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Unformatted text preview: (d) Introduce the variables p(t) = x(t) + y(t) and q(t) = x(t) y(t). Express the constants of the motion in p and q and find p(t) and q(t), and so x(t) and y(t), using the constants of the motion. 1.4H A charged particle moving in a magnetic field is described by the equations d v /dt = v x z ^ , d r /dt = v , where r and v have their usual meanings, and we have assumed that the magnetic field is given by B = z ^ . z ^ is the unit vector along the z axis, and some constants have been set to unity. (a) suppose v z = 0 at t = 0. Prove that the subsequent motion is confined to a plane orthogonal to z . Let this be the x-y plane. (b) Now use polar coordinates, r and , to obtain the set of coupled differential equations satisfied by r(t) and (t). (c) Show, by inspection, that your equations admit a solution such that r(t) = C and d (t)/dt = D, where C, D are constants. What is the value of D?...
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