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Unformatted text preview: Chapter 17 Orbits Dynamics of manybody systems. Many mathematical models involve the dynamics of objects under the influ ence of both their mutual interaction and the surrounding environment. The objects might be planets, molecules, vehicles, or people. The ultimate goal of this chapter is to investigate the nbody problem in celestial mechanics, which models the dynamics of a system of planets, such as our solar system. But first, we look at two simpler models and programs, a bouncing ball and Brownian motion. The EXM program bouncer is a model of a bouncing ball. The ball is tossed into the air and reacts to the pull of the earth’s gravitation force. There is a corresponding pull of the ball on the earth, but the earth is so massive that we can neglect its motion. Mathematically, we let v ( t ) and z ( t ) denote the velocity and the height of the ball. Both are functions of time. High school physics provides formulas for v ( t ) and z ( t ), but we choose not to use them because we are anticipating more complicated problems where such formulas are not available. Instead, we take small steps of size δ in time, computing the velocity and height at each step. After the initial toss, gravity causes the velocity to decrease at a constant rate, g . So each step updates v ( t ) with v ( t + δ ) = v ( t ) δ g The velocity is the rate of change of the height. So each step updates z ( t ) with z ( t + δ ) = z ( t ) + δ v ( t ) Here is the core of bouncer.m . Copyright c 2009 Cleve Moler Matlab R is a registered trademark of The MathWorks, Inc. TM August 9, 2009 1 2 Chapter 17. Orbits [z0,h] = initialize_bouncer; g = 9.8; c = 0.75; delta = 0.005; v0 = 21; while v0 >= 1 v = v0; z = z0; while all(z >= 0) set(h,’zdata’,z) drawnow v = v  delta*g; z = z + delta*v; end v0 = c*v0; end finalize_bouncer The first statement [z0,h] = initialize_bouncer; generates the plot of a sphere shown in figure 17.1 and returns z0 , the zcoordinates of the sphere, and h , the Handle Graphics “handle” for the plot. One of the exer cises has you investigate the details of initialize_bouncer . The figure shows the situation at both the start and the end of the simulation. The ball is at rest and so the picture is pretty boring. To see what happens during the simulation, you have to actually run bouncer . The next four statements in bouncer.m are g = 9.8; c = 0.75; delta = 0.005; v0 = 21; These statrements set the values of the acceleration of gravity g , an elasticity coef ficient c , the small time step delta , and the initial velocity for the ball, v0 . All the computation in bouncer is done within a doubly nested while loop. The outer loop involves the initial velocity v0 ....
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 Spring '11
 Adams
 Math, Acceleration, Orbits, Brownian Motion, Velocity

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