MATH10222For a particle in motion in a central field of force, you are given that the path equation is \frac{d^{2} u}{d \theta^{2}}+u=-\frac{1}{H_{O}^{2} u^{2}} f(1 / u) \text { Here } u=1 / r, H_{O} \text { is the constant angular momentum and } f(r) \text { defines the form of the fiell } (i) If a solution to the path equation exists in the form of an ellipse with u=1+\beta \cos \theta for some constant B 0, show that the function f(r) must be an attracting inverse-square law. Consider the case f(r) = -gamma/r3, where y > 0. By solving the path equation subject to u(0) = 1/d,u' (0) = 0, sketch the path in each of the three cases: \text { (a) } H_{O}^{2}=4 \gamma / 3 \text {, } \text { (b) } H_{O}^{2}=\gamma \text {, } \text { (c) } H_{O}^{2}=4 \gamma / 5

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