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PS1 - (d Introduce the variables p(t = x(t y(t and q(t =...

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Phys601/F11/Problem Set 1 Due 09/12/11 Problem 1 COVARIANCE (posted separately, to be renamed to 1.1H) 1.2H A particle mass m = 1 moves in a force field F = r /x, where (x,y) are Cartesian coordinates and x(t) is never zero. Initial conditions are specified at t = 1. These are x(1) = 1/2, y(1) = 0, z(1) = 0, (dx/dt)(1) = 1, (dy/dt)(1) = 3, (dz/dt)(1) = 0. (a) Find {x(t), y(t), z(t)} by direct solution of the 2 nd order ODE’s. (b) Find at least one constant of the motion and use this to solve for r (t). (You may still need to solve one 2 nd order ODE directly.) 1.3H A particle of mass m = 1 is moving in a force field F(x) = - U, where U = xy. x and y are Cartesian coordinates. At t=0, x(0)=1, y(0)=1, z(0) = 0, and v (0) = 0. (a) Find x(t), y(t), z(t) for all subsequent t by direct solution. (b) Identify a constant of the motion. (c) Show, by direct differentiation, that the combination M = (dx/dt)*(dy/dt) + (x 2 + y 2 )/2, is also a constant of the motion.
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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 φ , to obtain the set of coupled differential equations satisfied by r(t) and φ (t). (c) Show, by inspection, that your equations admit a solution such that r(t) = C and d φ (t)/dt = D, where C, D are constants. What is the value of D?...
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PS1 - (d Introduce the variables p(t = x(t y(t and q(t =...

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