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28 Pages

### CH03_N=2

Course: PHYS 221, Spring 2009
School: UCSD
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Word Count: 6204

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3 Two-Dimensional Chapter Phase Flows We've seen how, for one-dimensional 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 encountered. (That point may lie at infinity.) No oscillations are possible1 . For N = 2 phase flows, a...

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3 Two-Dimensional Chapter Phase Flows We've seen how, for one-dimensional 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 encountered. (That point may lie at infinity.) No oscillations are possible1 . For N = 2 phase flows, a richer set of possibilities arises, as we shall now see. 3.1 3.1.1 Harmonic Oscillator and Pendulum Simple harmonic oscillator A one-dimensional harmonic oscillator obeys the equation of motion, m d2 x = -kx , dt2 (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 where = known: x v = 0 1 - 2 0 x v = v - 2 x , (3.2) k/m has the dimensions of frequency (inverse time). The solution is well v0 sin(t) v(t) = v0 cos(t) - x0 sin(t) . x(t) = x0 cos(t) + (3.3) (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. TWO-DIMENSIONAL PHASE FLOWS Figure 3.1: Phase curves for the harmonic oscillator. The phase curves are ellipses: x2 (t) + -1 v 2 (t) = C , (3.5) 2 where the constant C = x2 + -1 v0 . A sketch of the phase curves and of the phase flow 0 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 = M , with = x x and M= 0 1 2 0 - (3.6) The formal solution to = M is (t) = eM t (0) . (3.7) What do we mean by the exponential of a matrix? We mean its Taylor series expansion: eM t = I + M t + Note that M2 = = hence M 2k = (- 2 )k I , M 2k+1 = (- 2 )k M . (3.10) 0 1 - 2 0 0 1 - 2 0 = - 2 I , (3.9) 1 2! M 2 t2 + 1 3! M 3 t3 + . . . . (3.8) - 2 0 0 - 2 3.1. HARMONIC OSCILLATOR AND PENDULUM 3 Thus, Mt e = k=0 1 (- 2 t2 )k I + (2k)! k=0 1 (- 2 t2 )k M t (2k + 1)! = cos(t) I + -1 sin(t) M = cos(t) -1 sin(t) - sin(t) cos(t) . (3.11) Plugging this into eqn. 3.7, we obtain the desired solution. For the damped harmonic oscillator, we have x + 2 x + 2 x = 0 = M= 0 1 2 -2 - . (3.12) The phase curves then spiral inward to the fixed point at (0, 0). 3.1.2 Pendulum Next, consider the simple pendulum, composed of a mass point m affixed to a massless rigid rod of length . m2 = -mg sin . This is equivalent to d dt = - 2 sin , (3.14) (3.13) where = is the angular velocity, and where = oscillations. g/ is the natural frequency of small The phase curves for the pendulum are shown in Fig. 3.2. The small oscillations of the pendulum are essentially the same as those of a harmonic oscillator. Indeed, within the small angle approximation, sin , and the pendulum equations of motion are exactly those of the harmonic oscillator. These oscillations are called librations. They involve a back-and-forth motion in real space, and the phase space motion is contractable to a point, in the topological sense. However, if the initial angular velocity is large enough, a qualitatively different kind of motion is observed, whose phase curves are rotations. In this case, the pendulum bob keeps swinging around in the same direction, because, as we'll see in a later lecture, the total energy is sufficiently large. The phase curve which separates these two topologically distinct motions is called a separatrix . 4 CHAPTER 3. TWO-DIMENSIONAL PHASE FLOWS Figure 3.2: Phase curves for the simple pendulum. The separatrix divides phase spac...
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Berkeley - ASTRO - 00312047
1.3250000 1.9250001 743.68880 220.61449 1.9250001 2.0750000 1777.0221 458.75330 2.0750000 2.4650000 850.98224 280.86813 2.4650000 3.2450000 596.09554
Berkeley - ASTRO - 00312047
