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University of Toronto St. George
Erich Poppitz

PHY254 Lecture 7 Oscillations in Phase Space and Dynamics ▯ Readings: For Oscillation dynamics: Morin 4.1 ▯ Phase space representations of oscillations, simple harmonic motion Begin with: show what happens when phi1 and phi2 aren’t 0. Show what happens when the amplitudes aren’t equal. Bring tuning forks to demonstrate beats. Also mention "carrier wave" de▯nition correction. Phase space representation ▯ A phase space plot is a parametric plot of velocity or momentum ver- sus position. These plots are used extensively in the study of ordinary di▯erential equations, dynamical systems, studies of chaos, and so on. In these notes, we want to develop techniques for calculating and in- terpreting phase space plots. ▯ Suppose we have a particle undergoing sinusoidal motion according to x = Acos!t. Its velocity is x _ = ▯!Asin!t. Say, for example, its a mass on a spring. We can plot the position and velocity as a function of time: 1.0 x/A v/(A\omega) 0.5 0.0 0.5 1.0 1 2 3 4 5 6 t/… 1 [The script used to create the plots in this section is phase space] ▯ But this plot contains a lot of redundant information and does not reveal a clear relationship between x and v. The phase plot takes x and v as coordinates in the plane, and plots the parameteric curve (x(t);v(t)) = (Acos!t;▯!Asin!t) (1) which describes an ellipse with horizontal and vertical axes. This curve is plotted below: 2.0 1.5 1.0 0.5 … / 0.0 v 0.5 1.0 1.5 2.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 x/… ▯ Notice that points where the spring is compressed or stretched maxi- mally are points where the velocity is 0, and points of maximum velocity occur at x=0. ▯ This curve describes the whole motion because it’s periodic. It proceeds clockwise. 2 2 2 ▯ To see why the shape is elliptical, we see that x = A cos (!t) and _ = ! A sin (!t). Thus x 2 _2 + = cos (!t) + sin (!t) = 1; (2) A 2 ! A 2 which is the equation of an ellipse in the x _-x plane. 2 ▯ We can plot phase curves for many types of motion. ▯ Example: Plot the z-z _ phase trajectory of a particle freely falling in Earth’s gravitational ▯eld, if it is initially projected upwards from z 0 with speed v . 0 ▯ Answer: The motion satis▯es z _ = v 0 gt and z = z + v 0 ▯ 0 1gt . 2 This means the z _ ▯ z curve must be a parabola. If you want proof, solve for t in the z_ equation and plug it into the z equation: 2 2 v0 1 2 v 0 _ z = z 0 (v0▯ z _) ▯ (v 0 z _) = z +0 ▯ (3) g 2g g 2g This represents a parabola opening to the left, as follows: 10 Phase plot for projectile m0tion: v =10.0 m/s 0 10 z 20 30 480 70 60 50 40 30 20 10 0 10 z This graph was made with the python program phase space ▯ Make sure you understand the relationship between the motion and the plot: The velocity decreases from v to zero as the particle approaches 0 its maximum height (the rightmost point), then as the particle falls the velocity becomes more negative. ▯ We will often plot systems of interest in phase space, as a way of sum- marizing their behaviour. 3 Oscillation Dynamics ▯ Oscillations generally occur in physics because of restoring forces. Restor- ing forces are usually dependent on the displacement from an equilib- rium position (i.e. they usually depend on "x" somehow). ▯ For example, imagine we displace a particle on a spring a distance x from its equilibrium. As long as the displacement isn’t too large, we ▯nd (or Hooke found) that the force is proportional to the displacement, and in the direction opposite to the displac
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