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MathGroup Archive 2006

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Re: Area of ellipse between major axis and ray through focus, given angle

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71673] Re: Area of ellipse between major axis and ray through focus, given angle
  • From: "Narasimham" <mathma18 at hotmail.com>
  • Date: Sun, 26 Nov 2006 03:48:59 -0500 (EST)
  • References: <ek96ij$fd5$1@smc.vnet.net>

Kelly Jones wrote:
> Given:
>
> 1) an ellipse with eccentricity "ec", one focus on the origin, and
> the major axis along the x-axis
>
> 2) a ray through the origin at angle theta to the x-axis
>
> Question:
>
> What Mathematica function gives the relation/inverse relation between
> the angle theta and the area of the ellipse between the x-axis and the ray?

dA/dt is constant as per Kepler's second Law.Theta to time t relation
is through Jacobi Elliptic integrals(Newton's differential equation of
planetary motion,not in terms of elementary functions).

(* Combine two equations  1) r^2 d theta = 2 dA ; 2) p/r = (1 +
ec*Cos[theta[A]]). Simplify output of Dsolve with Boundary Condns theta
= 0, A = 0 *)

p = 1 ; ec = .7071;
EQ = {theta'[A] == 2 ((1 + ec*Cos[theta[A]] )/p) ^2, theta[0] == 0} ;
NDSolve [EQ, theta, {A, 0, 10}]
TH[u_] = theta[u] /. First[%]
Plot[TH[A], {A, 0, 10}]

Clear[p, ec];
eq = {theta'[A] == 2 ((1 + ec*Cos[theta[A]] )/p) ^2} ;
DSolve [eq, theta, A];


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