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Unformatted text preview: Physics 316, Problem Set 9, due Thursday December 1 in class 1. conserved 2-form in accelerator A bunch of charged particles of unit mass enters an accelerator along the z direction. Their mutual interaction is negligible. Upon entry, no forces are acting. There is no special axial or other symmetry. These particles have a spread of x-velocity of v x and a spread of x positions x . Upon exit from the accelerator, the bunch is monitored at a point with no forces acting. Those particles which started at the same y, z, v x , v y , v z but differing by x now lie in a line along a co- ordinate q , which is a linear combination of x, y, z, v x , v y , v z . Likewise, particles that started with the same x, y, z, v y , v z , but differing by v x lie along a direction p which is another linear combination of x, y, z, p x , p y , and p z . Suppose for definiteness that particles separated by 1 mm in the x direction (with identical velocities) end up separated by 1 mm in y and 2 m/sec in v z , with no separation in x , z , v x or v y . Likewise, particles initially separated by 1 m/sec in v x end up separated by A mm in x , A m/sec in v x and 3 A m/sec in v y , with no separation in the other 3 phase space co-ordinates. (Im just making up the proportions of the different phase space components.) a) Suppose that the accelerator is purely electrostatic. The conservation of the symplectic form 2 dictates the value of A . What is it? b) Suppose that the accelerator contains static magnets, so that the forces depend on velocity. Is it still possible to determine the value of A ? What is it?...
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