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Unformatted text preview: Chapter 3 TwoDimensional Phase Flows We’ve seen how, for onedimensional dynamical systems ˙ u = f ( u ), the possibilities in terms of the behavior of the system are in fact quite limited. Starting from an arbitrary initial condition u (0), the phase flow is monotonically toward the first stable fixed point encoun tered. (That point may lie at infinity.) No oscillations are possible 1 . For N = 2 phase flows, a richer set of possibilities arises, as we shall now see. 3.1 Harmonic Oscillator and Pendulum 3.1.1 Simple harmonic oscillator A onedimensional harmonic oscillator obeys the equation of motion, m d 2 x dt 2 = − kx , (3.1) where m is the mass and k the force constant (of a spring). If we define v = ˙ x , this may be written as the N = 2 system, d dt parenleftbigg x v parenrightbigg = parenleftbigg 1 − Ω 2 parenrightbiggparenleftbigg x v parenrightbigg = parenleftbigg v − Ω 2 x parenrightbigg , (3.2) where Ω = radicalbig k/m has the dimensions of frequency (inverse time). The solution is well known: x ( t ) = x cos( Ωt ) + v Ω sin( Ωt ) (3.3) v ( t ) = v cos( Ωt ) − Ω x sin( Ωt ) . (3.4) 1 If phase space itself is multiply connected, e.g. a circle, then the system can oscillate by moving around the circle. 1 2 CHAPTER 3. TWODIMENSIONAL PHASE FLOWS Figure 3.1: Phase curves for the harmonic oscillator. The phase curves are ellipses: Ω x 2 ( t ) + Ω − 1 v 2 ( t ) = C , (3.5) where the constant C = Ω x 2 + Ω − 1 v 2 . A sketch of the phase curves and of the phase flow is shown in Fig. 3.1. Note that the x and v axes have different dimensions. Note also that the origin is a fixed point, however, unlike the N = 1 systems studied in the first lecture, here the phase flow can avoid the fixed points, and oscillations can occur. Incidentally, eqn. 3.2 is linear, and may be solved by the following method. Write the equation as ˙ ϕ = M ϕ , with ϕ = parenleftbigg x ˙ x parenrightbigg and M = parenleftbigg 1 − Ω 2 parenrightbigg (3.6) The formal solution to ˙ ϕ = M ϕ is ϕ ( t ) = e Mt ϕ (0) . (3.7) What do we mean by the exponential of a matrix? We mean its Taylor series expansion: e Mt = I + Mt + 1 2! M 2 t 2 + 1 3! M 3 t 3 + ... . (3.8) Note that M 2 = parenleftbigg 1 − Ω 2 parenrightbiggparenleftbigg 1 − Ω 2 parenrightbigg = parenleftbigg − Ω 2 − Ω 2 parenrightbigg = − Ω 2 I , (3.9) hence M 2 k = ( − Ω 2 ) k I , M 2 k +1 = ( − Ω 2 ) k M . (3.10) 3.1. HARMONIC OSCILLATOR AND PENDULUM 3 Thus, e Mt = ∞ summationdisplay k =0 1 (2 k )! ( − Ω 2 t 2 ) k · I + ∞ summationdisplay k =0 1 (2 k + 1)! ( − Ω 2 t 2 ) k · Mt = cos( Ωt ) · I + Ω − 1 sin( Ωt ) · M = parenleftbigg cos( Ωt ) Ω − 1 sin( Ωt ) − Ω sin( Ωt ) cos( Ωt ) parenrightbigg . (3.11) Plugging this into eqn. 3.7, we obtain the desired solution....
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This note was uploaded on 09/30/2011 for the course PHYS 221a taught by Professor Staff during the Fall '11 term at UCSD.
 Fall '11
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