RE: inverse square law attraction

• To: mathgroup at smc.vnet.net
• Subject: [mg34898] RE: [mg34884] inverse square law attraction
• From: "David Park" <djmp at earthlink.net>
• Date: Wed, 12 Jun 2002 02:15:28 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Dick,

Here are your equations...

Clear[x, y];
eqns =
{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}

This solves them. Solve for x and y, not x[t] anmd y[t].

sols = NDSolve[eqns, {x, y}, {t, 0, 15}][[1]]
{x -> InterpolatingFunction[],
y -> InterpolatingFunction[]}

This uses the solutions to create a parametrization...

{x[t_], y[t_]} = {x[t], y[t]} /. sols
{InterpolatingFunction[{{0., 15.}}, "<>"][t],
InterpolatingFunction[{{0., 15.}}, "<>"][t]}

This plots the orbit...

ParametricPlot[{x[t], y[t]}, {t, 0, 15}];

You can do calculus on InterpolatingFunctions, so this plots the
hodograph...

ParametricPlot[{x'[t], y'[t]}, {t, 0, 15}];

David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/

> From: Dick Boardman [mailto:boardman at onetel.net.uk]
To: mathgroup at smc.vnet.net
>
> 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?
>
> R.M.Boardman
>

```

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