Re: inverse square law attraction

• To: mathgroup at smc.vnet.net
• Subject: [mg34891] Re: [mg34884] inverse square law attraction
• From: BobHanlon at aol.com
• Date: Wed, 12 Jun 2002 02:15:18 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```In a message dated 6/11/02 6:38:44 AM, boardman at onetel.net.uk writes:

>I have a question I would be pleased if someone would answer.
>
>I am studying planetary motion under the inverse square law attraction.
>
>I use NDSolve to find the numerical solution to
>
>y''[t]====-y[t]/(x[t]^2+y[t]^2)^(3/2)
>x''[t]====-x[t]/(x[t]^2+y[t]^2)^93/2)
>x'[0]==0
>y'[0]==1.2
>y[0]==0
>x[0]==1
>
>And get a very satisfactory ellipse. However, I would like to check the
>hodograph
>( a parametric plot of x'[t] and y'[t]
>against time. NDSolve must have calculated values for x'[t] and y'[t] but
>I
> cannot find them. Where are they please?
>

Clear[x,y,t];

soln=NDSolve[{
y''[t]==-y[t]/(x[t]^2+y[t]^2)^(3/2),
x''[t]==-x[t]/(x[t]^2+y[t]^2)^(3/2),
x'[0]==0, y'[0]==1.2, y[0]==0, x[0]==1},
{x[t], y[t]}, {t, 0, 15}][[1]];

x[t_] := Evaluate[x[t]/.soln];
y[t_] := Evaluate[y[t] /.soln];

ParametricPlot[{x[t], y[t]}, {t, 0, 15}, AspectRatio->1];

Plot[{x[t], x'[t]},{t,0,15}];

Plot[{y[t], y'[t]},{t,0,15}];

ParametricPlot[{x'[t], y'[t]}, {t, 0, 15}, AspectRatio->1];

Bob Hanlon
Chantilly, VA  USA

```

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