Ordinary Differential Equations

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1 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations MCS 507 Lecture 17 Mathematical, Statistical and Scientific Software Jan Verschelde, 4 October 2013 Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

2 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

3 a simple pendulum Imagine a sphere attached to a massless rod oscilating back and forth due to gravity: where t is time, starting at 0; θ (t) + α sin(θ(t)) = 0, θ is the angle of deviation the rod makes from its (vertical) position at rest; α = g/l, where g is the gravitational constant and L is the length of the rod. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

4 using sympy For small angles, θ 0: sin(θ) θ. >>> from sympy import * >>> L, t, g = var( L, t, g ) >>> theta = Function( theta ) >>> eq = L*Derivative(theta(t),t,2) + g*t >>> dsolve(eq,theta(t)) theta(t) == C1 + C2*t - g*t**3/(6*l) Using more terms in a series approximation for sin(θ), we obtain more accurate solutions. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

5 first order equations Introducing an auxiliary variable v(t) = θ (t), we transform θ (t) + α sin(θ(t)) = 0 into a system of first order differential equations: θ (t) = v(t) v (t) = α sin(θ(t)) θ with initial conditions: θ(0) = π/6 and v(0) = 0. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

6 forward Euler method We use the definition θ θ(t + h) θ(t) (t) = lim h 0 h to discretize the time domain with step size t. At t = t k, we approximate θ(t k ) θ k and θ(t k+1 ) θ k+1 for and t k+1 = t k + t. So θ (t) = v(t) is approximated by θ k+1 θ k t Then we compute θ k+1 = θ k + tv k. Similarly, v (t) = α sin(θ(t)) is approximated by v k+1 = v k α t sin(θ k ). = v k. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

7 plotting the evolution Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

8 programming the Euler method import scipy as sp import matplotlib.pyplot as plt def pendulum(tmax, nstep, theta0, veloc0, alpha): Return the motion (theta, v, t) of a pendulum, governed by the ODE: theta (t) + alpha*sin(theta(t)) = 0, where the parameters are tmax : time t ranges from 0 to tmax, nstep : the number of time steps, theta0 : angle at t = 0, veloc0 : velocity at t = 0, alpha : value for the parameter. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

9 the code in pendulum tstep = tmax/float(nstep) tspan = sp.linspace(0, tmax, nstep+1) veloc = sp.zeros(nstep+1) theta = sp.zeros(nstep+1) veloc[0] = veloc0 theta[0] = theta0 for k in range(nstep): theta[k+1] = theta[k] + tstep*veloc[k] veloc[k+1] = veloc[k] - alpha*tstep*sp.sin(theta[k+1]) return theta, veloc, tspan Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

10 the constants def test_values(): Returns values for the input data: tmax, nstep, theta0, veloc0, and alpha. theta0 = sp.pi/6 nstep = 1000 tmax = 10 veloc0 = 0 alpha = 5 return tmax, nstep, theta0, veloc0, alpha Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

11 the function main def main(): Defines the test values and computes the trajectory of the pendulum. tmx, dim, pos0, vel0, alp = test_values() theta, vel, tspan = pendulum(tmx, dim, pos0, vel0, alp) fig = plt.figure() fig.add_subplot(211) plt.plot(tspan, theta) plt.title( angle as function of time ) fig.add_subplot(212) plt.plot(tspan, vel, label= velocity(t) ) plt.title( velocity as function of time ) plt.show() Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

12 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

13 calling ODEPACK do from scipy.integrate.odepack import odeint then help(odeint) shows odeint(func, y0, t,...) Solve a system of ordinary differential equations using lsoda from the FORTRAN library odepack. Solves the initial value problem for stiff or non-stiff systems of first order ode-s:: dy/dt = func(y,t0,...) where y can be a vector. ODEPACK is a FORTRAN77 library which implements Alan Hindmarsh s solvers for ordinary differential equations, available at Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

14 defining the right hand side import scipy as sp import matplotlib.pyplot as plt from scipy.integrate.odepack import odeint def fun(y, t): Returns the right hand side of the ODE dy/dt = fun(y,t). rhs = sp.array([0, 0], float) rhs[0] = y[1] rhs[1] = -5.0*sp.sin(y[0]) return rhs Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

15 the function main def main(): Computes the motion of a pendulum, governed by the ODE: theta (t) + alpha*sin(theta(t)) = 0, tmax = 10 nsteps = 1000 theta0 = sp.pi/6 veloc0 = 0 tspan = sp.linspace(0, tmax, nsteps+1) initc = sp.array([theta0, veloc0]) sol = odeint(fun, initc, tspan) theta = sol[:, 0] veloc = sol[:, 1] then plot (tspan,theta) and (tspan,veloc). Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

