Ordinary Differential Equations

Transcription

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 24 Mathematical, Statistical and Scientific Software Jan Verschelde, 22 October 2012 Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

7 plotting the evolution Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

8 programming the Euler method import scipy as sp import matplotlib.pyplot as plt def pendulum(t,n,theta0,v0,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 T : time t ranges from 0 to T, n : the number of time steps, theta0 : angle at t = 0, v0 : velocity at t = 0, alpha : value for the parameter. Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

9 the code in pendulum dt = T/float(n) t = sp.linspace(0,t,n+1) v = sp.zeros(n+1) theta = sp.zeros(n+1) v[0] = v0 theta[0] = theta0 for k in range(n): theta[k+1] = theta[k] + dt*v[k] v[k+1] = v[k] - alpha*dt*sp.sin(theta[k+1]) return theta, v, t Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

10 the constants def test_values(): Returns values for the input data: T, n, theta0, v0, and alpha. theta0 = sp.pi/6 n = 1000 T = 10 v0 = 0 alpha = 5 return T, n, theta0, v0, alpha Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

11 the function main def main(): Defines the test values and computes the trajectory of the pendulum. T,n,p0,v0,a = test_values() theta,v,t = pendulum(t,n,p0,v0,a) f = plt.figure() f.add_subplot(211) plt.plot(t,theta) plt.title( angle as function of time ) f.add_subplot(212) plt.plot(t,v,label= velocity(t) ) plt.title( velocity as function of time ) plt.show() Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

14 defining the right hand side import scipy as sp import matplotlib.pyplot as plt from scipy.integrate.odepack import odeint def f(y,t): Is the right hand side of the ODE dy/dt = f(y,t). r = sp.array([0,0],float) r[0] = y[1] r[1] = -5.0*sp.sin(y[0]) return r Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

15 the function main def main(): Computes the motion of a pendulum, governed by the ODE: theta (t) + alpha*sin(theta(t)) = 0, T = 10; n = 1000 theta0 = sp.pi/6; v0 = 0 tspan = sp.linspace(0,t,n+1) initc = sp.array([theta0,v0]) y = odeint(f,initc,tspan) theta = y[:,0] v = y[:,1] then plot (tspan,theta) and (tspan,v). Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

19 the figure eight Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

21 defining the system import scipy as sp import matplotlib.pyplot as plt from scipy.integrate.odepack import odeint def f(z,t): z is a vector with 12 entries ordered in blocks of 4: x[k](t),x [k](t),y[k](t),y [k](t) for k = 1,2,3. L = [0 for k in xrange(12)] r = sp.array(l,float) Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

22 relabeling # take three equal masses m = [1, 1, 1] # relabel input vector z x1 = z[0]; u1 = z[1]; y1 = z[2]; v1 = z[3] x2 = z[4]; u2 = z[5]; y2 = z[6]; v2 = z[7] x3 = z[8]; u3 = z[9]; y3 = z[10]; v3 = z[11] # u and v are first derivatives of x and y r[0] = u1; r[2] = v1 r[4] = u2; r[6] = v2 r[8] = u3; r[10] = v3 Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

23 computing distances # compute squared distances dx12 = x1 - x2; sdx12 = dx12**2 dx13 = x1 - x3; sdx13 = dx13**2 dx23 = x2 - x3; sdx23 = dx23**2 dy12 = y1 - y2; sdy12 = dy12**2 dy13 = y1 - y3; sdy13 = dy13**2 dy23 = y2 - y3; sdy23 = dy23**2 # denominators d12 = (sdx12 + sdy12)**1.5 d13 = (sdx13 + sdy13)**1.5 d23 = (sdx23 + sdy23)**1.5 Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

24 assigning the result # righthandside for second order r[1] = - m[1]*dx12/d12 - m[2]*dx13/d13; r[3] = - m[1]*dy12/d12 - m[2]*dy13/d13; r[5] = - m[0]*(-dx12)/d12 - m[2]*dx23/d23; r[7] = - m[0]*(-dy12)/d12 - m[2]*dy23/d23; r[9] = - m[0]*(-dx13)/d13 - m[1]*(-dx23)/d23; r[11] = - m[0]*(-dy13)/d13 - m[1]*(-dy23)/d23; return r Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

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) Ordinary Differential Equations 22 October / 38

26 solving the problem T = 2*sp.pi/3; n = 1000 tspan = sp.linspace(0,t,n+1) z = odeint(f,initz,tspan) # extracting the trajectories x1 = z[:,0]; y1 = z[:,2] x2 = z[:,4]; y2 = z[:,6] x3 = z[:,8]; y3 = z[:,10]; # plotting the trajectories fig = plt.figure() plt.plot(x1,y1, r,x2,y2, g,x3,y3, b ) plt.show() Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

27 an animation def animate(z,nbframes): Makes an animation of the trajectories computed by odeint in z, using nbframes. x1 = z[:,0]; x2 = z[:,4]; x3 = z[:,8] y1 = z[:,2]; y2 = z[:,6]; y3 = z[:,10] n = len(x1); deltaframe = n/nbframes; frame = deltaframe plt.ion(); fig = plt.figure() ax = fig.add_subplot(111) ax.set_xlim(-1.5,1.5); ax.set_ylim(-0.5,0.5) for i in range(nbframes): s1x = x1[1:frame]; s1y = y1[1:frame] s2x = x2[1:frame]; s2y = y2[1:frame] s3x = x3[1:frame]; s3y = y3[1:frame] ax.plot(s1x,s1y, r,s2x,s2y, g,s3x,s3y, b ) fig.canvas.draw(); plt.pause(1) frame = frame + deltaframe Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

28 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) Ordinary Differential Equations 22 October / 38

29 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) Ordinary Differential Equations 22 October / 38

30 tractor, trailer, and bar Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

31 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) Ordinary Differential Equations 22 October / 38

32 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) Ordinary Differential Equations 22 October / 38

33 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) Ordinary Differential Equations 22 October / 38

34 the tractor import scipy as sp import numpy as np import matplotlib.pyplot as plt from scipy.integrate.odepack import odeint def tractor(t): Returns the position (x,y) and velocity vector (u,v) for the tractor at time t. x = sp.cos(t); y = sp.sin(t) u = -sp.sin(t); v = sp.cos(t) return x,y,u,v Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

35 the trailer def trailer(y,t): Defines the right hand side of the system for the trailer. r = np.array([0,0],float) x1, x2, x1v, x2v = tractor(t) r[0] = y[0] - x1 r[1] = y[1] - x2 r = r/np.linalg.norm(r) d = x1v*r[0] + x2v*r[1] r = d*r return r Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

36 solving the IVP def main(): Defines the setup for the system for the trailer and solves it. T = 20; n = 100 tspan = sp.linspace(0,t,n+1) initc = sp.array([2,0]) path = odeint(trailer,initc,tspan) x = path[:,0]; y = path[:,1] x1, x2, x1v, x2v = tractor(tspan) fig = plt.figure() plt.plot(x1,x2, r,x,y, g ) for i in xrange(0,n,6): plt.plot(sp.array([x1[i],x[i]]), \ sp.array([x2[i],y[i]]), b ) plt.show() Scientific Software (MCS 507) Ordinary Differential Equations 22 October / 38

37 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) Ordinary Differential Equations 22 October / 38

38 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) Ordinary Differential Equations 22 October / 38