Monday, April 11).PROBLEM 8-1 (4 pts)Two identical oscillators are placed in viscous medium. They are coupled due to dissipative hydrodynamic in-teractions (γ-terms in the equations of motion below). One of the oscillator is subjected to periodic driving force,fei't.:¨x+δ˙x+'2x=γ˙y+fmei't;¨y+δ˙y+'2y=γ˙x.Find the amplitude of oscillations of the other oscillator (i.e. amplitude ofy).PROBLEM 8-2 (6 pts)Find thefxed points, analyze their stability and sketch the phase space behavior for the following systems:a) A dynamical system with the following equations of motion (consider all possible topologies of the phase space"portrait"):˙x=−¡x2+y2+a−1¢y˙y=¡x2+y2+a+ 1¢xb) A pendulum with dumpingδsubjected to constant torqueτ:˙pθ=τ−I'2sinθ−δpθ;˙θ=pθI.PROBLEM 8-3 (5 pts)a) Find the canonical transformation(q, p)−→(Q, P), upon which one of the new variables has the form of enegryof a linear oscillator:Q(q, p) =p22m+kq22b) Consider a system of two coupled oscillators:...
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