16 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

17 differential equations Consider three bodies with masses m 1, m 2, m 3 with positions (x 1 (t), y 1 (t)), (x 2 (t), y 2 (t)), (x 3 (t), y 3 (t)). For the first body: d 2 x 1 (t) dt 2 m = 2 (x 1 (t) x 2 (t)) ( (x1 (t) x 2 (t)) 2 + (y 1 (t) y 2 (t)) 2) 3/2 m 3 (x 1 (t) x 3 (t)) ( (x1 (t) x 3 (t)) 2 + (y 1 (t) y 3 (t)) 2) 3/2 d 2 y 1 (t) dt 2 =... Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

18 setting up the problem With u 1 (t) and v 1 (t), the velocities of x 1 (t) and y 1 (t): dx 1 (t) dt du 1 (t) dt dy 1 (t) dt dv 1 (t) dt = u 1 (t) =... = v 1 (t) =... Adding the equations for the other two bodies, we get 12 first order differential equations. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

19 the figure eight Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

20 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

21 defining the system import scipy as sp import matplotlib.pyplot as plt from scipy.integrate.odepack import odeint def fun(zvr, time): zvar is a vector with 12 entries arranged in blocks of 4: x[k](time), x [k](time), y[k](time), y [k](time) for the three masses k = 1, 2, 3. zeros = [0 for k in xrange(12)] rhs = sp.array(zeros, float) # take three equal masses mass = [1, 1, 1] Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

22 relabeling # relabel input vector zvr (x1p, u1x, y1p, v1y) = (zvr[0], zvr[1], zvr[2], zvr[3]) (x2p, u2x, y2p, v2y) = (zvr[4], zvr[5], zvr[6], zvr[7]) (x3p, u3x, y3p, v3y) = (zvr[8], zvr[9], zvr[10], zvr[11]) # u and v are first derivatives of x and y (rhs[0], rhs[2]) = (u1x, v1y) (rhs[4], rhs[6]) = (u2x, v2y) (rhs[8], rhs[10]) = (u3x, v3y) Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

23 computing distances # compute squared distances dx12 = x1p - x2p sdx12 = dx12**2 dx13 = x1p - x3p sdx13 = dx13**2 dx23 = x2p - x3p sdx23 = dx23**2 dy12 = y1p - y2p sdy12 = dy12**2 dy13 = y1p - y3p sdy13 = dy13**2 dy23 = y2p - y3p sdy23 = dy23**2 # denominators d12 = (sdx12 + sdy12)**1.5 d13 = (sdx13 + sdy13)**1.5 d23 = (sdx23 + sdy23)**1.5 Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

24 assigning the result # righthandside for second order rhs[1] = - mass[1]*dx12/d12 - mass[2]*dx13/d13 rhs[3] = - mass[1]*dy12/d12 - mass[2]*dy13/d13 rhs[5] = - mass[0]*(-dx12)/d12 - mass[2]*dx23/d23 rhs[7] = - mass[0]*(-dy12)/d12 - mass[2]*dy23/d23 rhs[9] = - mass[0]*(-dx13)/d13 - mass[1]*(-dx23)/d23 rhs[11] = - mass[0]*(-dy13)/d13 - mass[1]*(-dy23)/d23 return rhs Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

25 the function main def main(): Plots the trajectories of 3 equal masses forming a figure 8. # initial positions ip1 = [ , ] ip2 = [-ip1[0], -ip1[1]] ip3 = [0, 0] # initial velocities iv3 = [ , ] iv2 = [-iv3[0]/2, -iv3[1]/2] iv1 = iv2 # input for initial righthandside vector initz = [ip1[0], iv1[0], ip1[1], iv1[1], \ ip2[0], iv2[0], ip2[1], iv2[1], \ ip3[0], iv3[0], ip3[1], iv3[1]] Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

26 solving the problem # solving the IVP period = 2*sp.pi/3 dim = 1000 tspan = sp.linspace(0, period, dim+1) path = odeint(fun, initz, tspan) # extracing the trajectories (x1p, y1p) = (path[:, 0], path[:, 2]) (x2p, y2p) = (path[:, 4], path[:, 6]) (x3p, y3p) = (path[:, 8], path[:, 10]) # plotting the trajectories plt.figure() plt.plot(x1p, y1p, r, x2p, y2p, g, x3p, y3p, b ) plt.show() Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

27 an animation def animate(zvr, nbframes): Makes an animation of the trajectories computed by odeint in zvr, using nbframes. (x1p, x2p, x3p) = (zvr[:, 0], zvr[:, 4], zvr[:, 8]) (y1p, y2p, y3p) = (zvr[:, 2], zvr[:, 6], zvr[:, 10]) dim = len(x1p) deltaframe = dim/nbframes frame = deltaframe plt.ion() fig = plt.figure() axs = fig.add_subplot(111) axs.set_xlim(-1.5, 1.5) axs.set_ylim(-0.5, 0.5) Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