1.3250000 1.9250001 743.68880 220.61449 1.9250001 2.0750000 1777.0221 458.75330 2.0750000 2.4650000 850.98224 280.86813 2.4650000 3.2450000 596.09554
Berkeley - ASTRO - 00312047
# Time [days] Mag Magerr Band Uplim Ref 0.00122 19.59 0.12 white no GCN7758 0.00123 19.8 0.1 White no GCN7753 0.00244 20.26 0.38 v no GCN7758 0.0
Berkeley - ASTRO - 00312047
26280.3 23 0.5 R no
Berkeley - ASTRO - 00312047
26280.3 2.375476e-06 1.093947e-06 R no
Berkeley - ASTRO - 00312047
1.35501 1 1
Berkeley - ASTRO - 00312047
1.8899999 207.26900 8.0948969 -1.6837656 18792.640 207.26900 167549.64 4.4988381 -1.0247551 0.76540139
Berkeley - ASTRO - 00312047
chi^2/nu= 19.788363 / 15The fit is rejectable at 81.979747 % Confidence 1.92500 2.07500 1196.7708 2.07500 2.46500 1136.9505 2.46500 3.24500 1055.5888 3.24500 3.5150
Berkeley - ASTRO - 00312047
Berkeley - ASTRO - 00312047
104.995 133.405 1.41231 0.397191108.332 166 0.955355 0.188914166 208.624 0.375968 0.124193208.624 266.292 0.28387 0.0917293266.292 296.38 0.500004 0.166672296.38 374.107 0.212089 0.0680407374.107 454.34 0.223431 0.0691595454.34 567.169 0.14100
Berkeley - ASTRO - 00312047
Source Contamination: 1.16E-05 +/- 1.1E-06 cts/s
Berkeley - ASTRO - 00312047
output00312047000_999/sw00312047000xpcw2po_cl.evtoutput00312047001_999/sw00312047001xpcw2po_cl.evtoutput00312047002_999/sw00312047002xpcw2po_cl.evtoutput00312047003_999/sw00312047003xpcw2po_cl.evtoutput00312047003_999/sw00312047003xpcw3po_cl.evt
Berkeley - ASTRO - 00312047
# t1 t2 dt rad_min rad_max cts err scl bg bg_rat wt 0.104995 0.106968 0.001973 0. 16. 2.00 1.41 0.900829 0.000000 0.426667 1 0.108332 0.120868 0.012536 0. 16. 10.86
Berkeley - ASTRO - 00312047
# tmin tmax 0.108332 167.551 [ksec];instrument XRT;exposure 15685.167;xunit kev;bintype counts0.000000 0.010000 0.000000 0.0000000.010000 0.020000 0.000000 0.0000000.020000 0.030000 0.000000 0.0000000.030000 0.040000 0.000000 0
Berkeley - ASTRO - 00312047
# tmin tmax 0.10499500 5.41227 [ksec];instrument XRT;exposure 3.9661232;xunit kev;bintype counts0.000000 0.010000 0.000000 0.0000000.010000 0.020000 0.000000 0.0000000.020000 0.030000 0.000000 0.0000000.030000 0.040000 0.00000
Berkeley - ASTRO - 00312047
output00312047000_999/sw00312047000xwtw2po_cl.evtoutput00312047001_999/sw00312047001xwtw2po_cl.evtoutput00312047002_999/sw00312047002xwtw2po_cl.evt
Berkeley - ASTRO - 00312047
SIMPLE = T / file does conform to FITS standardBITPIX = 8 / number of bits per data pixelNAXIS = 0 / number of data axesEXTEND = T / FITS dataset may contain extensio
Berkeley - ASTRO - 00312047
# Ep dEp lprob lEiso dlEiso11.084 0.009 -1.05e-05 117.520 0.12311.094 0.010 -2.47e-05 117.495 0.12911.105 0.012 -3.51e-05 117.495 0.12911.117 0.013 -3.49e-05 117.495 0.12911.132 0.015 -2.96e-05 117.495 0.12911.148 0.018 -1.40e-04 117.495 0.129
Berkeley - ASTRO - 00312047
# Ep lEiso1.002 120.0651.004 119.3301.017 121.5141.025 121.2391.029 120.3521.038 119.4311.044 121.7941.049 122.2891.057 119.5291.057 119.5661.063 120.4801.072 120.0511.076 121.3501.083 120.7281.090 120.1471.093 119.2561.106 120.5191
Berkeley - ASTRO - 00312047
# Ep dEp lprob lNiso dlNiso11.084 0.009 -1.05e-05 135.543 0.21111.094 0.010 -2.02e-05 135.493 0.22011.105 0.012 -3.43e-05 135.493 0.21911.117 0.013 -4.78e-05 135.493 0.21911.132 0.015 -1.23e-05 135.493 0.21911.148 0.018 -1.40e-04 135.493 0.219
Berkeley - ASTRO - 00312047
# Ep lNiso1.002 139.3421.004 138.6201.017 140.7591.025 140.4871.029 139.6131.038 138.1861.044 141.0291.049 141.5151.057 138.7931.057 138.8301.063 139.7291.072 139.3021.076 140.5841.083 139.9681.090 139.3911.093 140.8941.106 139.7541
Berkeley - ASTRO - 00312047
# tmin tmax 10.0000 167.55140 [ksec];instrument XRT;exposure 13182.346;xunit kev;bintype counts0.000000 0.010000 0.000000 0.0000000.010000 0.020000 0.000000 0.0000000.020000 0.030000 0.000000 0.0000000.030000 0.040000 0.00000
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