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hw08soln - Ph-507 Homework 8(due Monday April 11 PROBLEM...

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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, fe i't .: ¨ x + δ ˙ x + ' 2 0 x = γ ˙ y + f m e i 't ; ¨ y + δ ˙ y + ' 2 0 y = γ ˙ x. Find the amplitude of oscillations of the other oscillator (i.e. amplitude of y ). PROBLEM 8-2 (6 pts) Find the fi xed 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 = ¡ x 2 + y 2 + a 1 ¢ y ˙ y = ¡ x 2 + y 2 + a + 1 ¢ x b) A pendulum with dumping δ subjected to constant torque τ 0 : ˙ p θ = τ 0 I' 2 0 sin θ δp θ ; ˙ θ = p θ I . 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 enegry of a linear oscillator: Q ( q, p ) = p 2 2
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