28 making the frames of the animation for i in range(nbframes): (s1x, s1y) = (x1p[1:frame], y1p[1:frame]) (s2x, s2y) = (x2p[1:frame], y2p[1:frame]) (s3x, s3y) = (x3p[1:frame], y3p[1:frame]) axs.plot(s1x, s1y, r, \ s2x, s2y, g, \ s3x, s3y, b ) fig.canvas.draw() plt.pause(1) frame = frame + deltaframe raw_input( hit enter to quit\n ) Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

29 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

30 the tractrix problem A tractor is connected to a trailer by a rigid bar of unit length. The tractor moves in a circle. The path of the trailer is the solution of a system of ordinary differential equations. Generalization: trailer is predator, tractor is prey. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

31 tractor, trailer, and bar Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

32 the model Given are (x 1 (t), x 2 (t)) defining the path of the tractor and L is the length of the rigid bar. Wanted: (y 1 (t), y 2 (t)), the path of the trailer. The velocity vector of the trailer is parallel to the direction of the bar: ( y 1 y 2 ) ( y1 x = λ 1 y 2 x 2 ), λ > 0. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

33 the ODE system The velocity vector of the trailer is the projection of the velocity vector of the tractor onto the direction of the bar: u = (y 1 x 1, y 2 x 2 ) (y 1 x 1, y 2 x 2 ) and v = (x 1, x 2 ), we compute the projection of the velocity vector as (v T u)u. Then the system of first-order equation that defines the path of the trailer is given by ( ) y 1 y 2 = (v T u)u. with u the normalized vector of the direction of the bar and v the velocity vector of the tractor. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

34 Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

35 the tractor import scipy as sp import numpy as np import matplotlib.pyplot as plt from scipy.integrate.odepack import odeint def tractor(time): Returns a 4-tuple xp, yp, xv, yv the coordinates xp, yp of the position and the components xv, yv of the velocity vector for the tractor at the given time. xpos = sp.cos(time) ypos = sp.sin(time) xvel = -sp.sin(time) yvel = sp.cos(time) return xpos, ypos, xvel, yvel Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

36 the trailer def trailer(yval, time): Defines the right hand side of the system for the trailer. rhs = np.array([0, 0], float) x1p, x2p, x1v, x2v = tractor(time) rhs[0] = yval[0] - x1p rhs[1] = yval[1] - x2p rhs = rhs/np.linalg.norm(rhs) vtu = x1v*rhs[0] + x2v*rhs[1] rhs = vtu*rhs return rhs Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

37 solving the IVP def main(): Defines the setup for the system for the trailer and solves it. endtime = 20 dim = 100 tspan = sp.linspace(0, endtime, dim+1) initc = sp.array([2, 0]) path = odeint(trailer, initc, tspan) xtrail = path[:, 0] ytrail = path[:, 1] xtrac, ytrac, x1v, x2v = tractor(tspan) plt.figure() plt.plot(xtrac, ytrac, r, xtrail, ytrail, g ) for i in xrange(0, dim, 6): plt.plot(sp.array([xtrac[i], xtrail[i]]), \ sp.array([ytrac[i], ytrail[i]]), b ) plt.show() Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

38 Summary + Exercises Appendices C, D, and E of the book are on differential equations. SciPy.integrate exports odeint() of odepack. A.C. Hindmarsh: ODEPACK, A Systematized Collection of ODE Solvers. In IMACS Transactions on Scientific Computation, Volume 1, pages 55-64, 1983, edited by R.S. Stepleman. 1 Extend the model for the n-body problem so it works for any number of bodies. 2 Instead of odeint() for the planar 3-body problem, write code for the forward Euler method. For which value(s) of the step size do you get the figure eight? 3 Make an animation of the trajectories of tractor, trailer, and the moving bar. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

39 more exercises 4 Solving y = s(t)w where s(t) is the speed and w the normalized direction vector of a predator chasing a prey gives in y the coordinates of the predator. Set up a model for a prey to move in a straight line and let s(t) be a large enough constant so the prey gets caught. Plot the trajectories of prey and predator. 5 Read the documentation about solving ODEs in Sage and use Sage to plot the trajectories either for the figure eight planar 3-body or the tractrix problem. Scientific Software (MCS 507 L-17) Ordinary Differential Equations 4 October / 39

Ordinary Differential Equations

Ordinary Differential Equations 1 An Oscillating Pendulum applying the forward Euler method 2 Celestial Mechanics simulating the n-body problem 3 The Tractrix Problem setting up the differential equations