Robert Harlander

Institut für Theoretische Teilchenphysik und Kosmologie
Fakultät für Mathematik, Informatik, Naturwissenschaften
RWTH Aachen University
52056 Aachen, Germany
phone: +49-241-80-27045
fax: +49-241-80-22187
harlander(at)physik.rwth-aachen.de
Büro: 28A414, Campus Melaten

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[Up: aximate] aximate: eom

Equations of motion

(thanks to contributions from Tim Kempkens, 2019)
While the evolve feature determines the movement of an object according to an explicit function of the "time" $t, you may also specify a differential equation, an "equation of motion" with a set of initial conditions. aximate will numerically solve this equation and move the object accordingly. So far, this works only for the coordinates of a point object though.

A simple example is given as follows:

 sub myeom {
    # This defines the equation of motion in (x,y).
    # We assume it to be of second order in the time t.
    my($dxdt,$dydt,$dvxdt,$dvydt,@y);
    my($t,$x,$y,$vx,$vy) = @_;

    # Defining dx/dt=vx and dy/dt=vy rewrites the eom into a 
    # first-order system of differential equations for (x,y,vx,vy):
    $dxdt = $vx;
    $dydt = $vy;

    # a parameter:
    $g = 100;
    
    # The actual differential equation. Try your own here!
    $dvxdt = -$g*$x/($x**2+$y**2)**(3/2);
    $dvydt = -$g*$y/($x**2+$y**2)**(3/2);
    
    # The output format is required to be this:
    @y = ($dxdt,$dydt,$dvxdt,$dvydt);
    return(@y);
}

sub manual19 {
    setparams({"steps" => 100, # number of frames
	       "circular" => 1, # 1: first=last frame
	       "size" => "(200,200)(-100,-100)",
	       "fbox" => 1,
	       "starttime" => 0,
	       "endtime" => 170,
	       "scale" => 1,
	       "loop" => 0 # run to end, then backwards
	      });
    # This just marks the center of gravity:
    point({"coordinates" => [[0,0]]});

    # The initial conditions are x(0)=-70, y(0)=0,
    # vx(0)=0,  vy(0)=0.7.
    # "eomname" is an arbitrary but unique label for the eom.
    point({"coordinates" => [[-70,0]],
       "color" => "Red",
       "eom" => ["eomname",\&myeom],
       "initv" => [0,.7]
  });
}
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aximate:eom (Robert Harlander)