657643098 - MAT H2255 Differential Equations Prof ll Due...

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MAT H2255: Differential Equations Extra Credit 1: Celestial Mechanics Name: Prof. ll Due: 1-Jul Introduction: Let P be a celestial body with some initial position (meters) and velocity (meters per second). Assume only the gravitational forces of S , the Sun positioned at the origin, may influence the trajectory of P . Also assume that the force of gravity always points in the direction from P to S and that its magnitude is inversely proportional to the square distance from P to S . The goal of this problem set is to derive from these assumptions alone the fact that the path of P is a conic section, i.e., is a circle, ellipse, parabola or hyperbola to be determined by the initial conditions. Notation: Let P ( t ) be a time-dependent parametrization of the trajectory of P . Let θ = θ ( t ) denote the angle the ray from S to P ( t ) makes with the rightward horizontal from S . S r ( θ ) P ( t ) θ Let r ( θ ) denote the distance of P ( t ) from S so that P ( t ) = r ( θ ) cos ( θ ) , sin ( θ ) . Note that P ( t ) is the trajectory of P whereas r ( θ ) geometrically describes the path (in polar coordinates) of P .
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  • Winter '15
  • Conic section

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