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Unformatted text preview: 1.1 Phase Diagrams The state of motion of a 1D oscillator will be complete specified as a function of time if two quantities are given at one instant of time: the initial conditions x ( t = 0) and x ( t = 0). We may consider x and x as coordinates of a point in a 2D space, called phase space . As the time varies, the point P ( x, x ) describing the state of the oscillator will trace a path in phase space, which will be different for different initial conditions. From the last section: x = A sin( t + ) (24) and x = A cos( t + ) (25) Eliminating t : x 2 A 2 + x 2 A 2 2 = 1 (26) or, since E = kA 2 / 2 for the oscillator, x 2 2 E/k + x 2 2 E/m = 1 (27) This is the equation for a family of ellipses, each phase path corresponding to a definite total energy of the oscillators (since the energy is conserved). No two phase paths can cross for this simple case. Figure 1: A plot of x 2 4 + y 2 1 = 1 , and x 2 2 + y 2 . 5 = 1 1.2 Damped oscillations The previous treatment of...
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This note was uploaded on 09/20/2010 for the course AMATH 261 taught by Professor Rogermelko during the Spring '10 term at Waterloo.
- Spring '10