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];